Department of


Seminar Calendar
for events the day of Tuesday, January 24, 2017.

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Tuesday, January 24, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, January 24, 2017

Boundary maps for some hierarchically hyperbolic spaces

Sarah Mousley (Illinois Math)

Abstract: There are natural embeddings of right-angled Artin groups $G$ into the mapping class group $Mod(S)$ of a surface $S$. The groups $G$ and $Mod(S)$ can each be equipped with a geometric structure called a hierarchically hyperbolic space (HHS) structure, and there is a notion of a boundary for such spaces. In this talk, we will explore the following question: does an embedding $\phi: G \rightarrow Mod(S)$ extend continuously to a boundary map $\partial G \rightarrow \partial Mod(S)$? That is, given two sequences $(g_n)$ and $(h_n)$ in $G$ that limit to the same point in $\partial G$, do $(\phi(g_n))$ and $(\phi(h_n))$ limit to the same point in $\partial Mod(S)$? No background in HHS structures is needed.

1:00 pm in 345 Altgeld Hall,Tuesday, January 24, 2017

Will the Nonstandard Analysis become the Analysis of Future?

Evgeny Gordon (Eastern Illinois Math)

Abstract: In 1973 Abraham Robinson gave a talk about the nonstandard analysis (NSA) at the Institute for Advanced Study. After his talk Kurt G\"odel made a comment, in which he predicted that "...there are good reasons to believe that Non-Standard Analysis in some version or other will be the analysis of the future". One has to admit that during the fifty years since this prediction, it did not come true. One of the reasons is that the most part of researchers in NSA considered it as a tool of obtaining new results in standard mathematics, instead of consider it as a more appropriate language, in which the "book of nature is written". Nowadays, the investigation of DE's that simulate processes in science and economy are based on computer (discrete) simulations of these DE's.
  In this talk I will try to justify the point of view that the language of NSA is more appropriate for investigation of the interaction between continuous models and their discrete simulations (or maybe vise versa - between discrete models and their continuous simulation, according to a popular among applied mathematicians point of view). The reason is that not well defined properties ("very big", "very small", "far enough of the boundaries of computer memory", etc.) can be introduced in the language of NSA on the level of rigor of Cantor's Set Theory. I will discuss some NSA theorems in algebra, calculus, ergodic theorem and quantum mechanics) concerning this problem that have intuitively clear sense and agree with computer experiments, while their formulation in the language of standard mathematics looks irrelevant and sometimes even unreadable.

2:00 pm in 241 Altgeld Hall,Tuesday, January 24, 2017

Poincaré sections for the horocycle flow in covers of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) and applications to Farey fraction statistics

Byron Heersink (UIUC)

Abstract: For a given finite index subgroup $H\subseteq$SL(2,$\mathbb{Z}$), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) found by Athreya and Cheung to the finite cover SL(2,$\mathbb{R}$)/$H$ of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$). We then relate the properties of this section to the gaps in Farey fractions and describe how the ergodic properties of the horocycle flow can be used to obtain certain statistical properties of various subsets of Farey fractions.

3:00 pm in 241 Altgeld Hall,Tuesday, January 24, 2017

Many $T$ copies in $H$-free subgraphs of random graphs

Abstract: For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $\chi(H)>m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\max_{H'\subset H, |V(H')|\geq 3}\left\{ \frac{e(H')-1}{v(H')-2} \right\}$. When $m_2(H)>m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can keep almost all copies of $K_m$ in an $H$-free subgraph of $G(n,p)$, or it is asymptotically best to take a $\chi(H)-1$ partite subgraph of $G(n,p)$. The transition between these two behaviors occurs at $p=n^{-1/m_2(H)}$. When $m_2(H)< m_2(K_m)$, the above cases still exist, however for $\delta>0$ small at $p=n^{-1/m_2(H)+\delta}$ one can typically still keep most of the copies of $K_m$. The reason for this is that although $K_m$ has the minimum average degree among the $m$-color-critical graphs, it does not have the smallest $m_2(G)$ among such graphs. This is joint work with N. Alon and C. Shikhelman.

3:00 pm in 243 Altgeld Hall,Tuesday, January 24, 2017

On the Behrend function and its motivic version in Donaldson-Thomas theory

Yungfeng Jiang (U Kansas Math)

Abstract: The Behrend function, introduced by K. Behrend, is a fundamental tool in the study of Donaldson-Thomas invariants. In his foundational paper K. Behrend proves that the weighted Euler characteristic of the Donaldson-Thomas moduli space weighted by the Behrend function is the Donaldson-Thomas invariants defined by R. Thomas using virtual fundamental cycles. This makes the Donaldson-Thomas invariants motivic. In this talk I will talk about the basic notion of the Behrend function and apply it to several other interesting geometries. If time permits, I will also talk about the motivic version of the Behrend function and the famous Joyce-Song formula of the Behrend function identities.

4:00 pm in 245 Altgeld Hall,Tuesday, January 24, 2017

A New Approach for Buffering Portfolio Returns in Investment-Linked Annuities

Daniel Linders (Technical University of Munich)

Abstract: This paper introduces a new class of investment-linked annuity contracts. To reduce payout volatility, we gradually adjust cash flows to portfolio returns. This contrasts with standard investment-linked annuity contracts in which cash flows immediately incorporate portfolio returns. To build a realistic risk-management framework, we consider a general financial market. Our framework allows to use various non-Gaussian distributions which incorporate stylized facts about portfolio returns. Furthermore, we show how to price and hedge the liabilities of our new annuity contract.

4:00 pm in 131 English Building,Tuesday, January 24, 2017

Organizational Meeting

(UIUC Math)

Abstract: A brief meeting to schedule speakers for the semester.