Mathematics

To Be Announced

Ronnie Chen (Caltech)

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.

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.

TBA

Anton Bernshteyn (Illinois Math)

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).