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
event
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place
MSB 110
group
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 Epstein-Schwede (2014).

Speaker: Peter McDonald, University of Utah
Geometry and Topology Seminar
event
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place
110 Math Science Building
group
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 data-driven era where meaningful interpretation of large data sets is a limiting factor. We focus first on quantifying the morphology of X-ray 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
event
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place
Room 110
group
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 Chevalley-Shephard-Todd 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 Chevalley-Shepard-Todd theorem, these results consist of a list of cofree representations, rather than a general group-theoretic 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
event
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place
110 Mathematical Science Building
group
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 sub-disciplines, 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 data-driven era where meaningful interpretation of large datasets is a limiting factor.

Speaker: Erik Amezquita Morataya (University of Missouri)
Algebra Seminar
event
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place
Room 110
group
TThe growth recurrence and Gelfand-Kirillov 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 skew-polynomial ring over a commutative graded algebra over a field in prime characteristic. The ideas are inspired from Gelfand-Kirillov 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
event
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place
Math Sci 110
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
$\{ e^{2\pi i \lambda.x} \}_{\lambda\in\Lambda}, $   where $\Lambda$ is a discrete set of $\Bbb{R}^d.$ The problem of proving (or disproving) the existence of such systems on measurable sets is still largely unsolved. For example, the existence of exponential bases on unbounded domains is proved only in very few special cases. Moreover, for most of the domains for which the existence of exponential bases is proved, no explicit expression of such systems is given.

In my talk, I will show explicit examples of exponential bases on finite or infinite unions of intervals. Also, I will describe newly established connections between Vandermonde matrices and exponential bases and prove a stability theorem  for Vandermonde matrices.

Speaker: Oleg Asipchuk (Florida International University)
Algebra Seminar
event
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place
Room 110
group
h-function of local rings of characteristic p

For a Noetherian local ring R of characteristic p, we will study a multiplicity-like object called h-function. It is a function of a real variable s that estimates the asymptotic behavior of the sum of ordinary power and Frobenius power. The h-function of a local ring can be viewed as a mixture of the Hilbert-Samuel multiplicity and the Hilbert-Kunz multiplicity. In this talk, we will prove the existence of h-function and the properties of h-function, including convexity, differentiability and additivity. If time permits, I will also mention how h-function recovers other invariants in characteristic p.

Speaker: Cheng Meng, Purdue University
Differential Equations Seminar
event
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place
Strickland Hall 310
group
The Nussbaum-Szkola distributions and their use

We will review the use of Nussbaum-Szkola 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/2308.02929https://arxiv.org/abs/2203.01964https://arxiv.org/abs/2303.03380.

Speaker: George Androulakis (University of South Carolina)
Geometry and Topology Seminar
event
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place
110 Mathematical Science Building
group
Sphere's theorem on Warped product submanifolds
Speaker: Jaewon Lee (Gyeongsang National University)
Differential Equations Seminar
event
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place
Strickland Hall 310
group
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)