Location and Time: Minard 116 at 3:00 PM (Refreshments at 2:30 in Minard 404)

*Special Colloquia or Tri-College Colloquia venues and times may vary, please consult the individual listing.

Tuesday, September 10        

Artem Novozhilov, NDSU

Teaching History of Mathematics (an admittedly biased perspective)

Abstract:  During the last 5 years I taught Math 478/678 History of Mathematics 3 times (Springs 2020, 22, 24). During this time period my idea of how to teach this course evolved from just a "follow the textbook" approach to a somewhat sophisticated structure with several equally important components. The key idea was to teach not "history of mathematics" but "mathematics and its history", thus focusing much more of the mathematical content of the course. In this talk I would like to share what worked for me in this course and what did not, ultimately looking for the faculty and graduate students input on what should/could be changed, replaced, removed, or added to such course.

Tuesday, September 24     

Jessica Striker, NDSU

Rotation-invariant web bases from hourglass plabic graphs and symmetrized six-vertex configurations

Abstract:  Many combinatorial objects with strikingly good enumerative formulae also have remarkable dynamical behavior and underlying algebraic structure. In this talk, we consider the promotion action on certain rectangular tableaux and explain its small, predictable order by reinterpreting promotion as a rotation in disguise. We show how the search for this visual explanation of combinatorial dynamics led to the solution of an algebraic problem involving web bases and a generalization of the six-vertex model of statistical physics. We find that this framework includes several unexpected combinatorial objects of interest: alternating sign matrices, plane partitions, and their symmetry classes (joint with Ashleigh Adams). This talk is based primarily on joint work with Christian Gaetz, Stephan Pfannerer, Oliver Pechenik, and Joshua P. Swanson.

Tuesday, October 8

*Special Tri-College Colloquium at NDSU

Doug Anderson, Concordia

Title: Hyers-Ulam Stability and the Harmonic Oscillator

Abstract:  Hyers-Ulam stability for differential equations and shadowing for dynamical systems both attempt to measure the error between a perturbation (approximate solution) and an actual solution to a given equation or system. If the error remains bounded the equation is stable and a minimum is sought. In this talk, we analyze the well-known harmonic oscillator, a second-order differential equation with constant and variable coefficients. In the constant coefficients case, the Hyers-Ulam stability depends on the solutions to the corresponding characteristic equation. In the variable coefficients case, a solution to a related Riccati equation is utilized to establish Hyers-Ulam stability and minimal error under certain conditions. Several examples are given, including the damped and undamped cases and the forced case, and application is made to Lane-Emden equations that blow up in finite time.

Tuesday, October 15    

Timothy Ryan, NDSU

Geometry of Hilbert schemes

Abstract:  Given a set of points in the plane, what is the minimal degree of a polynomial curve containing those points? This problem, known as the interpolation problem, dates back to at least Lagrange, whose work has applications to fields as diverse as physics, computer graphics, and data analysis.  Generalizations of the interpolation problem still motivate advances in algebraic geometry today, where their most natural setting is on a space known as the Hilbert scheme of points on the plane.  Hilbert schemes are some of the most classically studied spaces in algebraic geometry and have also proven to be important objects in representation theory, combinatorics, and symplectic geometry. In this talk, I will introduce the relevant concepts from algebraic geometry, formally define Hilbert schemes, and discuss my work on their geometry.

Tuesday, October 22     

Doğan Çömez, NDSU

Blum-Hanson property for non-additive processes

Abstract:  In ergodic theory, characterization of types of measure preserving transformations (ergodic, weak-mixing, mixing, K-authomorphism, etc.) is an important problem. The remarkable Blum-Hanson Theorem provides a characterization of strongly mixing measure preserving systems in terms of the mean ergodic theorem along all subsequences. Ever since it appeared in the literature, various similar characterizations as well as generalizations to other directions were obtained, and among these, N. Friedman proved a uniform version of the Blum-Hansen Theorem. Recently, various authors studied operators or function spaces for which this characterization, termed as Blum-Hanson property, is valid. In this talk we will extend Blum-Hanson property to the superadditive processes setting. In particular, a complete characterization is obtained for the admissible superaditive processes.

Tuesday, October 29    

Salem Selim, U of California, Irvine

Partial data inverse problems for magnetic Schrödinger operators with potentials of low regularity.

Abstract:  In this talk we discuss partial data inverse boundary problems for magnetic Schrödinger operators on bounded domains in the Euclidean space as well as some Riemannian manifolds with boundary. In particular, we show that the knowledge of the set of the Cauchy data on a portion of the boundary of a domain in the Euclidean space of dimension $n\ge 3$ for the magnetic Schrödinger operator with a magnetic potential of class $W^{1,n}\cap L^\infty$, and an electric potential of class $L^n$, determines uniquely the magnetic field as well as the electric potential. Our result is an extension of global uniqueness results of Dos Santos Ferreira--Kenig--Sjöstrand--Uhlmann (2007) and Knudsen--Salo (2007), to the case of less regular electromagnetic potentials. Our approach is based on boundary Carleman estimates for the magnetic Schrödinger operator, regularization arguments, as well as the invertibility of the geodesic X-ray transform. In this talk, we will also show an extension of our uniqueness result to non-admissible manifolds. This talk is based on joint-work with Lili Yan.

Tuesday, November 5   

Lili Yan, U of Minnesota

Inverse Problems for Partial Differential Equations

Abstract:  In an inverse boundary problem, one seeks to determine the coefficients of a PDE inside a domain, describing internal properties, from the knowledge of boundary values of solutions of the PDE, encoding boundary measurements. Applications of such problems range from medical imaging to non-destructive testing. In this talk, starting with the celebrated work on Calder\'on problem, we shall give an introduction to inverse problems for elliptic, hyperbolic, and transport equations, including partial data inverse problems, inverse problems on Riemannian manifolds, as well as inverse problems for nonlinear equations. 

Tuesday, November 12     

Florian Enescu, Georgia State University

Affine semigroups under rational twist

Abstract:  Motivated by the definition of the Frobenius complexity for a local ring of positive characteristic, we examine generating functions that can be associated to the twisted construction of a graded ring of positive characteristic. There is a large class of affine semigroup rings for which these generating functions are rational. We will discuss this class of rings and combinatorial aspects that arise in the study of the rationality of the complexity generating function.  Families of affine semigroups with rational twist will be presented. This work is joint with Yongwei Yao.