Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, January 30, 2018.

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Tuesday, January 30, 2018

11:00 am in 345 Altgeld Hall,Tuesday, January 30, 2018

Localizing the E_2 page of the Adams spectral sequence

Eva Belmont (MIT)

Abstract: The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. In this talk, I will discuss an approach for computing the Adams E_2 page at p = 3 in an infinite region, by computing its localization by the non-nilpotent element b_{10}. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory. This computation fits in the framework of chromatic homotopy theory in the stable category of comodules.

12:00 pm in 243 Altgeld Hall,Tuesday, January 30, 2018

Counting conjugacy classes of fully irreducibles in $Out(F_r)$

Ilya Kapovich   [email] (Illinois Math)

Abstract: Inspired by results of Eskin and Mirzakhani counting closed geodesics of length $\le L$ in the moduli space of a closed surface $\Sigma_g$ of genus $g\ge 2$, we consider a similar question in the $Out(F_r)$ setting. Let $h=6g-6$. The Eskin-Mirzakhani result, giving the asymptotics of $\frac{e^{hL}}{hL}$, can be equivalently stated in terms of counting the number of $MCG(\Sigma_g)$-conjugacy classes of pseudo-Anosovs $\phi\in MCG(\Sigma_g)$ with dilatation $\lambda(\phi)$ satisfying $\log\lambda(\phi)\le L$. For $L\ge 0$ let $\mathfrak N_r(L)$ denote the number of $Out(F_r)$-conjugacy classes of fully irreducibles $\phi\in Out(F_r)$ with dilatation $\lambda(\phi)$ satisfying $\log\lambda(\phi)\le L$. In a joint result with Catherine Pfaff, we prove for $r\ge 3$ that as $L\to\infty$, the number $\mathfrak N_r(L)$ has double exponential (in $L$) lower and upper bounds. We also obtain a companion result, joint with Michael Hull, and show that of distinct $Out(F_r)$-conjugacy classes of fully irreducibles $\phi$ from an $L$-ball in the Cayley graph of $Out(F_r)$ with $\log\lambda(\phi)$ on the order of $L$ grows exponentially in $L$.

1:00 pm in 345 Altgeld Hall,Tuesday, January 30, 2018

Canceled

2:00 pm in 347 Altgeld Hall,Tuesday, January 30, 2018

​Stochastic (partial) differential equations with singular coefficients

​Longjie Xie (Jiangsu Normal University & UIUC)

Abstract: It is a classical result that the ordinary differential equation is well-posed for Lipschitz coefficient but usually ill-posed if the coefficient is only Holder continuous. However, this dramatically changes if the system is perturbed by a noise. The purpose of this talk is to give a brief introduction and overview of the topic of regularization and well-posedness by noise for ordinary and partial differential equations.

3:00 pm in 241 Altgeld Hall,Tuesday, January 30, 2018

An improved upper bound for the (5,5)-coloring number of K_n

Emily Heath (Illinois Math)

Abstract: A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ in which each $p$-clique contains edges of at least $q$ distinct colors. We denote the minimum number of colors needed for a $(p,q)$-coloring of the complete graph $K_n$ by $f(n,p,q)$. In this talk, we will describe an explicit $(5,5)$-coloring of $K_n$ which proves that $f(n,5,5)\leq n^{1/3+o(1)}$ as $n\rightarrow\infty$, improving the best known probabilistic upper bound of $O(n^{1/2})$ given by Erdős and Gyrfs. This is joint work with Alex Cameron.

4:00 pm in Illini Hall 1,Tuesday, January 30, 2018

Riemann-Roch Formula and The Dimension of Our Universe

Lutian Zhao   [email] (UIUC)

Abstract: In this talk we'll introduce the classical Riemann-Roch formula, which appears as a vast generalization of the Euler-Maclaurin formula for the integrals. As an interesting application, the critical dimension for the bosonic string theory can be calculated by these formula to be d=26, which matches with the physical prediction using light-cone quantization. No basic knowledge on string theory and Riemann-Roch will be assumed.