Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, March 6, 2018.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 6, 2018

11:00 am in Psychology Building 21,Tuesday, March 6, 2018

#### The generalized homology of $BU\langle 2k\rangle$

###### Phillip Jedlovec

Abstract: In their 2001 paper, Elliptic spectra, the Witten genus and the theorem of the cube,'' Ando, Hopkins, and Strickland use an algebro-geometric perspective to give a partial description of the generalized homology of the connective covers of BU. For any complex-orientable cohomology theory, E, they define homology elements $b_{i_1, ..., i_k}$ in $E_*BU\langle 2k\rangle$, prove the so called cocycle relations'' and symmetry relations'' on these elements, and show that when $E=H\mathbb{Q}$ or $k=1, 2,$ or $3$, these are in fact the defining relations for $E_*BU\langle 2k\rangle$. In this talk, I will sketch a new proof of these results which uses very little algebraic geometry, but instead uses facts about Hopf rings and the work of Ravenel and Wilson on the homology of the spaces in the $\Omega$-spectrum for Brown-Peterson cohomology. Time permitting, I will also discuss how this approach might be used to prove the Ando-Hopkins-Strickland theorem for $k>3$ and $E=H\mathbb{Z}_{(2)}$.

12:00 pm in Room 214 Ceramics Building,Tuesday, March 6, 2018

#### Stable subgroups and Morse subgroups of mapping class groups

###### Heejoung Kim (Illinois Math)

Abstract: The notion of a quasiconvex'' subgroup plays of a word-hyperbolic group $G$ plays an important role in the theory of hyperbolic groups. This notion has several equivalent characterizations in that context, in terms of being undistorted", in terms of the action on the boundary, in terms of being rational" with respect to automatic structures on $G$, in terms of the contracting properties of the projection maps, etc. For an arbitrary finitely generated group $G$, there are two recent generalizations of the notion of a quasiconvex subgroup: a stable'' subgroup and a Morse'' subgroup. In this talk, we will discuss these two notions and their different properties. We prove that the properties of being Morse and being stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic.

1:00 pm in 113 Speech & Hearing Bldg at Daniel and 6th,Tuesday, March 6, 2018

#### Some aspects of simple proofs informed by history, ancient and modern

###### Victor Pambuccian (Arizona State University Math)

Abstract: We'll look at (i) the existence of a direct proof (even within intuitionistic logic) of the Steiner-Lehmus theorem, stating that a triangle with two congruent angle bisectors must be isosceles, (ii) the most primitive arithmetic capable of proving the irrationality of $\sqrt{2}$, going back to the early Pythagoreans, and why it cannot prove the irrationality of $\sqrt{17}$, the case Theodorus of Cyrene apparently stumbled upon, and (iii) the simplest proof for the fact that 30 is the greatest number all of whose totitives are prime.

2:00 pm in 255 Armory,Tuesday, March 6, 2018

#### Local density estimate for a hypoelliptic SDE

###### Cheng Ouyang (UIC Math)

Abstract: In a series of three papers in the 80’s, Kusuoka and Stroock developed a probabilistic program in order to obtain sharp bounds for the density function of a hypoelliptic SDE driven by a Brownian motion. We aim to investigate how their method can be used to study rough SDEs driven by fractional Brownian motions. In this talk, I will outline Kusuoka and Stroock’s approach and point out where the difficulties are in our current setting. The talk is based on an ongoing project with Xi Geng and Samy Tindel.

3:00 pm in Illini Hall 2,Tuesday, March 6, 2018

#### (normally ordered) Tensor product of Tate objects and decomposition of higher adeles

###### Aron Heleodoro (Northwestern University)

Abstract: In this talk I will introduce the different tensor products that exist on Tate objects over vector spaces (or more generally coherent sheaves on a given scheme). As an application, I will explain how these can be used to describe higher adeles on an n-dimensional smooth scheme. Both Tate objects and higher adeles would be introduced in the talk. (This is based on joint work with Braunling, Groechenig and Wolfson.)

3:00 pm in Talbot 104,Tuesday, March 6, 2018

#### Some important combinatorial sequences

###### Zoltan Furedi (Renyi Institute of Mathematics, Budapest, Hungary and UIUC)

Abstract: The sequence $a(1), a(2), a(3), \dots$ of reals is called subadditive if $a(n+m) \leq a(n)+ a(m)$ for all integers $n,m$. Fekete's lemma states that the sequence $\{a(n)/n\}$ has a limit (it could be negative infinity). Let $f(n)$ be a non-negative, non-decreasing sequence. De Bruijn and Erdos (1952) called the sequence $(a(n))$ nearly $f$-subadditive if $a(n+m) \leq a(n)+ a(m) + f(n+m)$ holds for all $n\leq m \leq 2n$. They showed that if the error term $f$ is small, $\sum_{ n=1}^{\infty} f(n)/n^2$ is finite, then the limit $a(n)/n$ still exists. Their results is listed in the Bollobas-Riordan book (2006) as one of the most useful tools in Percolation Theory. Among other things we show that the de Bruijn-Erdos condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. If $\sum_{ n=1}^{\infty} f(n)/n^2 =\infty$, then there exists an nearly $f$-subadditive sequence $(b(n))$, such that the sequence of slopes $(b(n)/n)$ takes every rational number. This is a joint work with I. Ruzsa.

4:00 pm in 1 Illini Hall,Tuesday, March 6, 2018

#### Introduction to GIT, II

###### Itziar Ochoa de Alaiza Gracia

Abstract: The aim of this talk is to give the motivation for the GIT quotient. We will do so by introducing different notions of quotients, illustrated by examples. Finally we will define the Affine and projective GIT quotients.