Department of

Mathematics


Seminar Calendar
for events the day of Thursday, March 10, 2016.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, March 10, 2016

11:00 am in 241 Altgeld Hall,Thursday, March 10, 2016

On p-adic strengthenings of the Manin-Mumford conjecture

Vlad Serban (Northwestern University)

Abstract: Let $G$ be an abelian variety or a product of multiplicative groups $\mathbb{G}_m^n$ and let $C$ be an embedded curve. The Manin-Mumford conjecture (a theorem by work of Lang, Raynaud et al.) states that only finitely many torsion points of $G$ can lie on $C$ unless $C$ is in fact a subgroup of $G$. I will show how these purely algebraic statements extend to suitable analytic functions on open $p$-adic unit poly-disks. These disks occur naturally as weight spaces parametrizing families of $p$-adic automorphic forms for $GL(2)$ over a number field $F$. When $F=\mathbb{Q}$, the "Hida families" in question play a crucial role in the study of modular forms. When $F$ is imaginary quadratic, I will explain how our results show that Bianchi modular forms are sparse in these $p$-adic families.

1:00 pm in 345 Altgeld Hall,Thursday, March 10, 2016

A stochastic model of eye lens growth: the case of a mouse

Hrvoje Sikic (Washington U. School of Medicine)

Abstract: A lens focuses light on the retina. It needs to be built with a precise shape and size. We develop a stochastic model relating the rates of cell proliferation and death in various regions of the lens epithelium to deposition of fiber cells and radial lens growth. Epithelial population dynamics were modeled as a branching process with emigration and immigration between proliferative zones. We show that a stochastic engine can produce the smooth and precise growth necessary for lens function.

2:00 pm in Altgeld Hall 241,Thursday, March 10, 2016

Quadratic nonresidues below the Burgess bound

Victor Guo (University of Missouri, Math)

Abstract: For any odd prime number $p$, let $(\cdot|p)$ be the Legendre symbol, and let $n_1(p)< n_2(p)< \cdots $ be the sequence of positive nonresidues modulo $p$, i.e., $(n_k|p)=-1$ for each $k$. In 1957, Burgess showed that the upper bound $n_1(p)\ll_\varepsilon p^{(4\sqrt{e})^{-1}+\varepsilon}$ holds for any fixed $\varepsilon>0$. We prove that the stronger bound $$ n_k(p)\ll p^{(4\sqrt{e})^{-1}}\exp\big(\sqrt{e^{-1}\log p\log\log p}\,\big) $$ holds for all odd primes $p$, where the implied constant is absolute, provided that \[ k\le p^{(8\sqrt{e})^{-1}} \exp\big(\tfrac12\sqrt{e^{-1}\log p\log\log p}-\tfrac12\log\log p\big). \] For fixed $\varepsilon\in(0,\frac{\pi-2}{9\pi-2}]$ we also show that there is a number $c=c(\varepsilon)>0$ such that for all odd primes $p$, there are $\gg_\varepsilon y/(\log y)^\varepsilon$ natural numbers $n\le y$ with $(n|p)=-1$ provided that $$ y\ge p^{(4\sqrt{e})^{-1}}\exp\big(c(\log p)^{1-\varepsilon}\big). $$

2:00 pm in 243 Altgeld Hall,Thursday, March 10, 2016

The rigorous derivation of focusing NLS from quantum many-body evolutions

Xuwen Chen (University of Rochester)

Abstract: The rigorous justification of mean-field type equations (Boltzmann, Vlasov-Poisson, NLS...) from the many-body systems they are supposed to describe is a vast and fundamental subject. In this talk, we talk about recent advances in this area on the derivation of focusing nonlinear Schrodinger equations (NLS) from quantum many-body evolutions in the context of Bose-Einstein condensation, which has been one of the most active areas of contemporary research since the Nobel prize winning experiments. We survey the background and the evolution of the results and techniques in the field during the talk.

3:00 pm in 241 Altgeld Hall,Thursday, March 10, 2016

From Hilbert's theorem on ternary quartics to varieties of minimal degree

Daniel Plaumann (Universitšt Konstanz)

Abstract: Hilbert classified all pairs of numbers (n,2d) such that every real non-negative form of degree 2d in n variables is a sum of squares of forms of degree d. Regarding the proof, the exceptional case of ternary quartics (n,2d)=(3,4) is especially interesting. Recently, Blekherman, Smith, and Velasco extended Hilbert's classification by relating it to the classification of projective varieties of minimal degree. In this talk, we will discuss refinements of their result concerning the minimal number of squares needed and the number of representations up to orthogonal equivalence. (Joint work with Greg Blekherman, Rainer Sinn, and Cynthia Vinzant)

4:00 pm in 245 Altgeld Hall,Thursday, March 10, 2016

Building Natural Lyapunov Functions and stablization by noise

Jonathan Mattingly (Duke University)

Abstract: I will discuss a number of stochastic systems where question of existence of a stochastic steady state (and invariant measure) or the convergence to equilibrium can be reduced to the proving the existence of an appropriate Lyapunov function. This will lead us to consider the questions: How does one build a Natural Lyapunov Function? Can this be done in a systematic way? I will consider a number of illustrative examples including the stabilization by noise of an unstable planer vector field and the convergence to equilibrium of a Hamiltonian oscillator with a singular potential, such as a Lennard-Jones potential. Along the way, I will make connections to stochastic averaging, indeterminacy and hypercoercivity.