Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, October 23, 2014.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2014          October 2014          November 2014
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1  2  3  4  5  6             1  2  3  4                      1
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30


Thursday, October 23, 2014

10:00 am in 143 Altgeld Hall,Thursday, October 23, 2014

#### Ergodic theory and the distribution of the Farey and Stern-Brocot sequences

###### Byron Heersink (UIUC Math)

Abstract: This talk will first discuss how to lift a cross section of the horocycle flow on SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) found by Athreya and Cheung to finite covers SL(2,$\mathbb{R}$)/$H$, $H$ a finite index subgroup of SL(2,$\mathbb{Z}$). As an application, we will establish the limiting gap distribution of various subsets of Farey fractions using the ergodic properties of the horocycle flow. We will then discuss the work of Kessebohmer and Stratmann in applying infinite ergodic theory to the distribution of the Stern-Brocot sequence, and give new related results using more elementary methods.

11:00 am in 241 Altgeld Hall,Thursday, October 23, 2014

#### The Erdős-Straus Conjecture

Abstract: This talk will outline my recent work on the Erdős-Straus conjecture. Simply put, the Erdős-Straus conjecture states that for every natural number $n$ greater than 1, there exist natural numbers $x$,$y$ and $z$ such that $4/n = 1/x + 1/y + 1/z$. I will give the historical background of the problem and briefly discuss a few partial results before outlining my attempts to prove the conjecture. I will also provide a few conjectures of my own about the problem to open a discussion about possible future work. This problem is very easy to understand, yet very difficult to solve. I would suggest that my talk is accessible to both undergraduate students and advanced researchers alike.

11:30 am in Urbana Country Club,Thursday, October 23, 2014

1:00 pm in Altgeld Hall 243,Thursday, October 23, 2014

#### No seminar today, because of the departmental retirees' luncheon

2:00 pm in 243 Altgeld Hall,Thursday, October 23, 2014

#### Koshliakov transforms and modular-type transformations

###### Atul Dixit (Tulane University Math)

Abstract: In 1938, N. S. Koshliakov obtained two remarkable identities which show that the modified Bessel function $K_{\nu}(x)$ is self-reciprocal in two kernels, one of which plays a prominent role in a generalization of the Voronoi summation formula. Motivated by these results, in a joint work with Bruce C. Berndt, Arindam Roy and Alexandru Zaharescu, we consider two integrals transforms which we call the first and second Koshlikov transforms of a function. Using a result from a recent joint work with Victor H. Moll giving conditions for a function to equal its first Koshliakov transform, we obtain a modular-type transformation involving infinite series of a modified Lommel function. Results involving such series are extremely rare. The motivating factor for this result was an incorrect identity on page 336 in Ramanujan's Lost Notebook. We will also show paucity of such results associated with the second Koshliakov transform.

3:00 pm in 243 Altgeld Hall,Thursday, October 23, 2014