Mathematics

Kohn's algorithm for subelliptic multipliers

Martino Fassina

Abstract: We will discuss some recent work of Kim-Zaitsev on effectiveness for Kohn's algorithm.

Strong conceptual completeness for $\mathcal L_{\omega_1\omega}$

Ronnie Chen (Caltech)

Abstract: A strong conceptual completeness (SCC) theorem for a logic allows the syntax of a theory to be canonically recovered from its space of models equipped with suitable structure. SCC theorems are known for finitary first-order logic (Makkai, who introduced the name SCC) and fragments thereof (Gabriel–Ulmer, Lawvere, and others). In this talk, I will present a SCC theorem for $\mathcal L_{\omega_1\omega}$: a countable $\mathcal L_{\omega_1\omega}$-theory can be recovered from its standard Borel groupoid of countable models. As a consequence, we obtain a generalization of a recent result of Harrison-Trainor, Miller, and Montalbán.

The Effectively Linear Behavior of the Nonlinear Schr\"odinger Equation

Katelyn Leisman (Illinois Math)

Abstract: The linear part of the Nonlinear Schr\"odinger Equation (NLS) ($iq_t=q_{xx}$) has dispersion relation $\omega=k^2$. We don't expect solutions to the fully nonlinear equation to behave nicely or have any kind of effective dispersion relation like this. However, I have seen that solutions to the NLS are actually weakly coupled and are often nearly sinusoidal in time with a dominant frequency, often behaving similarly to modulated plane waves. In fact, these highly nonlinear solutions eventually end up behaving more and more linearly.

Integro-differential equations arising in free boundary problems

Nestor Guillen (University of Massachusetts at Amherst )

Abstract: Many evolving interface models involve the solution of an elliptic equation in a region enclosed by the interface, the solution being zero along said interface, and its velocity in turn determined from the gradient of the solution. this talk, we will discuss how such a free boundary problem can be recast as a parabolic integro-differential equation, how weak solutions of the latter correspond to weak solutions of the former, and the implications this observation has when studying smoothness and singularities of the free boundary.

Singularities of semisimple Hessenberg varieties.

Erik Insko (Florida Gulf Coast University)

Abstract: Semisimple Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. Like Schubert varieties, the structure of semisimple Hessenberg varieties can be studied using the combinatorics of the symmetric group. In this talk, we will define these varieties and give a combinatorial criterion for identifying singular points in certain semisimple Hessenberg varieties. This is based on joint work with Martha Precup. At 3-4pm in Algeld 245 there will be an IGL/ICLUE seminar by Insko

Valuation of Large Variable Annuity Portfolios: Challenges and Potential Solutions

Guojin Gan (Department of Mathematics, University of Connecticut)

Abstract: In the past two decades, lots of variable annuity contracts have been sold by insurance companies. Insurers with large blocks of variable annuity business have faced great challenges especially when it comes to valuing the complex guarantees embedded in these products. The financial risks associated with guarantees embedded in variable annuities cannot be adequately addressed by traditional actuarial approaches. In practice, dynamic hedging is usually adopted by insurers and the hedging is done on the whole portfolio of VA contracts. Since the guarantees embedded in VA contracts sold by insurance companies are complex, insurers resort to Monte Carlo simulation to calculate the Greeks required by dynamic hedging but this method is extremely time-consuming when applied to a large portfolio of VA contracts. In this talk, I will talk about two major computational problems associated with dynamic hedging and present some potential solutions based on statistical learning to address these computational problems.

Baire measurable colorings of group actions

Anton Bernshteyn (Illinois Math)

Abstract: A typical combinatorial problem is that of coloring, i.e., assigning a "color" to each element of a given structure in a way that fulfills a specified set of constraints. For instance, one might want to color the vertices of a graph so that adjacent vertices receive different colors. Standard compactness arguments usually reduce the general situation to the case when the underlying structure is finite. However, as compactness is inherently dependent upon the Axiom of Choice, this approach is "non-constructive" and the colorings obtained in this way can be quite "pathological," for example, non-measurable. Nonetheless, maybe not all is lost: It is conceivable that the existence of a "well-behaved" coloring is equivalent to a stronger assertion in the finite case, perhaps that a finite coloring can be found via an algorithm of a certain form. In this talk I will describe a result that confirms this suspicion when the underlying notion of "well-behavedness" is Baire measurability and the structure that we wish to color is induced by the shift action of a countable group.

Bounds on the roots of peak and descent polynomials

Erik Insko (Florida Gulf Coast University)

Abstract: In 2012, Billey, Burdzy, and Sagan showed that given a positive integer $n$ and a subset $S \subset \{1,2, \ldots, n\}$ the number of permutations of length $n$ with peak set $S$ is $2^{n-|S|-1}p_S(n)$, where $p_S$ is a polynomial (now called the peak polynomial corresponding to $S$). In 2014, Billey, Fahrbach, and Talmage conjectured that the complex roots of peak polynomials of degree $m-1$ all lie within a circle of radius $m$, and they have real parts greater than $-3$. In this talk I will define descent polynomials, share a conjecture that states the roots of descent polynomials of degree $m-1$ satisfy the same bounds as those identified by Billey, Fahrbach, and Talmage, and discuss a number of partial results in support of these two conjectures. This is based on joint work with Alexander Diaz-Lopez, Pamela E. Harris. Mohamed Omar, and Bruce Sagan. In the process of explaining these conjectures, I will also share some experiences from my time working with undergraduate students on research at FGCU and organizing the Underrepresented Students in Topology and Algebra Research Symposium (USTARS).