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

Differential Equations Seminar  Stability of the compacton waves for the degenerate KDV and NLS models This talk is based on the degenerate semilinear Schrödinger and Kortewegde Vries equations in one spatial dimension. We construct variationally special solutions of the two models, that is, standing wave solutions of NLS and traveling waves for KDV, which turn out to have compact support, hence compactons. We show that the compactons are unique bellshaped solutions of the corresponding PDE's and for appropriate variational problems as well. We also provide a complete spectral characterization of such waves, for all values of \(p\). Namely, we show that all waves are spectrally stable for \(2<p\leq 8\), while a single mode instability occurs for \(p>8\). This extends the previous work of Germain, HarropGriffiths and Marzuola, who have previously established orbital stability for some specific waves, in the range \(p<8\). This is a joint work with Atanas stefanov and Sevdhan Hakkaev. Speaker: Abba Ramadan (University of Alabama) 

Analysis Seminar  Estimates for elliptic and parabolic measures for operators satisfying an oscillation condition It has been known for quite some time that various conditions on the oscillation of the matrix $A$ in elliptic/parabolic operators of the form $L = div A \nabla$ or $L = \partial_t  \div A \nabla$ are sufficient to guarantee either the $L^p$ solvability of the Dirichlet problem or the logarithm of the density of the elliptic/parabolic measure, $k$, is in BMO or (locally) Hölder continuous. Conditions such as the global Hölder continuity of $A$ are classical, whereas other conditions quantifying the oscillation of $A$ in terms of Carleson measures are more recent. Only very recently in joint works with Toro and Zhao, and later with Egert and Saari, was it shown that the BMO norm of $\log k$ could be controlled by these Carleson conditions. These new results are elliptic and heavily relied on work of David, Li and Mayboroda.
In forthcoming work with Egert and Saari, we adapt these new results (B., Toro, Zhao and B. Egert, Saari) to the parabolic setting. Therefore we needed to also adapt the work of David, Li and Mayboroda to the parabolic setting and, in doing so, sharpened their results. It seems very likely that these sharper estimates will allow one to treat the classical results (Hölder continuous coefficients) and the modern results (Carleson conditions) with a unified method.
I will delve into the ideas underpinning these results. Speaker: Simon Bortz (University of Alabama) 

Algebra Seminar  Multiplier ideals and klt singularities via (derived) splittings Thanks to the Direct Summand Theorem, splinter conditions have emerged as a way of studying singularities in commutative algebra and algebraic geometry. In characteristic zero, work of Kovács (2000) and Bhatt (2012) characterizes rational singularities as derived splinters. In this talk, I will present an analogous characterization of klt singularities by imposing additional conditions on the derived splinter property. This follows from a new characterization of the multiplier ideal, an object that measures the severity of the singularities of a variety, viewing it as a sum of trace ideals. This perspective also gives rise analogous description of the test ideal in characteristic as a corollary to a result of EpsteinSchwede (2014). Speaker: Peter McDonald, University of Utah 

Geometry and Topology Seminar  The Euler Characteristic Transform, or how a topologist and a plant biologist meet for a beer Shape is foundational to biology. Observing and documenting shape has fueled biological understanding as the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. The vision of Topological Data Analysis (TDA), that data is shape and shape is data, will be relevant as biology transitions into a datadriven era where meaningful interpretation of large data sets is a limiting factor. We focus first on quantifying the morphology of Xray CT scans of barley spikes and seeds using topological descriptors based on the Euler Characteristic Transform. We then successfully train a support vector machine to distinguish and classify 28 different varieties of barley based solely on the 3D shape of their grains. This shape characterization will allow us later to link genotype with phenotype, furthering our understanding on how the physical shape is genetically specified in DNA. Speaker: Erik Amezquita Morataya (University of Missouri) 

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) 