UIUC Dept. of Mathematics Seminar Calendar

    December 2011           January 2012          February 2012    
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              1  2  3    1  2  3  4  5  6  7             1  2  3  4
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Thursday, January 19, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, January 19, 2012

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Kevin Ford (UIUC Math)
Values of Euler's function not divisible by a given prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
Abstract: For a give prime $q>2$, we investigate the first and second order terms in the asymptotic series for the counting function of $n$ with $q\nmid \phi(n)$. Part of the analysis involves the Euler-Kronecker constant $EK(q)$ for the cyclotomic field $Q(e^{2 \pi i/q})$. One of our theorems gives a (conditional on the prime $k$-tuples conjecture) disproof of conjectures of Ihara concerning the distribution of $EK(q)$. The distribution of primes in the arithmetic progression $1\mod q$ plays a central role in the results. This is joint work with Florian Luca and Pieter Moree.

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, January 19, 2012

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Atul Dixit (UIUC Math)
Convexity of quotients of theta functions
Abstract: For fixed $u$ and $v$ such that $0 \leq u < v < 1/2 $, the monotonicity of the quotients of Jacobi theta functions, namely, $\theta_{j}(u|i\pi t)/\theta_{j}(v|i\pi t)$, $j=1, 2, 3, 4$, on $0 < t < \infty$ has been established in the previous works of A.Yu. Solynin, K. Schiefermayr, and Solynin and the author. In the present paper, we show that the quotients $\theta_{2}(u|i\pi t)/\theta_{2}(v|i\pi t)$ and $\theta_{3}(u|i\pi t)/\theta_{3}(v|i\pi t)$ are convex on $0 < t < \infty$. This is joint work with Arindam Roy and Alexandru Zaharescu.

Mathematics Colloquium - Special Lecture 2011-12
2:00 pm in 345 Altgeld Hall, Thursday, January 19, 2012

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Runhuan Feng (University of Wisconsin -- Milwaukee)
Modeling Investment Guarantees: From Asian option To Titanic option
Abstract: Over the past three decades, the challenge of pricing exotic options such as Asian option in the financial market has inspired theorists and practitioners to pursue innovative techniques. Among many technological breakthroughs on this subject, the work by Marc Yor and co-authors has led to the study of exponential functional of geometric Brownian motion and its integral. Around the same time the insurance market has also experienced dramatic shift from traditional life products to investment-combined products. In order to compete with mutual funds, nearly all major variable annuity writers offer various forms of investment guarantees, nicknamed by some as Titanic options. Although there is no apparent resemblance in the payoffs of Titanic option and Asian option, we found that the analysis on the integral of geometric Brownian motion can be extended to determine risk measures for variable annuity guarantees, which were only known previously by Monte Carlo simulations. In this work, we present for the first time in the actuarial literature a very efficient and accurate analytical approach to determine risk capitals as an alternative to the current market practice of Monte Carlo simulations. The talk will provide an introduction to Asian option, variable annuity guaranteed benefits as well as basics of risk measures. It should be easily accessible to students who have taken MATH 476 or MATH 567.

Mathematics Colloquium - Special Lecture 2011-12
4:00 pm in 245 Altgeld Hall, Thursday, January 19, 2012

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John Francis (Northwestern University)
Factorization homology of topological manifolds
Abstract: Factorization homology, or the topological chiral homology of Lurie, is a homology theory for manifolds conceived as a topological analogue of Beilinson & Drinfeld's algebraic theory of factorization algebras. I'll describe an axiomatic characterization of factorization homology, à la Eilenberg-Steenrod. The excision property of factorization homology allows one to see factorization homology as a simultaneous generalization of singular homology, the cohomology of mapping spaces, and Hochschild homology. Excision for factorization homology also facilitates a short proof of the nonabelian Poincaré duality of Salvatore and Lurie; this proof generalizes to give a nonabelian Poincaré duality for stratified manifolds, joint work with David Ayala & Hiro Tanaka. Finally, I'll outline work in progress with Kevin Costello, expressing quantum invariants of knots and 3-manifolds in terms of factorization homology.