Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, March 10, 2015.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 10, 2015

11:00 am in 241 Altgeld Hall,Tuesday, March 10, 2015

#### Asymptotics of Multidimensional Partitions

###### Daniel Hirsbrunner (Penn State)

Abstract: Although MacMahon’s conjecture about the generating function for multidimensional partitions was disproved by Atkin, et al. in 1967, there has been renewed interest in the asymptotic accuracy of this conjecture among physicists since the mid 1990s. Many of the resulting publications are computational in nature, providing very suggestive data. Others make headway in rigorously establishing the asymptotics of the number of multidimensional partitions. The best known result is that $\log p_d(n)$ is asymptotically equivalent to $n^{d/(d+1)}$.

1:00 pm in Altgeld Hall 243,Tuesday, March 10, 2015

#### How can you have dynamics when all you have is a category?

###### James Pascaleff (UIUC Math)

Abstract: One of the main objects in symplectic topology is the Fukaya Category F(M) of a symplectic manifold M, whose objects are Lagrangian submanifolds. It has recently been advocated (most strongly by P. Seidel), that interesting results in symplectic topology can be obtained by considering "continuous symmetries" of F(M). These symmetries are not related to any kind of group action on the manifold M, but are "hidden" in the category itself. The key (if imprecise) phrase is "dynamics of a vector field on the moduli space of objects in F(M)." The linearization of the vector field around a fixed point gives rise to the notion of an equivariant Lagrangian submanifold, and the closed orbits are also of interest for applications. This talk will not assume prior exposure to the Fukaya category.

1:00 pm in 345 Altgeld Hall,Tuesday, March 10, 2015

#### Algebraically closed fields with a generic character, Part 1

###### Tigran Hakobyan (UIUC)

Abstract: We (joint work with Tran M.) study the class $\mathcal{C}$ of two sorted structures $(F,K;\chi)$, where $F$ and $K$ are algebraically closed fields, $K$ has characteristic 0, and $\chi:F\to K$ is a generic multiplicative character which means $\chi$ is injective, multiplication preserving, and takes multiplicatively independent elements to algebraically independent elements over $\mathbb{Q}$. The examples of main interest are when $F=\mathbb{F}_{p}^{ac}$ and $K=\mathbb{Q}^{ac}$. We will discuss the following: $\mathcal{C}$ is an elementary class with a natural axiomatization denoted by $\text{ACFC}$; for $$p$$ prime or $$p=0$$, $\text{ACFC}_p=\text{ACFC}\cup \{\text{char}(F)=p\}$ has a categoricity property, is complete, has relative quantifier elimination, is $\omega$-stable, has all models algebraically bounded. This is the first of the two talks on this subject

1:00 pm in 347 Altgeld Hall,Tuesday, March 10, 2015

#### Random data Cauchy problems for nonlinear Schr\”odinger and wave equations

###### Aynur Bulut (Michigan Ann Arbor)

Abstract: We report on some recent progress on probabilistic well-posedness results for nonlinear Schrodinger and wave equations. Treating randomly chosen initial data which is distributed as a Gaussian process, and which has supercritical regularity in terms of the scaling of the nonlinearity, we obtain local and global well-posedness results, holding almost surely with respect to the the randomization. In particular, these results treat data which is almost surely in the ill-posed regime for the initial value problems, and probabilistic considerations are therefore essential. Tools used in the approach include sharp a priori bounds for the nonlinear evolutions and associated linearizations, algebraic structure arising from the Hamiltonian nature of the problems, and careful analysis of frequency interactions.

2:00 pm in 347 Altgeld Hall,Tuesday, March 10, 2015

#### Fluctuations for polymer models in intermediate disorder

###### Arjun Krishnan (Univ of Utah)

Abstract: Directed polymer models are finite-temperature versions of first- and last-passage percolation on the lattice. In 1+1 dimensions, the free-energy of the directed polymer is conjecturally in the Tracy-Widom universality class at all finite temperatures. Tracy-Widom universality has only been proven for a small class of polymers - the so-called solvable models that include Seppalainen's gamma polymers and the O'Connell-Yor semi-discrete polymer - with special shapes and edge-weight distributions. We present some new fluctuation results towards the universality conjecture for polymers in the intermediate disorder scaling regime. (joint work with Jeremy Quastel).

3:00 pm in 241 Altgeld Hall,Tuesday, March 10, 2015

#### Injective labelings of graphs

###### Jacques Verstraete   [email] (Department of Mathematics, UC San Diego)

Abstract: In this talk I will discuss the problem of coloring the vertices of a graph with $k$ colors such that the neighborhood $N(v)$ contains all $k$ colors for every vertex $v$ in the graph. The problem is to maximize the value of $k$ for which such a coloring is possible. We show that if $G$ is a $d$-regular graph, then the maximum is $k = (1 + o(1))d/log d$ and that almost every d-regular graph required at least $(1 + o(1))d/log d$ colors. This problem has connections to coding theory, and with this in mind we discuss colorings of the $q$-ary Hamming cube graphs of dimension $n$. Some open problems will be given. Joint work with Bob Chen, Jeong Han Kim and Mike Tait

4:00 pm in 165 Everitt,Tuesday, March 10, 2015

#### The Yule-Coalescent Hierarchical Model

###### Dr. James Degnan (University of New Mexico)

Abstract: Evolutionary biologists work with data at multiple levels which can be integrated into a hierarchical model: branching processes to model populations splitting (speciation), genes forming trees of ancestry within these populations, and mutations in genetic lineages. Although much work has been done on each of these levels separately, there is much to understand regarding how the levels interact. By averaging effects at intermediate levels, we can study impacts of higher level processes, such as speciation and extinction rates, on lower levels, such as patterns in gene trees and DNA sequences, and in turn estimate parameters from higher levels using data collected at the lower levels. In this talk I will examine some consequences of the Yule branching process model of speciation on distances between evolutionary trees and measures of tree balance.

4:00 pm in 243 Altgeld Hall,Tuesday, March 10, 2015

#### Smoothing and Dynamics of the Majda-Biello System

###### Erin Compaan (UIUC Math)

Abstract: This talk will discuss smoothing and dynamical properties of the periodic Majda-Biello system, a coupled KdV-type system. I'll begin with some background aimed at a general audience. Then I'll present some recent results concerned with the smoothness of the nonlinear part of the Majda-Biello evolution and long-time dynamics of the system, concluding with a brief look at the proof techniques.