Department of

# Mathematics

Seminar Calendar
for events the day of Friday, September 20, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, September 20, 2019

2:00 pm in 147 Altgeld Hall,Friday, September 20, 2019

#### Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carathéodory theory

###### Meredith Sargent (University of Arkansas)

Abstract: Let $f: D \rightarrow \Omega$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f(z)$ to the boundary of $\Omega$ and the distance of $z$ to the boundary of $D.$ A classical Julia-Carathéodory type theorem states that if there is a sequence tending to $\tau$ in the boundary of $D$ along which the Julia quotient is bounded, then the function $f$ can be extended to $\tau$ such that $f$ is nontangentially continuous and differentiable at $\tau$ and $f(\tau)$ is in the boundary of $\Omega.$ We develop an extended theory when $D$ and $\Omega$ are taken to be the upper half plane which corresponds to amortized boundedness of the Julia quotient on sets of controlled tangential approach, so-called $\lambda$-Stolz regions, and higher order regularity, including but not limited to higher order differentiability, which we measure using $\gamma$-regularity. I will discuss the proof, along with some applications, including moment theory and the fractional Laplacian. This is joint work with J.E. Pascoe and Ryan Tully-Doyle.

2:00 pm in 347 Altgeld Hall,Friday, September 20, 2019

#### Hopf Ore Extensions

###### Hongdi Huang (University of Waterloo (visiting UIUC for F19))

Abstract: Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure that is compatible with that of $R$. In fact, they gave a complete characterization of which $\sigma$ and $\delta$ can occur under the hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a, b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$, with $\beta$ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when $R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains $R$ that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.

3:00 pm in 341 Altgeld Hall,Friday, September 20, 2019

#### Quasiconformal and Quasisymmetric maps on metric spaces

###### Stathis Chrontsios (UIUC Math)

Abstract: The talk will be a quick introduction to quasiconformal and quasisymmetric maps on metric spaces. I will start by describing how quasiconformal maps first appeared as generalizations of conformal maps on the complex plane and how they were generalized in arbitrary metric spaces. In addition, I will present how they gave rise to quasisymmetric maps on the real line and their later generalization in metric spaces. Moreover, I will discuss interesting quasisymmetric invariants and the definition of the conformal gauge. Last but not least, I will mention some applications this theory has had in Geometric Group Theory and some open problems.

3:00 pm in 1 Illini Hall,Friday, September 20, 2019

#### The prime number theorem through the Ingham-Karamata Tauberian theorem

###### Gregory Debruyne (Illinois Math)

Abstract: It is well-known that the prime number theorem can be deduced from certain Tauberian theorems. In this talk, we shall present a Tauberian approach that is perhaps not that well-known through the Ingham-Karamata theorem. Moreover, we will give a recently discovered "simple" proof of a so-called one-sided version of this theorem. We will also discuss some recent developments related to the Ingham-Karamata theorem. The talk is based on work in collaboration with Jasson Vindas.

4:00 pm in 345 Altgeld Hall,Friday, September 20, 2019

#### A Logician's Introduction to the Problem of P vs. NP

###### Alexi Block Gorman (UIUC Math)

Abstract: Central to much of computer science, and some areas of mathematics, are questions about various problems' computability and complexity (whether the problem can be solved "algorithmically," and how "hard" it is to do so). In this talk, I will first give an overview of the complexity hierarchy for machines (from finite automata to Turing machines) and the mathematical properties of the space of languages that we associate with them. Next, I will discuss the relationship of deterministic and non-deterministic machines, which will allow us to segue from questions of computability to that of complexity. Finally, I will give a precise formulation of the problem of P vs. NP, and try to illustrate why the problem remains rather elusive. This talk does not require any background in logic or computer science, and should be accessible to all graduate students.

4:00 pm in 347 Altgeld Hall,Friday, September 20, 2019

#### Classifying Homomorphisms from the Braid and Symmetric Groups

###### Alice Chudnovsky (UIUC Math)

Abstract: One goal of group theory is to understand all possible homomorphisms between two groups. My group characterized maps from the symmetric group and braid group to solvable groups, abelian groups, dihedral groups, free groups, and other groups. This process was facilitated by the concept of Totally Symmetric Sets (TSS), a “basis” of sorts for a group, developed by Margalit, Kordek and Chen. TSS induces a correspondence: given a group homomorphism φ: G → H and a totally symmetric set S of size n in G, φ(S) is either a totally symmetric set of size n in H or of size 1. By determining two groups’ totally symmetric sets, we were able to prove for many cases that all possible homomorphisms factor through a cyclic group. Moreover, for those groups, we drew conclusions on the structure of their totally symmetric sets, and gave upper bounds for their possible size.

4:00 pm in 141 Altgeld Hall,Friday, September 20, 2019

#### A Geometric Proof of Lie's Third Theorem

###### Shuyu Xiao (UIUC)

Abstract: There are three basic results in Lie theory known as Lie's three theorems. These theorems together tell us that: up to isomorphism, there is a one-to-one correspondence between finite-dimensional Lie algebras and simply connected Lie groups. While the first two theorems are easy to prove with the most basic differential geometry knowledge, the third one is somehow a deeper result which needs relatively advanced tools. In this talk, I will go over the proof given by Van Est, in which he identifies any finite-dimensional Lie algebra with a semi-direct product of its center and its adjoint Lie algebra. I will introduce Lie group cohomology, Lie algebra cohomology and how they classify the abelian extensions of Lie groups and Lie algebras and thus determine the Lie algebra structure on the semi-direct product.