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 \({\rm SL}_n\times{\rm SL}_m\).

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.

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.

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.

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.

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/2308.02929, https://arxiv.org/abs/2203.01964, https://arxiv.org/abs/2303.03380.

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.

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 

1. Plane sections which have a given constant area

   (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.

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.