Department of

Mathematics


Seminar Calendar
for events the day of Thursday, October 15, 2015.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, October 15, 2015

11:00 am in 241 Altgeld Hall,Thursday, October 15, 2015

Kloosterman sums and Maass cusp forms of half integral weight for the modular group

Nickolas Andersen (UIUC Math)

Abstract: We estimate the sums $\sum_{c\leq x} \frac{S(m,n,c,\chi)}{c},$ where $S(m,n,c,\chi)$ are Kloosterman sums of half-integral weight on the modular group associated to the multiplier system $\chi$ for the Dedekind eta function. Our estimates are uniform in $m,n,$ and $x$ in analogy with Sarnak and Tsimerman's improvement of Kuznetsov's bound for the ordinary Kloosterman sums. We approach the problem via an analogue of Kuznetsov's trace formula; among other things this requires us to develop mean value estimates for coefficients of Maass cusp forms of weight $1/2$ and uniform estimates for $K$-Bessel integral transforms. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error term in Rademacher's formula for the partition function $p(n)$.

12:30 pm in 464 Loomis Laboratory,Thursday, October 15, 2015

The Quantum Geometry of 2d Gauge Theories

Hans Jockers (Bonn Physics)

Abstract: At specific points in the moduli space of certain supersymmetric 2d gauge theories the gauge theory ground states realize Calabi-Yau manifolds. In weakly coupled regimes of the gauge theory the Calabi-Yau geometries appear through standard symplectic reduction, whereas in strongly coupled regions a systematic study of the ground state variety remains challenging. In this talk I present a class of 2d gauge theories, in which the strongly coupled regime can be analyzed with gauge theory dualities. For a particular illustrating example, I propose that the ground state variety at both weak and strong coupling realizes respectively two Calabi-Yau threefolds of the same Euler characteristic that are not birational but derived equivalent to each other. I present evidence in favor of this proposal by matching and analyzing the $S^2$ partition function of dual pairs of gauge theories. Due to the $S^2$ partition function correspondence, these functions encode geometric invariants of the Calabi-Yau manifolds such as their genus zero Gromov-Witten invariants.

1:00 pm in 243 Altgeld Hall,Thursday, October 15, 2015

How to make predictions in topology (resp. arithmetic) using arithmetic (resp. topology)

Benson Farb (University of Chicago)

Abstract: Weil, Grothendieck, Deligne and others built an amazing bridge between topology and arithmetic. In this talk I will describe some recent attempts (some successful, some still conjectural) to add planks to this bridge. I hope to convince the audience that the question: "Why is $1/\zeta(n)$ the same as the 2-fold loop space of $CP^{n-1}$ ?" is not completely crazy. This is joint work with Jesse Wolfson.

2:00 pm in 260 Everitt ,Thursday, October 15, 2015

LÚvy Subordinated Hierarchical Archimedean Copulas (LSHAC): Theory and Empirical Applications

Wenjun Zhu (School of Finance, Nankai University)

Abstract: LÚvy subordinated hierarchical Archimedean copulas (LSHAC) are flexible models in high dimensional multivariate estimation. However, there are limited literature discussing their applications, largely due to the challenges in estimating the structures and parameters. In this talk, we introduce multi-level LSHACs and the corresponding sampling algorithm. In terms of estimation, we propose an augmenting inference for margin (AIFM) with a three-stage procedure to efficiently determine the hierarchical structure and the parameters of a LSHAC. Empirical evidence shows that LSHACs have better fitting performances with less parameters and are more flexible in modeling tail dependence. We empirically examine the modeling performance of LSHAC models with United States Exchange-Traded Funds (ETFs). The LSHACs provide more conservative estimations of the probabilities of extreme downward co-movements in the financial market. This talk will also introduce an application example of the LSHAC approach in multi-country longevity risk modeling. In particular, for the first time, we provide significant evidence of geographical correlation of mortality dependence. This finding provides important implications for longevity risk management and decision making.

2:00 pm in 241 Altgeld Hall,Thursday, October 15, 2015

Applications of Graph Containers

Michelle Delcourt (UIUC Math)

Abstract: Many results, such as Szemeredi's theorem on arithmetic progressions can be expressed as statements about families of independent sets in certain uniform hypergraphs. The method of counting independent sets in hypergraphs (developed independently by Balogh-Morris-Samotij and Saxton-Thomason) has gained popularity and has been used to solve a number of longstanding open problems. In this talk, I will introduce the simpler method for the case of graphs and present several applications to number theory and combinatorics. This includes joint work with Jozsef Balogh, Shagnik Das, Hong Liu, and Maryam Sharifzadeh.

2:00 pm in 347 Altgeld Hall,Thursday, October 15, 2015

Stabilizing Tensor Functors

Juan Villeta-Garcia (UIUC Math)

Abstract: Algebraic K-Theory is often thought of as ''the'' universal additive invariant of rings (or more generally, exact categories). Often, however, functors from exact categories to chain complexes don't satisfy additivity. We will describe a procedure (due to McCarthy) that constructs a functor's universal additive approximation. We will then apply it to different local coefficient systems, recovering known invariants of rings (K-Theory, THH, etc.). We will talk about what happens when push these coefficient systems to the world of spectra, and tie in work of Lindenstrauss and McCarthy.

3:00 pm in 243 Altgeld Hall,Thursday, October 15, 2015

Rees Algebras and Almost Linearly Presented Ideals

Vivek Mukundan (Purdue University)

Abstract: Consider a grade 2 perfect ideal $I$ in $R = k[x_1,\cdots, x_d]$ which is generated by forms of the same degree. Assume that the presentation matrix $\phi$ is almost linear, that is, all but the last column of $\phi$ consist of entries which are linear. For such ideals, we find explicit forms of the defining ideal of the Rees algebra $\mathcal{R}(I)$. We also introduce the notion of iterated Jacobian duals and present properties such as Cohen-Macaulayness, regularity, relation type of the Rees algebra of ideals whose second analytic deviation is one. Also, recent results on the special fiber equation associated to birational maps will be presented.

4:00 pm in 245 Altgeld Hall,Thursday, October 15, 2015

Spaces of polynomials (or: Everything I know I learned in kindergarten and EGA)

Benson Farb (University of Chicago)

Abstract: In this talk I will tell a story about polynomials. It can be hard to solve problems about polynomials (e.g. solving by radicals, or giving algorithms to find roots). Why? I will explain how Arnol'd found an answer in topology. I will then try to explain how this point of view connects to algebraic geometry, number theory, and more. I will try to make most (ok, at least some) of this talk accessible to advanced undergraduates.