Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, January 22, 2015.

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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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1  2  3  4  5  6                1  2  3    1  2  3  4  5  6  7
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Thursday, January 22, 2015

11:00 am in 241 Altgeld Hall,Thursday, January 22, 2015

#### 1. Iteration of an arithmetic function & 2. Pointwise multiplication in the Cantor set

###### Bruce Reznick   [email] (UIUC Math)

Abstract: 1. Let $f$ be the arithmetic function so that $f(p^k) = p^{k-1}(p+1)$. THM: For any integer $m$, there exists $N$ so that if $n \ge N$, then the $n$-th iterate $f^n(m) = 2^{n+a(m)}3^{b(m)}.$ Similar results hold if 1" is replaced by d".

2. Let $C$ denote the standard Cantor set. Since $C \subset [0,1/3] \cup [2/3,1]$, if $x \in (1/3,4/9)$, then $x$ is not a product of two members of $C$. THM: If $x \in [0,1]$, then $x$ is a product of three members of $C$. This work and its generalizations are joint with Jayadev Athreya and Jeremy Tyson.

1:00 pm in 243 Altgeld Hall,Thursday, January 22, 2015

#### Constructing surface homeomorphisms with given stretching factors

Abstract: Homeomorphisms from a compact surface to itself were classified by Thurston, and he associated to each such map an algebraic integer, called the dilatation - or the stretching factor. The question of which numbers can be realized as the dilatation of a pseudo-Anosov surface homeomorphism has a long history. A well-known necessary condition is that the number must be strictly greater in absolute value than all its Galois conjugates. We give a sufficient conditions for an algebraic number to be a pseudo-Anosov dilatation of a compact surface and describe an explicit construction of the surface and the map when this condition is met. View talk at http://youtu.be/EjBDwx3RvyE

2:00 pm in 140 Henry Administration Bldg,Thursday, January 22, 2015

#### Class Field Theory in Function Fields

###### Ravi Kiran Donepudi (UIUC Math)

Abstract: The famous Kronecker-Weber theorem asserts that every abelian extension of the rational numbers is contained in a cyclotomic extension. Since the field of rational functions over a finite field shares many properties with the rational numbers, it is natural to ask whether a similar phenomenon happens in this case. Carlitz (1938) found a class of function field extensions with properties strikingly similar to the cyclotomic extensions. Hayes proved in the 1970s that these extensions form a large chunk of the maximal abelian extension of the rational function field. In our talk we will focus on highlighting the similarities between these fields and how the geometry of the function fields often simplifies results imported from the rational numbers. We will also prove the Riemann Hypothesis (for Rational Function Fields).

4:00 pm in 245 Altgeld Hall,Thursday, January 22, 2015

#### Impact of Flexible Periodic Premiums on Variable Annuity Guarantees

###### Zhenyu Cui (Brooklyn College of the City University of New York)

Abstract: In this talk, I will first give an overview of my previous research and publications in stochastic models and their applications in actuarial science and mathematical finance. Then I shall talk about the pricing of variable annuities, which are popular long-term investment products that generate retirement income. Previous literature focus on single premium variable annuities(SPVA), where the annuitant makes a single contribution at start of the contract. Here I focus on flexible premium variable annuities(FPVA), where the annuitant is allowed to make additional discrete periodic contributions. A fast and accurate approximation is developed by drawing on its connection to arithmetic Asian options and by using conditioning technique and properties of comonotonic sums. It is shown that the fair fee charged by the insurer for FPVA is higher than that of SPVA. The study can help insurers estimate the magnitude of underpricing when offering FPVAs.

5:00 pm in 243 Altgeld Hall,Thursday, January 22, 2015

#### Khintchine inequalities

###### Minyu Zhao (UIUC)

Abstract: We will have a brief organization meeting and then start with a review (and later) new results on noncommutative Khintchine inequalities.