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.

A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid — think of it as a spinning bubble of air in water. In this talk, I will describe a general method for desingularizing non-degenerate translating, rotating, or stationary point vortex configurations into collections of steady hollow vortices. Through global bifurcation theory, moreover, these families can be extended to maximal curves of solutions that continue until the onset of a singularity. As specific applications, this machinery gives the first existence theory for co-rotating hollow vortex pairs and stationary hollow vortex tripoles, as well as a new construction of Pocklington’s classical co-translating hollow vortex pairs. All of these families extend into the non-perturbative regime, and we obtain a rather complete characterization of the limiting behavior along the global bifurcation curve.

This is joint work with Ming Chen (University of Pittsburgh) and Miles H. Wheeler (University of Bath).

While the research on water waves modeled by Euler's equations has a long history, mainly in the last two decades traveling periodic rotational waves have been constructed rigorously by means of bifurcation theorems. After introducing the problem, I will present a new reformulation in two dimensions in the pure-gravity case, where the problem is equivalently cast into the form “identity plus compact”, which is amenable to Rabinowitz's global bifurcation theorem. The main advantages (and the novelty) of this new reformulation are that no simplifying restrictions on the geometry of the surface profile and no simplifying assumptions on the vorticity distribution (and thus no assumptions regarding the absence of stagnation points or critical layers) have to be made. Within the scope of this new formulation, global families of solutions, bifurcating from laminar flows with a flat surface, are constructed. Moreover, I will discuss the possible alternatives for the global set of solutions, as well as their nodal properties. This is joint work with Erik Wahlén.

Tuesday 1-2 PM,  Zoom. 

Please see the schedule here.