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.

In this talk, we discuss anti-plane shear deformations on a semi-infinite slab with a non-linear mixed traction displacement boundary condition. We apply global bifurcation theoretic methods and deduce extreme behavior at the terminal end solution curves. It is shown that arbitrarily large strains are encountered for a class of idealized materials. We also consider degenerate materials, prove that ellipticity breaks down, and show that a concurrent blow-up in the second derivative occurs.

The analytic spread of a module M is the minimal number of generators of a submodule that has the same integral closure as M. In this talk, we will present a result that expresses the analytic spread of a decomposable module in terms of the analytic spread of its component ideals. In the second part of the talk, we will show an upper bound for the symbolic analytic spread of ideals of small dimension. The latter notion is the analogue of analytic spread for symbolic powers. These results are joint work with Carles Bivià-Ausina and Hailong Dao, respectively.

The G-transform to a data series is the extension of the Fourier transform from the unit circle to the entire complex plane.I shall introduce the Padé approximant to the G-transform and discuss some of its properties as regard its poles, zeros, and the residues. In particular, I’ll show examples of superresolution with respect to the Nyquist limit, numerical evidence of universality for the behavior of poles and zeros associated with noise and how the presence of signals alters that behavior. I’ll conclude showing a couple of applications. In particular, work in progress on brain waves.