Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, January 28, 2016.

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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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Thursday, January 28, 2016

11:00 am in 241 Altgeld Hall,Thursday, January 28, 2016

#### Feynman integrals and special values of $L$-functions

###### Detchat Samart (UIUC Math)

Abstract: Over the past few decades, Feynman integrals have been studied extensively in order to formulate the standard model of particle physics (i.e., the theory explaining how fundamental particles interact). In certain cases, these integrals can be expressed as periods associated to algebraic varieties and are conjectured to encode some interesting arithmetic information about the varieties. More precisely, numerical evidence suggests that their evaluations are, up to simple factors, special values of $L$-functions. In this talk, we will briefly explain what Feynman integrals are and present a recent result on Feynman integral evaluations related to critical values of $L$-functions of $K3$ surfaces, which was discovered numerically by D. Broadhurst.

12:00 pm in Altgeld Hall 243,Thursday, January 28, 2016

#### Extending the $\log (2k-1)$-Theorem

###### Rosemary Guzman (UIUC Math)

Abstract: In this talk, I discuss current work that expands the scope of the $\log (2k-1)$-Theorem of Anderson, Canary, Culler and Shalen. This was a seminal result in that it articulated a relationship between a set of $k$ freely-generating isometries of hyperbolic 3-space and how they interacted with points in hyperbolic 3-space; namely, under certain conditions, at least one of the given isometries must move a point $P$ by a distance $\ge \log(2k-1)$. The result lay the foundation for future novel geometric-topological results. Here I discuss an expansion of the theorem, wherein we consider sets of length-$n$ words contained in a rank-2 free group $\Xi$ on 2 letters (one can consider $\ge 2$ letters via the same methods), and present a generalized version that restricts how these isometries displace points in hyperbolic 3-space. This has application to classifying certain hyperbolic 3-manifolds in that the volume of the resulting manifold $M$ gotten by quotient of hyperbolic 3-space with $\Xi$, is expected to have a bounded volume which is improved from known volume bounds.

2:00 pm in 241 Altgeld Hall ,Thursday, January 28, 2016

#### Roth's theorem

###### George Shakan (UIUC Math)

Abstract: Szemeredi's theorem asserts that any dense subset of the natural numbers contains arbitrarily long arithmetic progressions. We will prove in full detail Roth's theorem, which is Szemeredi's theorem in the first nontrivial case of 3 term arithmetic progressions. The proof uses tools from exponential sums, as well as a useful density increment argument. If time permits, which it probably will, I'll explain how this is related to some of professor Zaharescu's previous work on approximation of real numbers by rationals with square denominator's. This will be a two-part talk.

3:00 pm in 243 Altgeld Hall,Thursday, January 28, 2016