Analysis, Applied Math and Geometry Seminar

Links to Previous Semesters:

Fall 2023 Spring 2024
Fall 2022  Spring 2023
Fall 2021  Spring 2022
Fall 2020  Spring 2021
Fall 2019  Spring 2020
Fall 2018  Spring 2019
Fall 2017   Spring 2018
Fall 2016  Spring 2017
Fall 2015  Spring 2016
Fall 2014  Spring 2015
Fall 2013  Spring 2014
Fall 2012  Spring 2013
Fall 2011  Spring 2012
Fall 2010  Spring 2011
Fall 2009  Spring 2010
                 Spring 2009

Spring 2024

23 April 2024


Artem Novozhilov: Systems with inheritance and their mathematical analysis (Part 2)
16 April 2024

Vincent Homlund: Totally symmetric self-complementary plane partition matrices: enumerative properties and polytopes

Plane partitions in the totally symmetric self-complementary symmetry class (TSSCPP) are known to be equinumerous with alternating sign matrices, but no explicit bijection is known. In this talk I will discuss a set of {0,1,-1}-matrices, called Magog matrices, which are in bijection with TSSCPP. I will explore their enumerative properties and compare them to those of ASMs. I will also look at them from a geometric perspective by discussing two different polytopes that are formed from TSSCPP.
23 April 2024

Artem Novozhilov: Systems with inheritance and their mathematical analysis (Part 2)
9 April 2024

Artem Novozhilov: Systems with inheritance and their mathematical analysis

My plan is to introduce a class of differential equations, often called selection systems or systems with inheritance, and discuss known properties of their solutions. These differential equations can be considered as a somewhat intermediate mathematical object between the classical ordinary differential equations and first order partial differential equations (transport equations). The analysis and applications of these equations have a long history (they were studied systematically for at least 40 years to the best of my knowledge), and yet there are quite a few interesting (in my biased opinion) open mathematical questions, which I plan to discuss. I will start with the simplest examples, and the talk will be mostly example driven, no previous knowledge about such equations will be assumed, so I encourage the graduate students to attend the talk.
2 April 2024

Lane Morrison: Wall Stabilization (Part 2)
26 March 2024

Lane Morrison: Wall Stabilization

In 1964 C. T. C. Wall proved the following result: Two smooth simply connected 4-manifolds that are h-cobordant are diffeomorphic after enough connected sums with the sphere bundle S2×S2. The proof itself is done using some basic yet important results from handlebody theory. This theorem provides a notion of what a "stable" exotic structure of a manifold might be. My goal in this talk is to discuss the handlebody properties necessary to prove this theorem and then give a justification of it. Afterward, I would like to discuss some of the research that has emerged which utilizes Wall's theorem as a jumping off point.

19 March 2024

Chase Reuter: A Local Result Relating Centroid and Intersection Bodies (Part 2)
12 March 2024

Chase Reuter: A Local Result Relating Centroid and Intersection Bodies

This will be the first of two talks discussing a local version of an open question in geometry. We will discuss the relevant background to state the problem. The question at the heart of the talk will be a conjecture on whether the fixed point of some non-linear operator on convex bodies is the Euclidean ball. In the process we will also explore one route to applying analysis techniques to geometric questions.

Fall 2023

5 December 2023

Azer Akhmedov: Bi-orderability in one-relator groups
28 November 2023

Azer Akhmedov: Bi-orderability in one-relator groups (Part 2)
14 November 2023

Azer Akhmedov: Bi-orderability in one-relator groups

A left order on a group is a linear order which is invariant under left-translations; a bi-order is a linear order which is invariant under both left and right translations. I'll mention several examples from topology about the use of left orders and bi-orders. Then I'll concentrate on one-relator groups and describe an example of a non-bi-orderable one-relator group which has no generalized torsion. This example is from a recent joint work with Jimmy Thorne.

31 October 2023

Doğan Çömez: Recurrence and universally good weights defined by a class of superadditive processes (Part 3)
24 October 2023

Doğan Çömez: Recurrence and universally good weights defined by a class of superadditive processes (Part 2)
17 October 2023

Doğan Çömez: Recurrence and universally good weights defined by a class of superadditive processes

In ergodic theory recurrence plays an important role in both characterizing and constructing sequences that are good for convergence of ergodic averages.  A large class of superadditive processes have recurrence property; hence, sequences defined by such processes can be considered as weights for ergodic or superadditive averages. In this two talk series, first we will investigate some recurrence theorems for both additive and superadditive processes.  It turns out that these recurrence properties pave way to Hartman almost periodic sequences. In particular, we will obtain conditions under which such sequences are good weights for the a.e. convergence of ergodic averages.
10 October 2023

Morgan O'Brien: An excursion into noncommutative ergodic theory (Part 4)
3 October 2023

Morgan O'Brien: An excursion into noncommutative ergodic theory (Part 3)
26 September 2023

Morgan O'Brien: An excursion into noncommutative ergodic theory (Part 2)
19 September 2023

Morgan O'Brien: An excursion into noncommutative ergodic theory

Motivated by problems in quantum physics, quantum information theory, and spectral theory, noncommutative analysis studies what happens one replaces spaces of continuous and measurable functions in analysis with spaces of linear operators on a Hilbert space (in particular, C*-algebras or von Neumann algebras).  In the first two talks, we will discuss some of these motivations for going from classical analysis to noncommutative analysis in more detail, and along the way we will also introduce definitions and some basic results in the theory of von Neumann algebras and their affiliated spaces. The third talk will focus on noncommutative weighted ergodic theory, where we will discuss some recent results that generalize known theorems from classical ergodic theory to the von Neumann algebra setting, and in doing so we will mention some of the intricacies that arise when this generalization is made.

Spring 2023
11 April 2023

Syed Husain:  Nyquist density thresholds for different Paley-Wiener spaces

Donoho and Logan used the L_1 reconstruction method for perfect recovery of a noisy signal, where they consider the 'window' to be a compact interval and the signals are also bandlimited to a compact interval. In this talk, I will discuss the general framework of the L_1 reconstruction method. We will find concentration inequalities for a ball window and a cube window where the signals are assumed to be bandlimited to a cube. Consquently, we will find the Nyquist density thresholds for the two problems and do a comparison between the two thresholds using some approximations of Bessel and Gamma functions. In the end, we will discuss a result for signals that are bandlimited to a ball in dimension 2.
28 March 2023

Morgan O'Brien:  Factorization and Nikishin's Theorem (Part 2)
21 March 2023

Morgan O'Brien:  Factorization and Nikishin's Theorem

The study of L_p-spaces and the linear operators on them is an important part of analysis. One aspect of this is the factorization of operators between these spaces. For example, Nikishin's Theorem states that any continuous linear operator from an L_p-space to L_0 of a finite measure space can be factored through a weak L_q-space; in other words, the operator actually maps the L_p space to a weak L_q-space which is then included in L_0 (though the measures may be different). In this talk we will discuss this result, the tools used to prove it, and some results that follow from it.

7 March 2023

Lane Morrison:  Handle Theory Basics (Part 2)
28 February 2023

Lane Morrison:  Handle Theory Basics

Handlebody Theory or Kirby Calculus is a well-known tool for studying 4 manifolds. Every smooth connected closed 4 manifold is diffeomorphic to a finite gluing of disks. We call the disks handles and the gluing a handlebody of the manifold. These handles can be manipulated while still preserving the diffeomorphism type of the manifold and give us simple objects to work with if we would like to perform surgeries. I plan on defining what a handlebody is, defining handle moves, looking at some examples, and then applying these techniques.
21 February 2023

Pratyush Mishra:  Can groups be ordered like real numbers? (Part 2)
14 February 2023

Pratyush Mishra:  Can groups be ordered like real numbers?

Orderable groups (groups admitting a translation-invariant left/right total ordering) have attracted interests in group theory, dynamical systems, and low-dimensional topology. While studying group actions on manifolds, the first and the simplest case that comes to mind is understanding groups acting on the real line. For countable groups, left orderability is the same as admitting a faithful action by homeomorphism on the real line.

Over the last few decades, there has been some substantial work on this by many to understand if there are any interesting actions of "big groups" on "small manifolds". In a series of talks, we will try to understand this by stating some of the related conjectures and then formulating it into the language of left-orderable groups to better understand them. We will see a nice proof of the fact that finite index subgroups of SL(n,Z) for n > 2 do not have nice actions on circles and hence on the real line (by D.W. Morris in 2008), which is a simple case of a more general conjecture that lattices in SL(n,R) for n > 2 do not have faithful action on circles. A wide generalization of this conjecture has been recently announced (by B. Deroin and S. Hurtado in 2020).
7 February 2023

Azer Akhmedov: Girth Alternative for Groups (Part 2)
31 January 2023

Azer Akhmedov: Girth Alternative for Groups

The well-known Tits Alternative (proved by Jacques Titis for finitely generated linear groups over any field) is a property of a class of groups saying that any group from this class is either virtually solvable or contains a copy of a non-abelian free group. Tits Alternative holds for many other classes of groups and (sometimes interestingly) fails for some others. In my Ph.D. thesis, I introduced the so-called Girth Alternative. The girth of a finitely generated group is a positive integer or infinity. We say that a class C of group satisfies Girth Alternative if any group in C is either virtually solvable or has infinite girth. Over the past 20 years, it has been proved or disproved (by me and others) for various classes of groups.  In one of the recent joint works with Pratyush Mishra, we prove it for large classes of HNN extensions and amalgamated free products.  I'll summarize the known results and list several open questions.

24 January 2023

Seppi Dorfmeister: Minimal Genus and Circle Sums

The minimal genus problem asks what the minimal genus is of a connected, embedded surface S representing the second homology class A in a 4-manifold M. 
One way to attempt to attack this problem is by constructing submanifolds from known examples.  A well-known technique is the connected sum.  Another is the circle sum, first introduced by B. H. Li and T. J. Li.
 
I will describe the minimal genus problem, the two sum techniques and try to highlight strengths and weaknesses of each.  Time permitting, I will describe how this is applied in the case of the 4-torus.

Fall 2022


29 November 2022

Chase Reuter: Local solutions to some uniqueness problems in convex geometry (Part 2)
 
15 November 2022

Chase Reuter: Local solutions to some uniqueness problems in convex geometry

Characterizing Euclidean spaces was one of the goals of Busemann and Petty in the 1950's.  We will present several uniqueness problems that have been solved locally, and survey the techniques used to obtain the local solutions. 

8 November 2022

Mariangel Alfonseca: A negative solution of Ulam's floating body problem (Part 2)
1 November 2022

Mariangel Alfonseca: A negative solution of Ulam's floating body problem

Problem 19 in the Scottish Book was posed by Ulam and asks if a convex body of constant density which floats in equilibrium in any orientation must be an Euclidean ball. I will present the main ideas of the recent counterexample by Ryabogin.
18 October 2022

Michael Preheim: Student Confidence and Certainty in Comprehension of a Proof by Induction

Researchers typically utilize response correctness to interpret student proficiency in proof comprehension. However, student metacognition offers important information about performance behavior but has not been simultaneously analyzed alongside correctness to determine student competency in proof comprehension. The primary objective of this study is to investigate the accuracy of student confidence and certainty levels at local and holistic aspects of proof comprehension regarding a proof by induction. Students were given a three-factor proof comprehension assessment at the beginning and end of an undergraduate transition-to-proof course that collected student confidence, correctness, and certainty at each tier of an established proof comprehension framework. Results of this study highlight a critical distinction between high and low performers’ metacognition throughout the host course. Additionally, one outlying assessment item especially illuminates additional considerations for future application of metacognition in proof comprehension research.




Spring 2022

3 May 2022

Michael Roysdon (Tel Aviv University): Measure theoretic inequalities and projection bodies

This talk will detail two recent papers concerning the Rogers-Shephard difference body inequality and Zhang's inequality for various classes of measures. The covariogram of of a convex body w.r.t. a measure plays an essential role in the proofs of each of these inequalities. In particular, we will discuss a variational formula concerning the covariogram resulting in a measure theoretic version of the projection body operator. If time permits, we will discuss how these results imply some reverse isoperimetric inequalities.

Joint work with: 1) D. Alonso-G, M. H. Cifre, J. Yepes-N, and A. Zvavitch and 2) D. Langharst and A. Zvavitch.

26 April 2022

Adam Erickson: A Rokhlin lemma for noninvertible totally-ordered measure-preserving dynamical systems

Suppose (X,ℱ(<),μ,T) is a non-invertible MPDS with σ-algebra ℱ generated by the intervals in the total order <. In this second of two talks, we describe a paper by the speaker, recently submitted to Real Analysis Exchange, proving that the Rokhlin lemma applies to such a system if a strengthened form of aperiodicity is assumed, utilizing an adaptation of the technique introduced by Heinemann and Schmitt to prove Rohklin's lemma for non-invertible measure-preserving systems on separable nonatomic aperiodic spaces.

19 April 2022

Adam Erickson: A survey of the history of the Rokhlin lemma and its variants

The Rokhlin lemma tells us that many measure-preserving dynamical systems (X,ℱ,μ,T) can "almost" be viewed as a disjoint union of preimages of some E ∈ ℱ. Beyond its original use in the theory of generators, this idea and its variants turns out to have tremendous utility throughout ergodic theory. In this first talk of two, we will introduce the many victories of the Rokhlin lemma in constructing examples and proving a wide variety of ergodic theorems.

12 April 2022

Pratyush Mishra: Girth alternative for HNN extension (Part 2)

5 April 2022

Pratyush Mishra: Girth alternative for HNN extension

The notion of a girth was first introduced by S. Schleimer in 2003. Later, a substantial amount of work on the girth of finitely generated groups was done by A. Akhmedov, where he introduced the so-called Girth Alternative and proved it for certain classes of groups, e.g. hyperbolic, linear, one-relator, PL_+(I) etc. Girth Alternative is like the well-known Tits Alternative in spirit; therefore, it is natural to study it for classes of groups for which Tits Alternative has been investigated. In this talk, we will explore the girth of HNN extensions of finitely generated groups in its broadest sense by considering cases where the underlying subgroups are either full or proper subgroups. We will present a sub-class for which Girth Alternative holds. We will also produce counterexamples to show that beyond our class, the alternative fails in general. The talk will be based on joint work with Azer Akhmedov.

29 March 2022

Morgan O'Brien: Dilations of contractions on Hilbert and L_p-spaces (Part 2)
22 March 2022

Morgan O'Brien: Dilations of contractions on Hilbert and L_p-spaces

Contractions are a very common and one of the most basic types of bounded operators on a Banach space, since they are exactly those operators in the unit ball of all bounded operators on the space. Though sometimes this may be enough for some purposes, unsurprisingly one frequently imposes more conditions on the operators they use to do things. For example, unitary operators on a Hilbert space are also contractions, but they are also invertible operators whose adjoint is their inverse, and so the methods of spectral theory may be applied to them. Although unitary operators seem much more restrictive to work with, the two types of operators are surprisingly a lot more related than one would think.

In the first part of this talk, we will discuss a dilation theorem for arbitrary contractions on a Hilbert space that allows one to study a contraction by extending it to a unitary operator on a larger Hilbert space. This will then be used to prove some mean weighted ergodic theorems for such operators. In the second part, we will a similar type of result that extends positive contractions on L_p-spaces to positive isometries on “larger” L_p-spaces, and discuss how this can be used to prove some pointwise ergodic theorems.

8 March 2022

Azer Akhmedov: The geometry of Banach spaces (Part 3)

1 March 2022

Azer Akhmedov: The geometry of Banach spaces (Part 2)

15 February 2022

 Azer Akhmedov: The geometry of Banach spaces

This will be a series of 2 (or 3) talks aimed at a general audience. In the first talk, I'll review some classical results from theory of Banach spaces. We will take a (perhaps) somewhat non-traditional view, concentrating on the study of topology of Banach spaces. In the second talk, I'll discuss recent results on the geometry of Banach spaces.   



Fall 2021


16 November 2021
Josef Dorfmeister: Minimal Genus in Rational Manifolds (Part 2)


9 November 2021
Josef Dorfmeister: Minimal Genus in Rational Manifolds

The minimal genus problem asks what the minimal genus is of a connected, embedded surface S representing the second homology class A in a 4-manifold M.  There are very few manifolds for which this problem has a complete solution, and only a few more for which estimates of any kind are known.

I will describe the problem in detail, describe techniques that have been used to study this problem and then describe recent results for rational manifolds.  On such manifolds, there exists a well-understood group action whose fundamental domain is partially known (Tits cone of a Kac-Moody algebra); I will describe the structure of the full fundamental domain.  This involves reducing the problem to solving a Diophantine set of equations.  Finally, studying the minimal genus question for this fundamental domain leads to new results.

2 November 2021
Pratyush Mishra: Growth of groups (Part 2)


26 October 2021
Pratyush Mishra: Growth of groups

The study of growth rate dates back to one of the famous theorems of Milnor & Schwarz in Riemannian geometry, which states that "The volume growth of the universal cover of a complete Riemannian Manifold M is equivalent to the growth rate of the fundamental group of M". Growth rates are of great interest in geometry, group theory and dynamical systems. In these talks, we shall study how the growth rate is slow for some class of groups while its fast for another class. We shall see some concrete examples of groups arising from geometry and compute their growth rate. 


19 October 2021
Frankie Chan: Profinite Rigidity and Fuchsian Groups (Part 2)


12 October 2021
Frankie Chan: Profinite Rigidity and Fuchsian Groups

Inspired by a result from Bridson--Conder--Reid, my work produces an effective construction for distinguishing the collection of finite quotients of a triangle group with that of a non-isomorphic Fuchsian lattice. With an aim of balancing motivation and technical details, I will begin with some background and exposition on profinite groups and surface theory. This is joint work with Ryan Spitler (Rice University).

5 October 2021
Doğan Çömez: Ergodic Theorems in Fully Symmetric Banach Spaces (Part 2)

28 September 2021
Doğan Çömez: Ergodic Theorems in Fully Symmetric Banach Spaces

Fully symmetric Banach spaces are large function spaces that include classical Banach spaces. Some well-known examples of spaces of this kind are Orlicz Spaces, Lorentz Spaces and Marcinkiewicz Spaces. Although these spaces are studied extensively in the functional analysis literature, investigation of convergence of ergodic averages in such spaces is fairly recent. In this talk, first we will provide a brief review of symmetric Banach spaces, then we will give necessary and sufficient conditions for almost uniform convergence of ergodic averages. These, in turn, will be utilized in extending several ergodic theorems to the setting of fully symmetric Banach spaces.
21 September 2021
Morgan O'Brien: A Non-Commutative Return Times Theorem (Part 2)

14 September 2021
Morgan O'Brien: A Non-Commutative Return Times Theorem (Part 2)

7 September 2021

 Morgan O'Brien: A Non-Commutative Return Times Theorem 

Weighted and subsequential ergodic theorems are frequent subjects of study in ergodic theory. An important version of these types of results are the Wiener-Wintner type ergodic theorems, which allow for averages of operators weighted by a collection of sequences to be considered simultaneously. In classical ergodic theory, these types of results are known for numerous classes of weights, while the noncommutative setting thus far has required all sequences to be bounded. In these talks, we will discuss a version of the Banach principle that has been specialized for proving such theorems for various types of weights in the von Neumann algebra setting. Using this, we will prove some Wiener-Wintner type ergodic theorems for some broad classes of Hartman sequences.


Spring 2021
 

27 April 2021

Doğan Çömez: Recurrence theorems and universally good sequences via superadditive processes  

Abstract: Poincaré Recurrence Theorem (PRT), proved in 1893, Birkhoff Ergodic Theorem, proved in 1931, and Wiener-Wintner Ergodic Theorem (WW-ET), proved in 1941, are three main pillars on which modern ergodic theory stands. Over the years all these results have been generalized to various settings; furthermore, each paved way to new avenues of research within ergodic theory as well as in other fields. In this talk the focus will be on a class of superadditive processes, whose averages are known to converge a.e., thereby providing a generalization of Birkhoff Ergodic Theorem. Particular attention will be on obtaining some recurrence theorems for such processes (connection to PRT), and study universally good sequences generated by them (connection with WW-ET).

Video of the talk
20 April 2021

Adam Buskirk: The Fourier transform of probability measures on non-abelian groups and shuffling

Abstract: The process of shuffling is commonly modeled as a random walk on the symmetric group $S_K$. In this talk, we explain a technique introduced by Diaconis derived from group representation theory using a version of the Fourier transform to compute the probabilities associated with a random walk after $n$ steps. We introduce an upper bound lemma derived by Diaconis and Shahshahani, and use this to obtain bounds on how rapidly the probabilities associated with a random walk will converge to uniformity.

Video of the talk
13 April 2021

Pratyush Mishra: Groups of piecewise linear homeomorphisms of the interval [0,1]. Talk 2.

Abstract: In this talk, we will continue studying groups of PL homeomorphisms of the interval.  We will mainly focus on the study of subgroup structures of these groups. The subgroup properties we are interested in have been well understood for linear groups. For the PL groups of homeomorphisms, we will see their story diverges from the one of the linear groups in a major way, yet we will present results which can be viewed as analogs of the results in the case of linear groups.

Video of the talk

6 April 2021

Pratyush Mishra: Groups of piecewise linear homeomorphisms of the interval [0,1]. Talk I: A Survey

Abstract: Group actions on manifolds have attracted people from different fields of mathematics such as group theory, low dimensional topology, dynamical systems, and differential geometry. Among group action on 1-manifolds, in this talk we will be looking at the group of piecewise linear homeomorphisms of the closed interval I= [0,1] (denoted PL_+(I)). We will study this infinite-dimensional group using algebraic and dynamical methods. By the end of the first talk, we will see a complete classification of solvable and non-solvable subgroups of PL_+(I) as appeared in the PhD thesis of Collin Bleak in 2005.

Video of the talk
23 March 2021

Jimmy Thorne: Finite Oscillation Stability

Abstract: This talk will explore the basics of the Ramsey-Dvoretzky-Milman phenomenon, which is also known as finitely oscillation stable.  Intuitively this phenomenon is the observation that uniformly continuous functions on high dimension structure are close to constant on all but a vanishing small measure set. This talk will start with a few basics of uniform spaces and formally define finitely oscillation stability.  We
then move on to give examples of spaces that exhibit this phenomenon, such as, S^\infty, S_\infty (infinite symmetric group), and extremely amenable groups.

Video of the talk
9 March 2021

Lane Morrison: Description of Morse Homology

Abstract: Morse theory is the study of critical points of real valued functions to understand the topology of smooth manifolds. If V is a compact smooth manifold and f is a "nice enough" smooth real valued function on V, then one can show V has the homotopy type of a CW-complex where each cell corresponds to a critical point of f. From this decomposition we can construct the Morse homology of V and use this construction to study the topology of V. In this talk we will define "nice enough" and give an overview of the construction of this homology.

Video of the talk

2 March 2021
Chase Reuter: The Fifth and Eighth Busemann-Petty Problems (Part 3)

Video of the talk

23 February 2021
Chase Reuter: The Fifth and Eighth Busemann-Petty Problems (Part 2)

Video of the talk
16 February 2021
Chase Reuter: The Fifth and Eighth Busemann-Petty Problems (Part 1)

Abstract: In 1956, Busemann-Petty were studying finite dimensional norm spaces from a geometric point of view and sought to characterize Euclidean spaces.  They posed ten conjectures of which only one has been resolved relatively recently, in 1999.  The first talk will focus on motivating and introducing the basic notation and machinery used to obtain partial results in two of the remaining conjectures.  Some of the topics introduced and explored will be the Gaussian curvature, the isotropic position, and the space of spherical harmonics. The second and third talks will outline the solution of the problems.

Video of the talk

9 February 2021

 Morgan O'Brien: A Subsequential Individual Ergodic Theorem on von Neumann algebras 

Abstract: When studying an operator on a vector space, one might need to know the long-term behavior of the iterations of that operator on a fixed element. One way to do this is to study some type of convergence of the sequence of running averages that one obtains from these iterations. However, one might find themselves needing or wanting to ignore a number of terms and consider the averages of a subsequence of the iterations - for example, maybe an error was made somewhere that forces some terms to be ignored. Numerous results of this nature are known for pointwise and norm convergence for various types of operators on the Lp-spaces associated to a measure space, but not much is known for these types of problems in the von Neumann algebra setting. In this talk, we will look at the almost uniform and bilateral almost uniform convergence of the subsequential averages of a Dunford-Schwartz operator acting on noncommutative Lp-spaces associated to a semifinite von Neumann algebra along sequences of density 1.

Video of the talk
2 February 2021

 Doğan Çömez: Quantization for infinite affine transformations (Part 2)

Video of the talk

Slides

26 January 2021

 Doğan Çömez: Quantization for infinite affine transformations (Part 1) 

Click here for the Abstract

Video of the talk

Slides


Fall 2020
 

24 November 2020
Seljon Akhmedli: Preimage Cardinalities of Continuous Functions 

We find all subsets of N which occur as the set of possible cardinalities of preimages of a continuous function. We also study and answer this question for various subclasses of continuous functions. 
17 November 2020
Gulnar Aghabalayeva: Sums, Projections, and Sections of Lattice Sets, and the Discrete Covariogram(Part 2)

10 November 2020
Gulnar Aghabalayeva: Sums, Projections, and Sections of Lattice Sets, and the Discrete Covariogram

We present discrete versions (in Z^n) of known results for convex bodies in R^n, in particular about reconstruction from information about their sections or projections.
27 October 2020
Alex Kokot (University of Notre Dame), Brooke Dippold (Longwood University), Ian Klein (Carleton College), Jose Agudelo (NDSU):
Curve Reconstructions in Equi-affine and Euclidean Geometries as Applications in Computer Vision

Equi-affine and Euclidean geometry supply different advantages in image processing. Equi-affine geometry is considered a more appropriate setting to compare handwriting while Euclidean geometry is appropriate for more common, everyday objects. Reconstructing curves from a given curvature in Euclidean or equi-affine geometry is then an important geometric application in computer vision. For certain curvatures this is impossible. It is also often the case that processing experiences slight perturbations, causing deformations in reconstructed curves. Here, we supply methods to approximate these reconstructions for both analytic and non-analytic curvatures. We also study slightly deformed curvatures and their corresponding reconstructions and propose metrics with which to compare these deformations to the original curve. 

20 October 2020
Syed Husain: Applications of inequalities for bandlimited functions in signal recovery (Part 2)

13 October 2020
Syed Husain: Applications of inequalities for bandlimited functions in signal recovery

There has been development in analytic number theory related to inequalities for entire functions of exponential type that have a finite L_p norm (1 \leq p \leq \infty) on the real axis (also called bandlimited functions). These inequalities can be used to find conditions under which perfect recovery of a signal, which has missing data or noise over a period of time, is possible. Logan's phenomenon gives one such condition which was considerably improved by the concept of Nyquist density introduced by Donoho and Logan (1990). They also developed some inequalities for bandlimited functions in L_1 and L_2 settings that have applications in signal recovery. I am currently working on such kind of inequalities for different bandlimited functions that I will be discussing in my talks.

29 September 2020
Faraad Armwood: More Circle Sum (Part 2)

22 September 2020
Faraad Armwood: More Circle Sum

In this talk we'll introduce the circle sum and discuss cases in which the summation can be realized. 

Spring 2020


10 March 2020
Faraad Armwood: More about the Circle Sum

The circle sum is a surgery on oriented surfaces S(a) and S(b) where a,b denote the genus. Denoting the summation by "+", then S(a) + S(b) is an oriented surface of genus a + b - 1. The surgery requires a smooth annulus (f) with a nonzero normal section (e). In this talk, we'll give a case in which f,e can be topologically, and smoothly constructed. 
3 March 2020
Morgan O'Brien: Some Ergodic Theorems on von Neumann Algebras (Part 3)

25 February 2020
Morgan O'Brien: Some Ergodic Theorems on von Neumann Algebras (Part 2)

18 February 2020
Morgan O'Brien: Some Ergodic Theorems on von Neumann Algebras
 
In the context of noncommutative geometry, the study of von Neumann algebras can be viewed as the noncommutative analogue of measure theory. As an important area of study in classical measure theory is ergodic theory, it is natural to ask what the noncommutative analogue of ergodic theory would look like. With this in mind, during the first talk we will discuss some important properties and give some examples of von Neumann algebras, and in the second talk we will discuss and prove some ergodic theorems that apply to these algebras.

4 February 2020
Mariangel Alfonseca: A local solution for Busemann-Petty Problem number 8 
 
The 8th Busemann-Petty problem, posed in 1956 and still open, asks the following question: "Are ellipsoids characterized by the fact that the Gauss curvature at a point of contact with a tangent plane parallel to a hyperplane H is proportional to the -(n+1) power of the volume of their section by H?"
The Minkowskian interpretation of the question is: "Is the geometry Euclidean if the Minkowski sphere has constant Minkowski curvature?"  The answer is known only in dimension 2, proven by Petty in 1955. We will present an overview of the proof that, for bodies close to the Euclidean ball in the Banach-Mazur distance, the answer to the question is affirmative. This is joint work with F. Nazarov, D. Ryabogin and V. Yaskin.

Fall 2019

3 December 2019

Pratyush Mishra, NDSU: Mostow's rigidity 
 
A basic problem in topology is to decide up to what extent homotopy equivalence between two spaces decides whether they are homeomorphic or not? Mostow's rigidity has a strong answer to this for hyperbolic manifold, which states that "Given two complete hyperbolic manifolds M and N(of dim >=3) of finite volume, any homotopy equivalence between M and N is homotopic to an isometry(hence a homeomorphism) from M to N". In this talk, I will also state the algebraic formulation of this rigidity theorem in terms of lattices in Lie group SO(n,1). I will build the necessary ingredients in my first talk and shall prove the Mostow's rigidity for compact case by the end of second talk. The theorem was originally proved by G.D. Mostow (in 1968) for closed manifolds and latter extended by Margulis, Prasad, Gromov and many others. I will present Gromov's (in 1982) proof of the theorem, which also appeared in Thruston's lecture notes (in 1979).

19 November 2019

Adam Buskirk, NDSU: Riffle Shuffle Dynamics
 
There has been a substantial amount of research into the dynamics of shuffling cards. One avenue of attack upon the problem follows a deterministic approach involving perfect shuffling, which reduces the problem to purely one of algebra, especially analyzing the characteristic of a single element of a symmetric group. However, this approach, while useful for sleight-of-hand and capturing certain intuitions about shuffling dynamics, neglects the purpose of shuffling: introducing randomness, i.e. denying all observers any certain information about the relative locations of the cards in the deck. In this seminar, we introduce the other side of the study of riffle shuffling, following research established by Dave Bayer and Persi Diaconis in their seminal paper Trailing the Dovetail Shuffle to its Lair, and explore four different models of riffle shuffling and results derived thereby. With an eye towards future research in extending two of these four models utilizing ideas from dynamical systems which underlie them, a new approach and moderate adaptations to the notations used by Bayer and Diaconis are utilized to facilitate a slightly more complete algebraic comprehension of shuffling dynamics.
12 November 2019

Faraad Armwood, NDSU: The Circle Sum (Part 2)
5 November 2019

Faraad Armwood, NDSU: The Circle Sum (Part 1)
 
The circle sum is a (smooth) surgery of manifolds discovered by Bang-He Li and Tian-Jun Li.  It was used to resolve a case concerning the minimal genus problem for non-trivial sphere bundles over surfaces. Outside of its previous use, this surgery becomes an interesting one within its own right. In this talk we'll, (1) discuss the prerequisites for the sum and (2) give a sketch of the construction.

29 October 2019

Michael Preheim, NDSU: An Introduction to Mapping Class Groups of Surfaces (Part 2)

22 October 2019

Michael Preheim, NDSU: An Introduction to Mapping Class Groups of Surfaces (Part 1)

The mapping class group of a surface S is the group of isotopy classes of orientation-preserving homeomorphisms on S. In the first talk, we will define this group and provide some examples which can be calculated “by hand” as well as some examples which are much more involved. The mapping class group of a genus g≥0 surface is generated by finitely many Dehn twists about non-separating simple closed curves in the surface. In the second talk, we will discuss Dehn twists and some important theorems regarding the kernels of the inclusion, cutting, and capping homomorphisms between mapping class groups.
15 October 2019

 Doğan Çömez: Quantization of singular condensation measures (Part 2)
8 October 2019

 Doğan Çömez: Quantization of singular condensation measures (Part 1)

Click here for the Abstract

1 October 2019

 Seppi Dorfmeister, NDSU: Curves in 4-Manifolds and the Bounded Negativity Conjecture (Part 2)
17 September 2019

 Seppi Dorfmeister, NDSU: Curves in 4-Manifolds and the Bounded Negativity Conjecture (Part 1)

A smooth 4-manifold M contains an abundance of smooth curves. If one adds structure to M, such as a complex or symplectic structure, it is an interesting question to study the set of curves compatible with this additional structure. This has led to the development of extremely powerful tools allowing us to distinguish manifolds using curves which are “positive’ enough. “Negative” curves are much harder to study. In the first talk I will introduce the basic tools involved in the study of such problems.  In the second I will describe some results that have been obtained and what the BNC is.


Spring 2019

 

23 April 2019

 Michael Cohen, Carleton College: Maximal pseudometrics and distortion of circle diffeomorphisms

I'll discuss "distortion" in two contexts: (1) as an informal notion in the study of dynamical systems (especially circle dynamics), and (2) as a quasi-isometry invariant in geometric group theory.  Then I'll propose a definition intended to unify the two: we will say that a circle diffeomorphism is C^k-distorted if the distance of the n-th iterate from the identity grows sublinearly with n, where the distance in question is the Cayley distance associated to a sufficiently small open neighborhood of identity in the topological group of all orientation-preserving C^k circle diffeomorphisms.  I will give a simple classification of all C^1-distorted diffeomorphisms, and describe the (apparently much more difficult) problem of understanding C^k-distortion for k greater than or equal to 2.

16 April 2019

Halley Fritze, NDSU: A brief survey of Lefschetz fibrations (Part 2)
9 April 2019

 Halley Fritze, NDSU: A brief survey of Lefschetz fibrations

A Lefschetz fibration is a fibration over a surface where we allow finitely many fibers in the interior of the surface to self intersect. This talk will look at the topology of these fibrations and their relationship to symplectic manifolds.
2 April 2019

 Hee Jung Kim, NDSU: Heegaard Floer Homology and Spines of 4-Manifolds

This is the continuation of the previous talk about Heegaard Floer Homology. Using the Heegaard Floer homology d-invariant, a recent work of Levine and Lidman showed that there are infinitely many 4-manifolds which are homotopy equivalent to 2-sphere but do not admit a piecewise linear embedding of 2-sphere, referred to a spine, that realizes the homotopy equivalence. This was an interesting open question in dimension 4 while there exist spines in other dimensions. However, our work jointed with D. Ruberman shows that some of their examples do admit a topological embedding of 2-sphere that realizes the homotopy equivalence.
26 March 2019

 Hee Jung Kim, NDSU: Heegaard Floer Homology and Invariants

Heegaard Floer Homology introduced by Ozsbath and Szabo has been rapidly developed due to its various contribution to low-dimensional topology, particularly knot theory. This talk will give an introduction of Heegaard Floer Homology and the invariants of 3-manifolds.
19 March 2019

 Dakota Ihli, University of Illinois: Genericity of Monothetic Subgroups of Polish Groups

Given a topological group G, we ask whether the group cl(<g>) has the same isomorphism type for "most" g ∈ G. More precisely, is there a group H such that the set { g ∈ G : cl(<g>) ≅ H } is dense? Comeagre? If so, can we identify this H? In this expository talk I will discuss known results and conjectures for certain Polish groups. Emphasis will be given to the case when G is the group of Lebesgue-measure preserving automorphisms of the unit interval.
5 March 2019

 Morgan O'Brien, NDSU: The Gelfand-Naimark Theorem (Part 2)
26 February 2019

 Morgan O'Brien, NDSU: The Gelfand-Naimark Theorem

In this talk, we will start by introducing the concept of C*-algebras and discuss some examples and important properties that they possess. After this, we will prove the Gelfand-Naimark Theorem, which classifies commutative C*-algebras with identity as the spaces of continuous functions on a compact space.

19 February 2019

 Chase Reuter, NDSU: Iterations of the projection body operator

Given a convex body K, the projection body Π(K) encodes information about shadows (projections) of the original.  It is useful in several inequalities; and, in such endeavors, it is often important to classify the fixed points of the operator in question (projection body operator).  To that end, we will classify one fixed point, namely the sphere, as a local attractor of this operator.  To do this we'll delve a bit into spherical harmonics, some fun operators, and other topics.

12 February 2019

Faraad Armwood, NDSU:  Symplectic Geometry and Hamiltonian Systems (Part 2)
5 February 2019

 Faraad Armwood, NDSU: Symplectic Geometry and Hamiltonian Systems

Symplectic manifolds are manifolds equipped with a closed, non-degenerate 2-form. The structure preserving morphisms of this geometry are symplectomorphisms. These morphisms are very important from a physical viewpoint as they are the canonical transformations of Hamilton's Equations, which is what we'll show. Why care? Such transformations aid in solving the Hamilton-Jacobi Equation (HJE) of your system. This equation is another formulation of classical mechanics, equivalent to Newtonian and Hamiltonian. The HJE is very useful for identifying conserved quantities in your system, sometimes doing so even when the system cannot be completely solved.

Fall 2018

Tuesdays, 11:00 am
Minard 208
27 November 2018

 Mariangel Alfonseca, NDSU: On Schäffer's conjecture about the girth of polar convex bodies (Part 2)
30 October 2018

 Mariangel Alfonseca, NDSU: On Schäffer's conjecture about the girth of polar convex bodies

In 1973, Schäffer proved that the perimeter of a planar convex body (measured in the norm whose unit disc is the given body), is equal to the perimeter of its dual body (measured in the dual norm). Schäffer then conjectured that in higher dimensions the girth of a convex body should equal the girth of its polar. The conjecture was proven in 2006 by Alvarez Paiva, using techniques from symplectic geometry. In the first talk we will show a geometric proof of Schäffer's two dimensional theorem. In the second talk, we will address the higher dimensional problem.
23 October 2018

 Semyon Litvinov, Pennsylvania State University Hazleton: Some recent advances in noncommutative ergodic theory

It is known that, for a positive Dunford-Schwartz operator in a noncommutative Lp-space, associated with a semifinite von Neumann algebra the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in every noncommutative fully symmetric space E such that μ_t(x)→0 as t→0 for every x in E, where μ_t(x)(x) is the non-increasing rearrangement of x. Further, we establish almost uniform convergence in a variety of noncommutative individual ergodic theorems such as weighted ergodic theorems with continuous time and Wienner-Wintner ergodic theorem.

25 September 2018

 Hee Jung Kim, NDSU: Exotic surfaces in 4-manifolds and stabilization (Part 2)
18 September 2018

 Hee Jung Kim, NDSU: Exotic surfaces in 4-manifolds and stabilization

It is well known in 4-dimensional topology due to Freedman and Donaldson that there are infinitely many simply-connected 4-manifolds that are homeomorphic but not diffeomorphic, and the fundamental principle of Wall shows that such manifolds become diffeomorphic after stabilization by connected sum with sufficiently many copies of a S^2-budle over S^2. In many cases of interest, a single stabilization suffices to obtain a diffeomorphism and so this is conjected to be always the case. Exotic phenomenon is also known in surfaces embedded in 4-manifolds, namely, exotic surfaces that are equivalent up to homeomorphism but not diffeomorphism. Recent works jointed with Dave Auckly, Paul Melvin, and Daniel Ruberman show that an analogous stabilization principle holds for exotic surfaces in 4-manifolds. In this talk, we will discuss the constructions of exotic surfaces and the dissolution of exotic smooth structures on surfaces after single stabilization.


Spring 2018

24 April 2018

 Michael Cohen, NDSU: Bounded variation, absolute continuity, and topologizing transformation groups (Part 2)


17 April 2018

 Michael Cohen, NDSU: Bounded variation, absolute continuity, and topologizing transformation groups

I'll recall and compare the classical conditions of bounded variation (BV) vs. absolute continuity (AC) in the context of real-valued functions of a real variable, and I'll mention some classical results which may not be well-known and which highlight the gap from BV to AC.  Then I'll apply this machinery to answer a question about topologizing groups:  Does there exist a separable complete metric group topology on the group of interval homeomorphisms which have BV derivatives, or respectively, AC derivatives?

27 March 2018

 Doğan Çömez, NDSU: Quantization of self-similar probability measures and optimal quantizers (Part 3)

Click here for the talk notes

20 March 2018

 Liz Sattler, Carleton College: S-limited shifts

S-gap shifts are well-studied shift spaces on the alphabet {0,1} defined by a subset, S, of the non-negative integers.  The set S defines the allowable number of 0s that can occur between two 1s in each infinite string in the subshift.   Building on this idea, we will construct an S-limited shift, which is a subshift on the alphabet {0,1,...p}. Infinite strings in this particular shift space satisfy the property that any block of a single letter must appear in length s, where s is an element of a predetermined subset of the natural numbers for that specific letter. We will calculate the entropy for these shift spaces and compare this calculation to known entropy calculations for other types of shift spaces.  As time permits, we may discuss other properties of these spaces, such as conjugacy conditions or mixing properties. 

6 March 2018

Doğan Çömez, NDSU: Quantization of self-similar probability measures and optimal quantizers (Part 2)

Click here for the talk notes

27 February 2018

Doğan Çömez, NDSU: Quantization of self-similar probability measures and optimal quantizers

Quantization of probability distributions concerns the best approximation of an n-dimensional probability distribution P by a discrete probability with a given number of supporting points (optimal quantizers).  Naturally, it deals with determining appropriate partitioning of the underlying space while ensuring that error in this approximation is minimal.  The theory has its origins in signal processing, where quantization refers to discretization of signals without significant loss of information.

The first part of this talk will focus on describing the mathematical theory in the general, but uniform, setting.  In the second part I will provide insight on some results obtained recently.

Click here for the talk notes

13 February 2018

MSRI Video Talk by Ramon van Handel: Gaussian Measures.

Fall 2017


28 November 2017
Bjorn Berntson, NDSU: Painlevé property for partial differential equations

Singularity analysis of ordinary differential equations in the complex plane has been successful in isolating the most important nonlinear, physically-relevant equations. The extension of this analysis to partial differential equations is discussed through key semilinear examples, including Burgers' equation and the Korteweg-de Vries equation. Outstanding issues in applying this method to more general classes of equations are described.

21 November 2017
Azer Akhmedov, NDSU: Groups not acting on compact metric spaces

We produce a rather simple criterion on a group to guarantee that it does not act on compact metric spaces by homeomorphisms. This criterion has numerous (very efficient) applications.

7 November 2017
Friedrich Littmann, NDSU: Viazovska's approach to sphere packing bounds (Part 4)
24 October 2017
Friedrich Littmann, NDSU: Viazovska's approach to sphere packing bounds (Part 3)
17 October 2017
Friedrich Littmann, NDSU: Viazovska's approach to sphere packing bounds (Part 2)
10 October 2017
Friedrich Littmann, NDSU: Viazovska's approach to sphere packing bounds

The sphere packing problems asks for the densest packing of spheres into Euclidean space. The solution for dimension 2 has been long known, while dimension 3 was solved by T.C. Hales with a very complicated proof in 1998. In 2016 M. Viazovska published a beautiful Fourier analytic proof for dimension 8, and it was shown shortly after that her method also yields dimension 24.

This talk will outline her approach. The main tool is an interpolation method that allows to simultaneously interpolate a radial Schwarz function and its Fourier transform in dimension 8 at the square roots of the even positive integers. I will use without proof Poisson summation on a lattice and Fourier transforms of multivariate Gaussians, as well as some basic facts about continuation of analytic functions.

3 October 2017
Mariangel Alfonseca, NDSU: On a local solution of the fifth Busemann-Petty problem (Part 3)
26 September 2017
Mariangel Alfonseca, NDSU: On a local solution of the fifth Busemann-Petty problem (Part 2)
19 September 2017
Mariangel Alfonseca, NDSU: On a local solution of the fifth Busemann-Petty problem

In 1956, Busemann and Petty proposed ten problems concerning sections of origin symmetric convex bodies, which originated as questions about the geometry of Minkowski spaces. The first of the problems (now known as the Busemann-Petty problem), was completely solved in 1999, after decades of contributions by a number of mathematicians, who in the process opened new directions in the study of convex bodies. The other nine problems remain open up to date. In these talks I will introduce the fifth Busemann-Petty problem, along with its geometric interpretation, and provide a local solution. This is a joint work with F. Nazarov, D. Ryabogin and V. Yaskin. 
Spring 2017
 

28 March 2017

Friedrich Littmann, NDSU: Extremal functions with transforms supported in convex bodies (Part 2)

A sequence in [0,1] is called uniformly distributed if the number of elements in A among the first N sequence elements is asymptotically equal to N length(A) for every interval A in [0, 1]. A criterion of H. Weyl relates this to the behavior of certain exponential sums. We will discuss applications of bandlimited majorants and minorants of characteristic functions that give in particular a quantitative version of Weyl's criterion ('Erdos-Turan inequality').

21 March 2017

Friedrich Littmann, NDSU: Extremal functions with transforms supported in convex bodies

This talk considers the problem of finding a majorant (or minorant) of the characteristic function of an origin symmetric convex body, where the majorant has Fourier transform supported in another convex body.

Effective constructions in dimension two or higher are known only for some very special choices of bodies (mainly spheres, ellipses, and rectangles). I will discuss the history, some applications, and several obstructions. The latter arise from the fact that some fundamental facts about Hilbert spaces of entire functions in one variable fail to be true in higher dimensions.

7 March 2017

Josef Dorfmeister, NDSU: An Excursion into 4-Dimensional Manifolds (Part 2) 

21 February 2017

Josef Dorfmeister, NDSU: An Excursion into 4-Dimensional Manifolds

Low Dimensional Topology generally means the study of manifolds of dimension less than 5.  I will give an introduction to some of the key concepts involved in the study of 4-dimensional manifolds and describe what some of the interesting (to me) questions are.  The talk is aimed at non-specialists, so I will try to explain all concepts, at least conceptually, as they appear. 

14 February 2017

Azer Akhmedov, NDSU: On non-embeddability of knot groups into the group of analytic diffeomorphisms of compact 1-manifolds (Part 2)

31 January 2017

Azer Akhmedov, NDSU: On non-embeddability of knot groups into the group of analytic diffeomorphisms of compact 1-manifolds
Every knot group is known to be left-orderable thus it embeds in Homeo+(I) - the group of orientation preserving homeomorphisms of the interval I = [0,1]. By far, not every knot group is bi-orderable. The bi-orderability of a knot group has interesting topological consequences for a Dehn filling of the knot. Embedding a group into Diff_{+}^{\omega }(I)  - the group of orientation preserving analytic diffeomorphisms of the interval would imply its bi-orderability. Thus the question of which knot groups embed into Diff_{+}^{\omega }(I) becomes interesting. It is also very interesting which knot groups embed in Diff_{+}^{\omega }(S^1)- the group of orientation preserving analytic diffeomorphisms of the circle. In a joint work with M.Cohen, we have classified all RAAGs which embed in the group of orientation preserving analytic diffeomorphisms of a compact 1-manifold. By extending the techniques of this work, we provide an answer to the embeddability question for the knot groups. This is a joint work with Cody Martin.

Fall 2016

22 November 2016 

Michael Cohen, NDSU: A problem in Haar null sets and some questions about Lie groups

I will describe a very simple but interesting problem in the theory of generalized Haar null sets that I worked on as a grad student and did not solve.  Then I'll describe an idea for possibly constructing a negative solution using Lie groups.  Toward the possible construction, I'll ask some questions about Lie groups of varying degrees of concreteness, which might be either easy or hard.  The answers probably boil down to performing computations on the Lie algebra.  If anybody knows the answers or has some other insight, we might be able to swiftly publish a paper. .

15 November 2016 

Doğan Çömez, NDSU: Shift dynamical systems, subshifts and beta-shifts (Part 5)

1 November 2016 

Benton Duncan, NDSU: Operator algebras and directed graphs: a survey (Part 2)

25 October 2016 

Benton Duncan, NDSU: Operator algebras and directed graphs: a survey

I will do a quick survey of the C*-algebras associated to directed graphs. I will then consider generalization given by introducing a coloring function to the edge set of the directed graph. I will focus on the question of when the full and reduced C*-algebras are the same.

18 October 2016 

Doğan Çömez, NDSU: Shift dynamical systems, subshifts and beta-shifts (Part 4)

11 October 2016 

Doğan Çömez, NDSU: Shift dynamical systems, subshifts and beta-shifts (Part 3)

4 October 2016 

Doğan Çömez, NDSU: Shift dynamical systems, subshifts and beta-shifts (Part 2)

27 September 2016 

Doğan Çömez, NDSU: Shift dynamical systems, subshifts and beta-shifts
This is a series of talks on one of the most fundamental class of dynamical systems: shift spaces. The first two talks will essentially be expository and accessible to all graduate students. To follow all the talks a rudimentary knowledge of measures and topology will be helpful, but not absolutely necessary.

The first talk will be on shift spaces and investigation of main features of such dynamical systems with plenty of illustrative examples. The second talk will be devoted to the classes of shift spaces known as subshifts of finite type and sofic subshifts. Again, the main features of these spaces will be exposed via some concrete examples.

The last talk will focus on the class of subshifts which are currently one of the hot topics of study in dynamical systems: beta-shifts. Starting with some well-known examples, we will explore the structure of beta-shifts and outline some of the latest results obtained. The talk will culminate in discussion on some open problems related to sofic shifts and beta shifts.

Spring 2016
19 April 2016 
Vlad Yaskin, University of Alberta: Uniqueness Questions in Geometric Tomography
We will discuss some results on the unique determination of convex bodies (and other objects) from various tomographic data.

5 April 2016 
Erin Brownlee, NDSU: Graph C*-algebras
Formal C*-algebras arose in the 1940's as a way to define an algebraic structure for operators on Hilbert spaces. In fact, every C*-algebra is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space (Gelfand-Naimark). In the 1980's, J. Cuntz and W. Krieger developed a family of C*-algebras which coincide with row-finite directed graphs. In this talk, we'll do a brief review of C*-algebras, and introduce the Cuntz-Krieger graph families. We'll consider several examples of such families, and discuss some of the main results of Cuntz and Krieger. Finally, I'll present some of my results up to this point.
22 March 2016 
Friedrich Littmann, NDSU: De Branges transform and the Hilbert inequality (Part II)
7 March 2016 
Friedrich Littmann, NDSU: De Branges transform and the Hilbert inequality
Abstract (click here)
1 March 2016 
Eder de Moraes Correa, Unicamp: Hamiltonian systems in coadjoint orbits

In this talk I am going to give an overview about the construction of Hamiltonian integrable systems in coadjoint orbits using the Thimm's trick as well as to explain its relations with Lax pair, quantum groups and representation theory.

16 February 2016 
Doğan Çömez, NDSU: Existence of local ergodic Hilbert transform for admissible processes

This presentation is on the existence of the local ergodic Hilbert transform (leHt) for a class of superadditive processes, known as  admissible processes.  The local ergodic Hilbert transform is known to exist a.e. with respect to invertible measure preserving flows; hence for all additive processes. We extend his result to the setting of bounded symmetric admissible processes relative to invertible measure preserving flows.  For this purpose, we obtain a version of the maximal ergodic inequality for the leHt applicable to admissible processes.

9 February 2016
Mariangel Alfonseca, NDSU: Convex bodies with directly congruent projections
It is known that a convex body L in R^n is completely determined if we know its orthogonal projection on every k-dimensional linear subspace of R^n, for a fixed k<n. In fact, knowing L's projections up to a translation on each subspace determines L up to a translation in R^n. However, if the projections are known up to an isometry on each subspace, it is an open problem whether L is determined up to an isometry. In this talk we will survey known results, and give a proof of the case k=3 when the isometry is a rotation. This is joint work with M. Cordier and D. Ryabogin. 

Fall 2015


1 December 2015
Chung-I Ho, National Tsing Hua University: Embedded non-oriented Lagrangian surfaces in symplectic 4-manifolds
The existence and construction of Lagrangian surfaces in symplectic 4-manifolds is an interesting topic in symplectic geometry and low dimension topology. In this talk, I will discuss this problem for non-oriented Lagrangian surfaces. In particular, I will give more detailed descriptions for rational and ruled manifolds. 
24 November 2015
Jessica Striker, NDSU: On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope

The Chan-Robbins-Yuen polytope is a polytope with a remarkable volume formula given by a product of Catalan numbers. This has been proved by Zeilberger in 1999, but a combinatorial proof remains elusive. In this talk, we introduce an alternating sign matrix analogue of the Chan-Robbins-Yuen polytope, which we call the ASM-CRY polytope. We show that this polytope also has a remarkable volume, namely, the number of standard Young tableaux of staircase shape. Furthermore, we are able to prove this combinatorially by showing that the ASM-CRY polytope is an order polytope, which is an especially nice class of polytopes with a combinatorial interpretation for the volume. This talk will be accessible to students, and is joint work with Alejandro Morales (UCLA) and Karola Meszaros (Cornell).

17 November 2015
Josef Dorfmeister, NDSU: Lefschetz Fibrations (Part II)
10 November 2015
Josef Dorfmeister, NDSU: Lefschetz Fibrations

This talk will be an introduction to Lefschetz fibrations. Such fibrations play a central role in low-dimensional topology, providing a wide range of examples of 4-manifolds.  They have deep connections to algebraic geometry, symplectic geometry and algebra.  I will try to highlight some of these connections, show where they fit into the modern picture and describe some open problems and recent results. 

27 October 2015
Azer Akhmedov, NDSU: Finiteness of the topological rank of diffeomorphism groups (Part II)
I will continue discussing the finiteness result for the topological rank of the group Diff^k _0(M) of diffeomorphisms  of compact smooth manifolds. One of the key tool used in the result is the perfectness of diffeomorphism group proved which I discussed in the first talk - this is due to Tsuobi in dimesion one, and various authors (Thurston, Mather, Epstein, Haller-Rubycki-Tiechmann) in higher dimensions. In the second talk, I'll discuss another major ingredient: the root approximation property of diffeomorphism groups. Like perfectness, this property is interesting for an arbitrary topological group. Some of the results which I'll state during the talk are from the joint work with Michael Cohen and Dogan Comez. 
20 October 2015
Eder de Moraes Correa, Unicamp: Symplectic geometry in coadjoint orbits (Part III) 
In this talk I will finish my exposition about symplectic geometry in coadjoint orbits. I have introduced coadjoint orbits as symplectic manifold in the first talk, in the second one I have shown how coadjoint orbits fit in the context of Hamiltonian Lie group action. In this last talk, I am going to show how the Lie-Poisson structure provides a good approach to study Hamiltonian dynamics in coadjoint orbits. 
13 October 2015

Azer Akhmedov, NDSU: Finiteness of the topological rank of diffeomorphism groups

I will state a finiteness result for the topological rank of the group Diff^k _0(M) of diffeomorphisms of compact smooth manifolds. I'll discuss some key ingredients of the proof which are interesting independently. 

6 October 2015

Michael Cohen, NDSU: On dense finitely generated subgroups (Part II)

29 September 2015
Michael Cohen, NDSU: On dense finitely generated subgroups 

By a classical theorem of Leopold Kronecker, almost every (n+1)-tuple of vectors generates a dense additive subgroup of R^n.  What is the situation in other uncountable topological groups-- do finitely generated dense subgroups exist, and "how common" are they (in the sense of measure or category)?  I'll discuss the history of this problem and the solution in a variety of groups, from compact Lie groups to massive infinite-dimensional groups.  In particular, I'll present recent joint results with Azer Akhmedov on the existence and genericity of finitely generated dense subgroups in the homeomorphism and C^1-diffeomorphms groups of the interval..

22 September 2015
Eder de Moraes Correa, Unicamp: Symplectic geometry in coadjoint orbits (Part II)

15 September 2015

Eder de Moraes Correa, Unicamp: Symplectic geometry in coadjoint orbits

In this first talk I will introduce the basic concepts of coadjoint orbits as symplectic manifolds and discuss how this kind of manifold provides a good "toy model" to study Hamiltonian Lie group actions, Poisson geometry and Integrable systems. The main idea will be to introduce the basic material and motivations to study symplectic geometry in coadjoint orbits, for this purpose, the basic knowledge required are the conceptions of diff erentiable manifolds and Lie groups.


Spring 2015

Location: South Engineering 208
Time: Tuesdays, 11:00-11:50 am

30 March 2015

Brian Chapman, NDSU: Adic transformations related to Pascal's triangle and the Catalan numbers

I will introduce adic transformations, which are dynamical systems on the space of infinite paths in certain graphs.  Properties of two adic transformations, whose underlying graphs are closely related to Pascal's triangle and the Catalan numbers, will then be discussed.

10 March 2015
Liz Sattler, NDSU: Connections between entropy and Hausdorff dimension

Let X be a collection of all infinite words on a finite alphabet.  A subshift of X is a subspace of X which is closed under the shift map.  Given a fractal in R^n constructed with a finite number of contractive maps, we can associate each point in the fractal with an infinite word from a symbolic space, X.  Similarly, we can define a subfractal by only considering the points in the fractal associated with an infinite word from a  given subshift.  In 1967, Furstenberg proved that the entropy of a subshift is equal to its Hausdorff dimension.  In this talk, we will discuss the​ importance of Furstenberg's result and how it can be utilized to calculate the Hausdorff dimension of some subfractals.

3 March 2015
Doğan Çömez, NDSU: Classical and new recurrence theorems and connections to combinatorial number theory (Part II)

24 February 2015
Doğan Çömez, NDSU: Classical and new recurrence theorems and connections to combinatorial number theory

In ergodic theory the celebrated Poincare Recurrence Theorem (PRT) has been an inspiration to many other recurrence results.  One such theorem, now a classic, is Furstenberg’s Multiple Recurrence Theorem (FMRT). This result, besides generalizing PRT, has some deep connections with some results in combinatorial number theory. Namely, it provides a (rather) short proof of the famous theorem of Szemeredi (as well as Van der Waerden) on arithmetic progressions. FMRT led to many new developments in ergodic theory as well as in combinatorial number theory. There are numerous variations and generalizations of FMRT; some of which have counterparts in number theory (such as Sarkozi’s Theorem on intersective polynomials), but some others do not.

In this talk (or, possibly two talks), I will begin with a brief survey of recurrence theorems in ergodic theory and sketch the proof of Szemeredi’s Theorem using FMRT. In the second part, I will talk about a new generalization of FMRT into admissible processes setting. The last part of the talk will concern the strong convergence of multiple recurrence for admissible processes.
10 February 2015
Friedrich Littmann, NDSU: Reconstruction of bandlimited functions(Part II)
3 February 2015
Friedrich Littmann, NDSU: Reconstruction of bandlimited functions

Let F be a function whose Fourier transform is supported in an interval (say, [-delta,delta]). Let N be an integrable function. In 1984, Donoho and Logan discovered that knowledge of N +F is sufficient  to recover F provided that N is "sparse enough". A motivation to study this problem comes from signal  rocessing; it is not uncommon that a bandlimited signal sent from a transmitter to a receiver is corrupted by some noise (which can be thought of as a sparse L1-function). The receiver needs to reconstruct the original signal from the corrupted version. In this talk we will describe Donoho and Logan's approach and discuss possible generalizations to higher dimensions.

Fall 2014

Location: Minard 208
Time: Tuesdays, 11:00-11:50 am
2 December 2014
Michael Cohen, NDSU: On the complexity of classification problems (Part II)
18 November 2014
Michael Cohen, NDSU: On the complexity of classification problems

When are two unitary n by n matrices unitarily equivalent? When are two Lebesgue measure-preserving transformations of [0,1] equivalent?  The first classification problem is known to be "easy" while the latter is known to be "hard."  Giving a rigorous mathematical meaning to common-sense notions of "difficulty" or "complexity" of classification problems is the main content of the field known as invariant descriptive set theory.  In this expository talk, I'll give an overview of this field, with an emphasis on equivalence relations arising from Polish group actions, and classifying ``how complicated they are'' from a rigorous perspective.  I'll give lots of examples of problems arising in analysis, low-dimensional topology, ergodic theory, and other classical fields.
4 November 2014
Leo Butler, NDSU: Nosé-Hoover Thermostats (Part II)

28 October 2014

 Leo Butler, NDSU: Nosé-Hoover Thermostats

Statistical mechanics attempts to predict the macroscopic behaviour of many-particle systems. On the other hand, the microscopic behaviour is governed by classical mechanics. The predictions of the two models may or may not coincide.

A thermostat is a system that models a particle system immersed in a heat bath. A fundamental question about such a system is whether it reaches thermal equilibrium with the heat bath. Even for the harmonic oscillator this question is non-trivial, and an answer is only known in the simplest case.
7 October 2014

 Caroline Turnage-Butterbaugh, NDSU: Large gaps between zeros of Dedekind zeta-functions of quadratic number fields

Let K be a quadratic number field with discriminant d. As we discussed last week, the Dedekind zeta-function attached to K, denoted $\zeta_K(s)$, factors as the product of the Riemann zeta-function and a Dirichlet L-function. Using the mixed second moments of $\zeta_K(1/2+it)$ and its derivatives, we prove the existence of gaps between consecutive zeros of $\zeta_K(s)$ on the critical line which are larger than the average spacing.
23 September 2014

 Caroline Turnage-Butterbaugh, NDSU: Introduction to the Riemann zeta-function and L-functions (Part II)

As a continuation of Part I, we will introduce some examples of L-functions, including Dirichlet L-functions and Dedekind zeta-functions of quadratic number fields. We will then use the moments of these functions to study the vertical distribution of their nontrivial zeros on the critical line.
16 September 2014

 Caroline Turnage-Butterbaugh, NDSU: Introduction to the Riemann zeta-function and L-functions
The Riemann zeta-function and its generalizations, called L-functions, are ubiquitous yet mysterious functions in number theory. These functions can be defined in association with a plethora of mathematical objects, including Dirichlet characters, number fields, and modular forms. In Part I, we discuss properties of the Riemann zeta-function and motivate the study of its moments. 


Spring 2014

Location: Minard 212
Time: Tuesdays, 2:30-3:20 pm
15 April 2014

 Mark Spanier, NDSU: Interpolations at Zeros of Laguerre-Polya Functions and Lı-approximations (Part II)

1 April 2014

 Mark Spanier, NDSU: Interpolations at Zeros of Laguerre-Polya Functions and Lı-approximations

For many nice real-valued functions f, it is known that a real entire function of exponential type pi is the best approximation in Lı-norm to f if it interpolates f at a translate of the integers. However, in the case of best approximations in Lı(dµ)-norm the optimal approximations may interpolate at sets other than a translate of the integers. In this talk, we will describe general interpolation results that allow us to construct entire functions that interpolate a given function at the zeros of Laguerre-Polya functions. As an application, we will construct best approximations in Lı(dµ)-norm for a wide class of functions.
11 March 2014

 Rob Hladky, NDSU: Special geometric structures and sub-Riemannian manifolds (Part III)

25 February 2014

Rob Hladky, NDSU: Special geometric structures and sub-Riemannian manifolds (Part II)

18 February 2014

Rob Hladky, NDSU: Special geometric structures and sub-Riemannian manifolds

 We’ll outline the idea of geometries associated to subgroups of GL(n,R) and review how this is related to the classical Riemannian notions of holonomy, curvature and special geometries. Then we’ll see how these ideas can be generalized to step 2, sub-Riemannian manifolds where there are new, intrinsic invariants that must be accounted for .


Fall 2013

Location: Minard 220
Time: Tuesday, 2:30-3:20 pm

3 December 2013

Abraham Ungar, NDSU: Relativistic Analytic Hyperbolic Geometry with Applications

Relativistic analytic hyperbolic geometry in n dimensions is our target, whose underlying motive is to cultivate a new interdisciplinary region between the hyperbolic geometry of Lobachevsky and Bolyai and the special theory of relativity of Einstein. The new interdisciplinary region involves analysis, algebra and geometry. As such, it proves to be a rich playground for hyperbolic geometry and relativistic physics, including relativistic quantum mechanics, resulting in the speaker’s six books.
26 November 2013

Michael Cohen, NDSU: Smallness in Polish Groups (Part II)

12 November 2013

Michael Cohen, NDSU: Smallness in Polish Groups

I will discuss three different notions of "smallness" which are well-defined in any Polish topological group:  the meager sets, the Haar null sets, and the openly Haar null sets.  A question of Darji asks:  Does every Polish group decompose into the disjoint union G=A \cup B of a meager set A and a Haar null set B?  This is well-known to be true in the locally compact case.

In joint work with Robert R. Kallman, we use a general method involving openly Haar null sets to answer Darji's question in the affirmative for a large class of natural Polish groups including: all abelian groups, all countable products of locally compact groups, the full unitary group of separable infinite-dimensional Hilbert space, the homeomorphism group of the circle, the group of permutations of $\mathbb{N}$, the group of order-preserving automorphisms of $\mathbb{Q}$, and many others.  We also demonstrate the limitations of our method, by identifying a peculiar property of two-sided translations in the group of orientation-preserving homeomorphisms of the interval [0,1] that renders our lemma inapplicable.  I will pose some open questions along the way.
29 October 2013
Azer Akhmedov, NDSU: Lattices of Lie groups (Part II)

22 October 2013

 Azer Akhmedov, NDSU: Lattices of Lie groups

This is an elementary introduction to the theory of lattices of Lie groups. I'll mainly concentrate on some important examples. I'll also introduce the notion of a lattice for the infinite-dimensional group Diff(I). The talk is aimed at a general audience.

15 October 2013

Josef Dorfmeister, NDSU: Spheres in Rational and Ruled Manifolds (Part II)

8 October 2013

Josef Dorfmeister, NDSU: Spheres in Rational and Ruled Manifolds

I will present some recent results on the existence of symplectic spheres in rational or ruled manifolds.  In the first talk I will state the main result and explain the language used.  I will define what a rational or ruled manifold is, describe a symplectic sphere and the necessary operations on homology/cohomology to determine the existence of such an object.  A rudimentary knowledge of what a smooth manifold is would be useful. In the second talk I will describe how to prove this result.  This makes use of some earlier results on the existence of certain symplectic forms and the theory of J-holomorphic curves (as well as some Calculus!).  I will describe the key steps in the proof. This is joint work with Tian-Jun Li and Weiwei Wu.

24 September 2013

Davis Cope, NDSU: Modeling the LGN neuron response

The lateral geniculate nucleus (LGN) is a component of mammalian visual systems that is intermediate between the retina and the visual cortex. This talk presents a nonlinear model for LGN neurons and an analysis of its properties. The model consists of a linear filter with a special type of contrast gain control
normalization. The model has the following properties:
(1) Explicit solutions can be obtained for sinusoidal grating stimuli and for circular spot stimuli.
(2) It produces the experimentally observed behavior of response saturation with increasing stimulus magnitude with the unique exception of homogeneous field stimuli.
(3) For homogeneous field stimuli, the response increases with increasing stimulus magnitude (saturation does not occur). This non-saturation (luxotonic behavior) is experimentally observed to hold over several orders of magnitude.
(4) It can be formulated as a pattern-to-pattern map and is thus potentially applicable to modeling the human visual system.
In addition:
(5) The model demonstrates an advantageous property of large (as opposed to small) gain pools. Gain pools cannot be observed directly but their properties can be inferred, and the ones that occur are large.
(6) The model provides a new and experimentally testable explanation for the occurrence of distorted contrast saturation curves.
(7) Saturated responses can be determined explicitly for many types of stimuli, thus making a new nonlinear regime accessible to experimental testing (complementing the usual linear regime for low stimulus magnitude).

17 September 2013

Davis Cope, NDSU: Modeling the human visual system: brightness illusions and other background

Mathematical models of the human visual system must, in addition to matching mathematical and physiological components, produce visual scenes that match the subjective experience of an objective scene. In other words, the crucial test for such models occurs when the subjectively perceived scene is most different from the original, that is, when a visual illusion occurs. Brightness illusions are particularly suitable as test cases for modeling because they are monocular, require only black/gray/white levels, and are stationary state. This talk provides an overview of factors and constraints relevant for brightness models of the human visual system. Relevant concepts must be drawn from experimental work and empirical conclusions because there is (as yet) no way to derive properties of perception from properties of individual neurons. Such concepts include the discovery of receptive fields for individual visual neurons and distinctive differences between the fields of neurons in different parts of the visual system, the use of neuron channels in formulating models, and the successful use of a channel model by Blakeslee and McCourt in 1999 to explain certain classes of brightness illusions.


Spring 2013


Location: Minard 212
Time: Tuesday, 2:30-3:20 pm

14 May 2013, 1 pm in Minard 136

Semyon Litvinov, Pennsylvania State University Hazleton: The Banach Principle for Semifinite Measure

We show that the Banach Principle on the almost uniform convergence of sequences of measurable operators does not entirely hold if the measure in question is not finite. Then we state and prove its proper extension to the case of semifinite measure.

9 April 2013

Mariangel Alfonseca, NDSU: Introduction to the Fourier theory of intersection bodies (Part II)

2 April 2013

Mariangel Alfonseca, NDSU: Introduction to the Fourier theory of intersection bodies

Intersection bodies appear naturally in the study of geometric tomography (reconstruction of bodies from their lower-dimensional information). Their definition is geometric, but most of their theory is developed using functional or Fourier analytic techniques, and as a result, only recently results on the geometric properties of these bodies (such as convexity) have been proven. In the first talk I will introduce intersection bodies and the basic Fourier-analytic techniques to study them. In following talks we will prove regularity and convexity results for intersection bodies of revolution.

26 February 2013

Jaegil Kim, Kent State University: On the local minimality of the volume product

Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in a fixed dimension. It is known that every Hanner polytope has the same volume product as the cube or the cross-polytope. We will talk about the local minimality of the volume product at the cube, the simplex, and the Hanner polytopes.
19 February 2013

Mariangel Alfonseca, NDSU: Introduction to convexity and the Mahler conjecture.

This talk is a preparation for next week's talk on the Mahler's conjecture. I will introduce the basic notation and results.

12 February 2013

Azer Akhmedov, NDSU: Tiles of groups with big Heesch number (Part II)

5 February 2013

 Azer Akhmedov, NDSU: Tiles of groups with big Heesch number

I'll briefly discuss several classical (open) problems about the tilings of the Euclidean or hyperbolic planes. These problems are interesting also for an abstract finitely generated group. I will present an example of a group which contains tiles of arbitrarily big Heesch number. Related to this, I'll also discuss the so-called einstein problem in groups. (has nothing to do with Albert Einstein; "einstein" stands for "ein stein"-"one tile", in German).

22 January 2013

Azer Akhmedov, NDSU:  A weak Zassenhaus-Margulis Lemma for discrete subgroups of Diff(I)

Zassenhaus-Margulis Lemma states that in a connected Lie group there exists an open neighborhood U of the identity such that any discrete subgroup generated by the elements of U is nilpotent. For example, if G is a simple Lie group (such as SL(2,R)), and \Gamma is a lattice in G, then \Gamma cannot be generated by elements too close to the identity. This lemma does not hold for Diff(I).  We prove a somewhat weaker version of Zassenhaus-Margulis Lemma for Diff(I). Built on the ideas of the proof we also prove that any C_0-discrete subgroup of Diff ^2(I) is metaabelian. 


Fall 2012

Location: ABEN 201
Time: Tuesday, 2:30-3:20 pm

14 December 2012, 2:00 PM in Minard 135

Emanuel Carneiro, IMPA, Rio de Janeiro  : On the regularity of maximal operators

In this talk I will briefly go over some of the most recent advances on the study of maximal operators in Sobolev and BV spaces, both in the continuous and discrete settings.

27 November 2012

Jayant Singh, NDSU: Stability Analysis of Discrete time Recurrent Neural Networks (Part II)

20 November 2012, 11:00 AM in Minard 336

Anar Akhmedov, University of Minnesota: Construction of symplectic 4-manifolds via Luttinger surgery

Luttinger surgery is a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold. The surgery was introduced by Karl Murad Luttinger in 1995, who used it to study to study Lagrangian tori in R^4. Luttinger surgery has been very effective tool recently for constructing exotic smooth structures on 4-manifolds. In this talk, I will present several new constructions of symplectic 4-manifolds via Luttinger surgery. This talk will be more elementary than the colloquium talk and is designed in part to provide background for the colloquium talk.

13 November 2012

Jayant Singh, NDSU: Stability Analysis of Discrete time Recurrent Neural Networks (Part I)

In this talk, we will go over the basics of Artificial Neural Networks(ANN). The famous universal approximator  problem will be introduced. I will introduce some of the useful applications of  ANN's.  And then,we will go through some of the well known approaches to solve the stability issue of Recurrent neural networks.

30 October 2012

Dmitriy Bilyk, University of Minnesota: Geometric Discrepancy Theory and Analysis.

Geometric discrepancy theory is a fascinating branch of mathematics
which studies different variations of the following question: how well can one approximate uniform distributions by  finite sets of points and how large are the errors that necessarily arise in such approximations? The apparently simple formulations of most of the questions in the theory are deceitful: these questions are deep and hard  and several basic problems are still open, especially in higher dimensions. Historically, the methods of functional and harmonic analysis (Fourier series, wavelets, Riesz products) have played a pivotal role in the development of the subject. In this talk, I will discuss the classical and recent results, methods, and ideas in this theory.

23 October 2012

Indranil SenGupta, NDSU:  Option Pricing in Financial Market (Part II)

9 October 2012

Indranil SenGupta, NDSU:  Option Pricing in Financial Market

Option pricing is one of the major challenges in financial mathematics. In this presentation I plan to present several basic aspects of appropriate pricing of options. I will go over the meaning and different examples of “options”. I will talk about introductory topics like “no arbitrage opportunity” (“there is no free lunch!”), delta hedging, binomial model and classical Black-Scholes equation. I will discuss about the concept of volatility in the financial market. I will conclude my presentation with the more recent concepts of stochastic volatility and indicate a possible research direction. This will be an introductory level talk in financial mathematics and no special background is assumed apart from basic calculus.

25 September 2012

Artem Novozhilov, NDSU:  Random Networks, Power Laws, and Epidemics (Part II)


11 September 2012

Artem Novozhilov, NDSU:  Random Networks, Power Laws, and Epidemics

Networks are the language that is used today in numerous interdisciplinary studies that unite mathematicians, physicists, biologists, engineers, computer scientists, etc. I plan to present several aspects of the network science that utilize mathematical methods, including various modeling approaches to random networks. As a particular application I am going to consider how network methods work to generate predictions about epidemic spread in a closed population.  I plan to conclude my presentation with a discussion on possible research direction. No special background is assumed,  the necessary prerequisites are  basic knowledge of classical probability and differential equations.


Spring 2012


Location: Dolve 115
Time: Tuesday, 2:30-3:20 pm

17 April 2012

Nikita Barabanov, NDSU:  Stability of discrete linear inclusions (Part III)

10 April 2012

Nikita Barabanov, NDSU:  Stability of discrete linear inclusions (Part II)

Abstract: This topic has much in common with linear algebra, real analysis, and dynamical systems. We consider main concepts: Lyapunov exponent, joint spectral radius, extremal norms and extremal solutions. Several stability criteria together with open problems will be presented and
discussed.
28 February 2012

Nikita Barabanov, NDSU:  Absolute stability of feedback systems and linear inclusions (Part I)

Abstract: Classical absolute stability problem for systems with sector time-varying nonlinearities may be reformulated as a stability problem for linear inclusions. The main concepts in this area: Lyapunov exponent, extremal norm, dual inclusions will be introduced, discussed and used to solve the problem for inclusions of order less than four.

21 February 2012

Weiyi Zhang, University of Michigan:   Nakai-Moishezon Theorem and Donaldson's question for almost complex four manifolds

Abstract: There are two interesting questions for almost complex four manifolds.The classical Nakai-Moishezon theorem (for surfaces) states
the duality between ample divisor cone and curve cone for projective surfaces. Demailly-Paun, Buchdahl and Lamari generalized this duality to
Kahler surfaces. It is natural to ask for such a duality between J-compatible symplectic cone and curve cone for almost Kahler surfaces. Another interesting question is raised by Donaldson. He asked that, in dimension four, if there is a J-tamed symplectic form, do we have a J-compatible symplectic form as well? I will discuss both questions and some recent developments (joint with Tian-Jun Li and partly with Tedi Draghici)

14 February 2012

Friedrich Littmann, NDSU:  Bandlimited functions and de Branges spaces (Part II)

7 February 2012

Friedrich Littmann, NDSU:  Bandlimited functions and de Branges spaces (Part I)

Abstract: A function is called bandlimited if its Fourier transform is supported in [-k,k] for some fixed positive k. The problem of approximation of a given function by bandlimited functions is closely related to reconstruction of a bandlimited function from its values at discrete sets on the real line.

Determination of these sets is not an easy task. In some cases the theory of de Branges spaces can be used to determine the correct reconstruction set. A de Branges space is a reproducing kernel Hilbert space of entire functions in which an analogue of the Shannon-Whittaker
interpolation theorem holds. (These spaces are generalizations of the Paley-Wiener space of bandlimited functions with the usual square integral norm.)

In this talk we explain the connection of the approximation problem above to de Branges spaces using the example of approximation to characteristic functions of arbitrary intervals.

Fall 2011

Location: Morrill 105
Time: Tuesday, 11:00-11:50 am

29 November2011

Azer Akhmedov, NDSU: Homogenous Spaces and Diophantine Approximation (Part III)

22 November2011

Azer Akhmedov, NDSU: Homogenous Spaces and Diophantine Approximation (Part II)

8 November2011

Azer Akhmedov, NDSU: Homogenous Spaces and Diophantine Approximation (Part I)

Abstract: I'll briefly review some major advances made in the area of analytic number/diophantine approximation by applying ergodic theoretic
methods in homogenous spaces. I'll concentrate around two problems, Oppenheim Conjecture and Littlewood Conjecture. The first problem has
been solved by G.Margulis in 1986. An important progress has been made recently towars the solution of the second conjecture (by Einsiedler,
Katok and Lindenstrauss). Both approaches are the special cases of the program-conjecture proposed by Margulis.

1 November2011

Nikita Barabanov, NDSU: Uniform distributions, 3/2 problem and p/q-representation of real numbers (Part II)

25 October 2011

Nikita Barabanov, NDSU: Uniform distributions, 3/2 problem and p/q-representation of real numbers (Part I)

Abstract: Asymptotic distributions of fractional parts of sequences of real numbers have been a subject of investigations of a number of prominent mathematicians including H.Weyl, I.Vinogradov, P.Erdos. We consider certain results on the uniform distribution modulo one. Then we turn to metric theorems and the famous 3/2 problem. The sequences of fractional parts determined by the so-called P.V. numbers and outputs of integer dynamical systems generate other types of asymptotic distributions. Relation with representation of real numbers in system with rational basis will be described and discussed.

18 October 2011

Josef Dorfmeister, NDSU: Relative Invariants in Symplectic Topology (Part III)

11 October 2011

Josef Dorfmeister, NDSU: Relative Invariants in Symplectic Topology (Part II)

27 September 2011

Josef Dorfmeister, NDSU: Relative Invariants in Symplectic Topology (Part I)

Abstract: A very elementary question in symplectic topology is the following:  What "different types" of symplectic manifolds exist? And can these be ordered in some fashion?  One method of answering these (and related) questions is to construct examples of symplectic manifolds and then compare them.  In order to distinguish different classes, differential-topological invariants (some of which turn out to be symplectic invariants) were developed. This was started by Donaldson and was given a massive push by the seminal work of Gromov, who recognized the importance of J-holomorphic curves.  Such curves have now been used to define a number of invariants and these curves are closely related to symplectic submanifolds.

In this talk I would like to give a brief introduction to the field of symplectic topology and then describe the standard method of defining differential topological invariants.

20 September 2011

Souvik Bhattacharya, NDSU: Size structured predator-prey model (Part III)

13 September 2011

Souvik Bhattacharya, NDSU: Size structured predator-prey model (Part II)

6 September 2011

Souvik Bhattacharya, NDSU: Size structured predator-prey model

Abstract: Predator-prey systems have been a matter of interest to mathematical biologists for a long time. In the talk, we will discuss a model where dependence of predation on the size of the prey population have been considered. Predators do not indiscriminately choose their prey but select them by size. The talk is focused on a partial differential equation model where we have investigated the conditions when the total number of prey and predator population will die after certain number of years, when only the prey population will persist after a long time and when they both will co-exist in nature. In the coexistence case, we have observed that under certain conditions both of them will have a fixed value in the long run and also there may be situations when the total number of prey and predator will increase and decrease in an oscillatory manner.


Spring 2011

Location: Minard 304A (Seminar Room)
Time: Tuesday, 11:00-11:50 am

3 May 2011

Doğan Çömez, NDSU: Adic Dynamical Systems and the Pascal-adic (Part II)

26 April 2011

Doğan Çömez, NDSU: Adic Dynamical Systems and the Pascal-adic

Abstract: One of the most effective methods of construction in dynamical systems is- so called- the "cutting and stacking method". In early 1980's A.Vershik introduced adic systems as a tool to describe the maps of the unit interval as symbolic dynamical systems. It turns out that adics have an additional feature of providing a nice combinatorial way to describe maps constructed by cutting and stacking. Adic maps are defined on the sets of infinite paths of the ordered Bratelli diagrams. During the talk(s) first I will outline basics of the cutting and stacking method and provide some interesting (and simple) examples. The adic counterparts of these examples will provide the set-up for the basic properties of the Pascal-adic.

5 April 2011

Ben Duncan, NDSU: Graph algebras (Part III)

29 March 2011

Ben Duncan, NDSU: Graph algebras (Part II)

22 March 2011

Ben Duncan, NDSU: Graph algebras

Abstract: We will introduce and discuss the fundamental basics of the theory of graph C*-algebras. We will also discuss related non-selfadjoint 
operator algebras.
22 February 2011

Azer Akhmedov, NDSU: On Large Scale Geometry of Groups (Part II)

15 February 2011

Azer Akhmedov, NDSU: On Large Scale Geometry of Groups


Fall 2010

Location: Minard 304A (Seminar Room)
Time: Tuesday, 11:00-11:50 am

16 November 2010

Azer Akhmedov, NDSU: On some applications of ergodic theory in number theory (Part III)

9 November 2010

Azer Akhmedov, NDSU: On some applications of ergodic theory in number theory (Part II)

2 November 2010

Azer Akhmedov, NDSU: On some applications of ergodic theory in number theory (Part I)

Abstract: Ergodic theoretic approach has proven to be fruitful in solving several difficult problems from number theory, especially in the additive number theory and in Diophantine approximation. I'll discuss the major ergodic theoretic ingredients of these approaches and, in some easier cases, present complete solutions. This talk is aimed at a general audience.

26 October 2010

Friedrich Littmann, NDSU: Bandlimited approximations to integral transforms (Part II)

Abstract: Click here

19 October 2010

Friedrich Littmann, NDSU: Bandlimited approximations to integral transforms (Part I)

Abstract: Click here

28 September 2010

Rob Hladky, NDSU: Introduction to sub-Riemannian Geometry (Part IV)

21 September 2010

Rob Hladky, NDSU: Introduction to sub-Riemannian Geometry (Part III)

14 September 2010

Rob Hladky, NDSU: Introduction to Riemannian Geometry (Part II)

7 September 2010

Rob Hladky, NDSU: Introduction to Riemannian Geometry


Spring 2010

Location: Minard 304A (Seminar Room)
Time: Wednesday, 4:00-4:50 pm

5 May 2010

Jim Olsen, NDSU: Kakutani-Rohlin tower constructions and their use in transferring maximal inequalities (Part II)

28 April 2010

Jim Olsen, NDSU: Kakutani-Rohlin tower constructions and their use in transferring maximal inequalities

21 April 2010

Mariangel Alfonseca, NDSU: Regularity and convexity properties of intersection bodies of revolution (Part II)

Abstract: Today we will show the role of intersection bodies in the solution of the Busemann-Petty problem. Then we will derive geometric and regularity properties of intersection bodies of revolution and direct sums of bodies, using the Radon and the Fourier transforms.

7 April 2010

Mariangel Alfonseca, NDSU: Regularity and convexity properties of intersection bodies of revolution

Abstract: In the first talk, we will define intersection bodies and talk about their history and basic properties. This talk will be accessible to graduate students. In the following talk we will study more in-depth the geometric and regularity properties of some intersection bodies using the Radon and the Fourier transforms.

31 March 2010

Doğan Çömez, NDSU: Sequences that are universally good for the ergodic Hilbert transform in some classes of dynamical systems (Part III)

24 March 2010

Doğan Çömez, NDSU: Sequences that are universally good for the ergodic Hilbert transform in some classes of dynamical systems (Part II)

3 March 2010

Doğan Çömez, NDSU: Sequences that are universally good for the ergodic Hilbert transform in some classes of dynamical systems

24 February 2010

Riley Casper, NDSU: The Spectral Theorem for Normal Operators on a Hilbert Space (Part II)

A current version of the notes for the talk is available on Riley's website

17 February 2010

Riley Casper, NDSU: The Spectral Theorem for Normal Operators on a Hilbert Space

A current version of the notes for the talk is available on Riley's website

10 February 2010

Azer Akhmedov, NDSU: On discrete subgroups of Diff(S1) (Part III)

3 February 2010

Azer Akhmedov, NDSU: On discrete subgroups of Diff(S1) (Part II)

27 January 2010

Azer Akhmedov, NDSU: On discrete subgroups of Diff(S1)

Abstract: The study of discrete subgroups of Lie groups has started in the works of Klein, Poincare, and has experienced enormous growth in recent decades due to Selberg, Mostow, Borel, Margulis, and many others. The study of discrete subgroups of the infinite-dimensional group Diff(S1) - the group of diffeomorphisms of the circle - however, is a very fresh subject, and despite some remarkable isolated results, many natural/elementary questions still remain unanswered. The later development of the subject will reveal how much the properties of these subgroups resemble the properties of discrete subgroups of Lie groups. In the first talk I'll provide some background mostly aimed at graduate students. Then I'll discuss some recent results of analytic flavor.


Fall 2009

Location: Minard 304A (Seminar Room)
Time: Wednesday, 4:00-4:50 pm

2 December 2009

Friedrich Littmann, NDSU: One-sided approximation (Part II)
 

25 November 2009

Friedrich Littmann, NDSU: One-sided approximation

Abstract: In this talk some questions concerning existence and uniqueness of best one-sided approximation by entire functions of finite exponential type will be discussed. We give an old example where uniqueness fails (certain characteristic functions of an interval), and discuss cases where the extremal function exists and is unique. We will compare this to the situation of L^1 (and L^p) -approximation (without the 1-sided condition)

28 October 2009

Cristina Popovici, NDSU: A decomposition result for sequences of gradients

Abstract: We will discuss a decomposition result for sequences of gradients of Sobolev functions which plays an important role in the proofs of a number of key results in the Calculus of Variations, including the lower semicontinuity result of Acerbi and Fusco, Kinderlehrer and Pedregal's characterization of gradient Young measures, and various relaxation results for nonconvex integrands. The proof uses L^p estimates for maximal functions, Lipschitz extensions of Sobolev functions, and Young measures.

21 October 2009

Marian Bocea, NDSU:  An introduction to Young measures (Part III)


14 October 2009

Marian Bocea, NDSU:  An introduction to Young measures (Part II)


7 October 2009

Marian Bocea, NDSU:  An introduction to Young measures

Abstract: The notoriously poor behavior of weak convergence with respect to nonlinear operations is a source of many difficulties in Nonlinear Analysis. Originally introduced by L.C. Young to study nonconvex problems in optimal control theory, Young measures (or parametrized probability measures) have been efficiently used in recent years to understand certain oscillatory phenomena in a more general Calculus of Variations and PDE framework. I will give an introduction to this concept outlining its main properties as well as some of its drawbacks.

23 September 2009

Rob Hladky, NDSU: CR manifolds and the tangential Cauchy-Riemann equations (Part III)


16 September 2009

Rob Hladky, NDSU: CR manifolds and the tangential Cauchy-Riemann equations (Part II)


9 September 2009

Rob Hladky, NDSU: CR manifolds and the tangential Cauchy-Riemann equations 

Abstract: The study of CR manifolds lies in the intersection of subRiemannian geometry and several complex variables. We'll look from a geometric perspective at what it means for a manifold to carry a complex structure and see how much of this complex structure survives when instead you look at submanifolds. We shall then define CR manifolds as a natural abstraction of this concept and look at the equivalent notions of holomorphic functions and forms; solutions to the tangential Cauchy-Riemann equations. Next we study the Kohn Laplacian, a natural analogue of the standard Laplacian, and see how the classical elliptic theory can be modified to study this sub-elliptic Laplacian and the tangential CR-equations.


Spring 2009

Location: Minard 304A (Seminar Room)
Time: Tuesday, 2:30-3:30 pm
 

5 May 2009

Brian Chapman, NDSU: Measure Theoretical and Topological Entropy (Part II)
 

28 April 2009

Brian Chapman, NDSU: Measure Theoretical and Topological Entropy

Abstract: A goal in the study of dynamical systems is to determine whether or not two systems are conjugate--that is, if they are essentially the "same." I will introduce the concept of entropy in both measure preserving and topological dynamical systems, as it plays an important role in determining conjugacy. I will calculate the entropy of some simple dynamical systems and give an example of how entropy can be used to show the conjugacy of two systems (or lack thereof).

14 April 2009

Jim Olsen, NDSU (Part V)
 
Abstract: We'll discuss the Return Times Theorem of Bourgain(and Furstenburg, Katznelson and Ornstein) along with a connection to the weighted ergodic theorem with Besicovitch weights.

7 April 2009

Jim Olsen, NDSU (Part IV)
 
Abstract: I intend to review the material briefly already discussed since it's been a while. Specifically, I'll discuss the weighted ergodic theorem or Dunford-Schwartz operators and Besicovitch weights as well as the relevant facts about Besicovitch sequences. Then the Return Times Theorem of Bourgain(and Furstenburg, katznelson and Ornstein) will be discussed along with a connection to the weighted ergodic theorem with Besicovitch weights.

3 March 2009

Jim Olsen, NDSU (Part III)
 

24 February 2009

Jim Olsen, NDSU (Part II)
 

17 February 2009

Jim Olsen, NDSU 

Abstract: We will discuss the basics of Besicovitch Almost Periodicity and their applications to weighted ergodic theorems for some general operators, which will also be described and discussed. Eventually, I would like to give a connection with the return times theorem.

3 February 2009

Doğan Çömez, NDSU: Structure of some sequence classes good for the modulated ergodic Hilbert transform (Part II)
 

27 January 2009

Doğan Çömez, NDSU: Structure of some sequence classes good for the modulated ergodic Hilbert transform 

Abstract: The class W_{alpha), 0<alpha <1, of (complex) sequences which have finite rated average will be studied. This class contains the class of sequences of Fourier coefficients of functions belonging to L_p [0,2pi], and some other sequence classes which are known to be universally good for the modulated ergodic Hilbert transform.