Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, December 5, 2017.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, December 5, 2017

11:00 am in 345 Altgeld Hall,Tuesday, December 5, 2017

#### Algebraic K-theory, polynomial functors, and lambda-rings

###### Akhil Mathew (University of Chicago)

Abstract: The Grothendieck group K_0 of a commutative ring is well-known to be a lambda-ring, via taking exterior powers of modules. In joint work in progress with Barwick, Glasman, and Nikolaus, we study space-level refinements of this structure. Namely, we show that the K-theory space of a category is naturally functorial for polynomial functors, and describe a universal property of the extended K-theory functor. This leads to a natural spectral refinement of the notion of a lambda-ring.

12:00 pm in 243 Altgeld Hall,Tuesday, December 5, 2017

#### Characteristic random subgroups and their applications

###### Rostyslav Kravchenko (Northwestern University)

Abstract: The invariant random subgroups (IRS) were implicitly used by Stuck and Zimmer in 1994 and defined explicitly by Abert, Glasner and Virag in 2012. They were actively studied since then. We define the notion of characteristic random subgroups (CRS) which are a natural analog of IRSs for the case of the group of all automorphisms. We determine CRS for free abelian groups and for free groups of finite rank. Using our results on CRS of free groups we show that for groups of geometrical nature (like hyperbolic groups, mapping class groups and outer automorphisms groups) there are infinitely many continuous ergodic IRS. This is a joint work with R. Grigorchuk and L. Bowen

1:00 pm in 347 Altgeld Hall ,Tuesday, December 5, 2017

#### A Spherical Maximal Function along the Primes

###### Theresa Anderson (University of Wisconsin-Madison)

Abstract: Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior.  The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example.  In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to.  We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory.  This is joint work with Cook, Hughes, and Kumchev.

2:00 pm in 241 Altgeld Hall,Tuesday, December 5, 2017

#### Zeros of Eisenstein series

###### Oscar Gonzalez Pagan (UIUC)

Abstract: The zeros of the classical Eisenstein series have been well studied over the years. In 1970 it was shown by F. Rankin and Swinnerton-Dyer that the zeros lie on the boundary of the standard fundamental domain. In this talk we will discuss this result as well as some previous results and related problems. We then study the zeros of the object obtained by applying a differential operator to the classical Eisenstein series.

3:00 pm in 241 Altgeld Hall,Tuesday, December 5, 2017

#### A short proof of a variant of the container lemma for 3-uniform hyper graphs

###### József Balogh (Illinois Math)

Abstract: Many important theorems and conjectures in combinatorics, such as the theorem of Szemerédi on arithmetic progressions and the Erdős-Stone Theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. These hypergraphs have a clustering phenomena, which can be summarized in a general theorem, called as Container Theorem (of Balogh-Morris-Samotij and Saxton-Thomason), and the method is the container method. The method seems to be surprisingly applicable for enumerating problems, extremal questions in random environment, and proving the existence of some combinatorial structures. In this talk we skip the applications, but provide a short, complete proof of a variant of the container lemma for 3-uniform hyper graphs.The pace of the proof will be slow and in a discussion style, the focus will be on to make sure that the audience understands it.

4:00 pm in 245 Altgeld Hall,Tuesday, December 5, 2017

#### Two Problems in Risk Management with Basis Risk

###### Jingong Zhang (University of Waterloo)

Abstract: Basis risk occurs naturally in a variety of finance and insurance applications, and introduces additional complexity to risk management. The theme of this presentation is to study risk management in the presence of basis risk under two settings: index insurance design and dynamic longevity hedge. In the first part of the talk, we study the problem of index insurance design under an expected utility maximization framework. We formally prove the existence and uniqueness of optimal contract for general utility functions, and obtain analytical expressions of the optimal indemnity function for exponential utility and quadratic utility functions. Our method is illustrated by a numerical example where weather index insurance is designed for protection against the adverse rice yield using temperature and precipitation as the underlying indexes. When compared to the linear-type contracts that have been advocated in the literature, the empirical results show that our proposed index-based contract is more efficient at reducing farmers’ basis risk. In the second part of the talk, from a pension plan sponsor’s perspective, we study dynamic hedging strategies for longevity risk using standardized securities in a discrete-time setting. The hedging securities are linked to a population which may differ from the underlying population of the pension plan, and thus basis risk arises. Drawing from the technique of dynamic programming, we develop a framework which allows us to obtain analytical optimal dynamic hedging strategies to achieve the minimum variance of hedging error. Extensive numerical experiments show that our hedging method significantly outperforms the standard “delta” hedging strategy which is commonly studied in the literature.