The Mathematics Department holds regular seminars on a variety of topics. Please see below for further details.
Seminars
Seminar  Meeting Details  Title & Abstract 

Algebra Seminar  Characterization of Cofree Representations of SL_n\times SL_m Given a finite dimensional representation \(V/k\) of a group \(G\), we consider the space \(k[V]^G\) of all polynomial functions which are invariant under the action of \(G\). At its heart, invariant theory is the study of \(k[V]^G\) and its interactions with \(k[V]\). We are particularly interested in the situation where \(k[V]\) is free as a \(k[V]^G\)module, and call such representations cofree. The classification of cofree representations is a motivating problem for a field of research that has been active for over 70 years. In the case when \(G\) is finite, the ChevalleyShephardTodd theorem says that \(V\) is cofree iff \(G\) is generated by pseudoreflections. Several classifications of cofree representations have been found for certain connected reductive groups, but unlike the ChevalleyShepardTodd theorem, these results consist of a list of cofree representations, rather than a general grouptheoretic characterization. In 2020, D.~Edidin, M.~Satriano, and S.~Whitehead stated a conjecture which intrinsically characterizes irreducible cofree representations of connected semisimple groups and verified it for simple Lie groups and tori. In this talk, we discuss this conjecture and the work towards verifying it for \({\rm SL}_n\times{\rm SL}_m\). Speaker: Nicole Kitt, University of Waterloo 

Geometry and Topology Seminar  The shape of things to come: Topological Data Analysis and biology, from molecules to organisms Shape is data and data is shape. Biologists are accustomed to thinking about how the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. Less often do we consider that data itself has shape and structure, or that it is possible to measure the shape of data and analyze it. Here, we review applications of Topological Data Analysis (TDA) to biology in a way accessible to biologists and applied mathematicians alike. TDA uses principles from algebraic topology to comprehensively measure shape in datasets. Using a function that relates the similarity of data points to each other, we can monitor the evolution of topological features—connected components, loops, and voids. This evolution, a topological signature, concisely summarizes large, complex datasets. We first provide a TDA primer for biologists before exploring the use of TDA across biological subdisciplines, spanning structural biology, molecular biology, evolution, and development. We end by comparing and contrasting different TDA approaches and the potential for their use in biology. The vision of TDA, that data is shape and shape is data, will be relevant as biology transitions into a datadriven era where meaningful interpretation of large datasets is a limiting factor. Speaker: Erik Amezquita Morataya (University of Missouri) 

Algebra Seminar  TThe growth recurrence and GelfandKirillov base for the cusp In this talk, I will discuss some joint work with Alan Dills on concepts devised to describe the size of the Frobenius skewpolynomial ring over a commutative graded algebra over a field in prime characteristic. The ideas are inspired from GelfandKirillov dimension theory. I will discuss what these notions are for the cusp and how to compute them. Speaker: Florian Enescu, Georgia State University 

Analysis Seminar  Exponential bases/frames on unbounded domains and Vandermonde matrices An exponential basis on a measurable domain of $\Bbb{R}^d$ is a Riesz basis in the form of Speaker: Oleg Asipchuk (Florida International University) 

Algebra Seminar  hfunction of local rings of characteristic p For a Noetherian local ring R of characteristic p, we will study a multiplicitylike object called hfunction. It is a function of a real variable s that estimates the asymptotic behavior of the sum of ordinary power and Frobenius power. The hfunction of a local ring can be viewed as a mixture of the HilbertSamuel multiplicity and the HilbertKunz multiplicity. In this talk, we will prove the existence of hfunction and the properties of hfunction, including convexity, differentiability and additivity. If time permits, I will also mention how hfunction recovers other invariants in characteristic p. Speaker: Cheng Meng, Purdue University 

Differential Equations Seminar  The NussbaumSzkola distributions and their use We will review the use of NussbaumSzkola distributions in quantum information and in particular in computing quantum divergences. The talk will be based on joint works with T.C. John https://arxiv.org/abs/2308.02929, https://arxiv.org/abs/2203.01964, https://arxiv.org/abs/2303.03380. Speaker: George Androulakis (University of South Carolina) 

Geometry and Topology Seminar  Sphere's theorem on Warped product submanifolds Speaker: Jaewon Lee (Gyeongsang National University) 

Differential Equations Seminar  Transmission of fast solitons for the NLS with an external potential We consider the dynamics of a boosted soliton evolving under the cubic NLS with an external potential. We show that for sufficiently large velocities, the soliton is effectively transmitted through the potential. This result extends work of Holmer, Marzuola, and Zworski, who considered the case of a delta potential with no bound states in their 2007 paper “Fast soliton scattering by delta impurities,” and the work of Datchev and Holmer, who considered the case of the delta potential with a linear bound state in their 2009 paper “Fast soliton scattering by attractive delta impurities.” This is joint work with Jason Murphy. Speaker: Christopher Hogan (MS&T) 

Analysis Seminar  Questions related to Ulam's floating body problem and to centroid bodies Abstract: Croft, Falconer and Guy posed a series of questions generalizing Ulam's floating body problem, as follows. Given a convex body K in R^3, we consider its plane sections with certain given properties,
(V): Plane sections which cut off a given constant volume
(I) Plane sections which have equal constant principal moments of inertia
Ulam's floating body problem is equivalent to problem (V,I): If all plane sections of the body K which cut off equal volumes have equal constant moments of inertial, must K be an Euclidean ball?
We give a negative answer to problem (V,A) following Ryabogin's counterexample to Ulam's floating body problem. We also give a positive answer to problem (A,I) in the class of bodies of revolution.
In addition, we prove several local fixed point results for the centroid body (the surface of buoyancy associated to Ulam's floating body problem when the density of K is 1/2).
This is joint work with Gulnar Aghabalayeva and Chase Reuter. Speaker: Maria Alfonseca (North Dakota State University) 

Algebra Seminar  Wilf’s Conjecture and More (and Less) Wilf’s conjecture establishes an inequality that relates three fundamental invariants of a numerical semigroup: the minimal number of generators (or the embedding dimension), the Frobenius number, and the number of gaps. Based on a preprint by Srinivasan and S, the talk will discuss the past, present, and future of this conjecture. We prove that this Wilf inequality is preserved under gluing of numerical semigroups. If the numerical semigroups minimally generated by \(A = \{ a_1, \ldots, a_p\}\) and \(B = \{ b_1, \ldots, b_q\}\) satisfy the Wilf inequality, then so does their gluing which is minimally generated by \(C =k_1A\sqcup k_2B\). We discuss the extended Wilf's Conjecture in higher dimensions and prove an analogous result. Speaker: Srishti Singh, University of Missouri 