Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, April 3, 2014.

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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, April 3, 2014

10:00 am in 1000 Micro and Nanotechnology Laboratory (208 North Wright St),Thursday, April 3, 2014

#### Local-to-Global: Topological Data Analysis

###### Robert Ghrist (University of Pennsylvania)

Abstract: Many contemporary challenges in the engineering sciences concern the inference of global features from local data. This passage from local-to-global data is as subtle as it is fundamental; however, it is not unprecedented. In the mathematical sciences, several types of local-to-global challenges were overcome with new techniques -- from topology, homological algebra, and sheaves. This talk will outline both the vision and the first steps of exporting homological and sheaf-theoretic tools for data in the engineering sciences, with examples drawn from sensor networks, signal processing, target tracking, and network optimization.

Prof. Ghrist has a BS in Mechanical Engineering from the University of Toledo, and an MS & Ph.D. in Applied Mathematics from Cornell University. He has held positions at the University of Texas, Austin; Georgia Institute of Technology; and the University of Illinois, Urbana-Champaign. Currently, he is the Andrea Mitchell University Professor of Mathematics and Electrical & Systems Engineering at the University of Pennsylvania. Prof. Ghrist’s work focuses on topological methods in applied mathematics, with applications in networks, robotics, sensing, and more. His work has been honored by an NSF PECASE award in 2004, a Scientific American "SciAm50 Top Research Innovation" award in 2007, and the Chauvenet prize in 2013.

11:00 am in 241 Altgeld Hall,Thursday, April 3, 2014

#### Properties and applications of Apéry-like numbers

###### Armin Straub (UIUC Math)

Abstract: Apéry-like numbers are special integer sequences, going back to Beukers and Zagier, which are modeled after and share many of the properties of the numbers that underly Apéry's proof of the irrationality of $\zeta(3)$. Among their remarkable properties are connections with modular forms and so-called supercongruences, some of which remain conjectural. In the course of several examples, we demonstrate how these numbers and their connection with modular forms feature in various, apparently unrelated, problems. The examples are taken from personal research of the speaker and include the theories of short random walks, binomial congruences, series for $1/\pi$, and positivity of rational functions. We go on to introduce a multivariate extension of the Apéry numbers by realizing them as the diagonal coefficients of a simple rational function in four variables. Finally, we prove that supercongruences hold for all coefficients of this rational function. This fresh perspective on supercongruences extends to other Apéry-like numbers.

12:30 pm in 464 Loomis Laboratory,Thursday, April 3, 2014

#### Wilson Loops and Riemann Theta Functions

###### Martin Kruczenski (Purdue Physics)

Abstract: The AdS/CFT correspondence relates the Wilson loop, a basic observable of gauge theories to the problem of finding a minimal area surface in hyperbolic space. Such problem turns out to be integrable. In this talk I'll describe how those integrability properties allowed us to find an infinite parameter family of analytical surfaces in terms of Riemann Theta functions. This leads also to an interesting result relating the area to the Schwarzian derivative of the contour where it ends.

1:00 pm in Altgeld Hall 347,Thursday, April 3, 2014

#### Geometric Statistics of Ford Circles

Abstract: We explore the statistics of various geometric quantities associated to Ford circles, using horocycle flow on the modular surfaces. This is joint work with S. Chaubey, A. Malik, and A. Zaharescu.

2:00 pm in 140 Henry Administration Building,Thursday, April 3, 2014

#### Bessel Function Series in connection with Ramanujan's Mathematics

###### Ranjan Jana (UIUC Math)

Abstract: Bessel function, also known as the circular cylinder function, is the most commonly used special function in the eld of mathematical physics. No other special functions have received such detailed treatment in readily available treaties (G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, London, 1958) as have the Bessel functions. Due to its importance in Mathematical, Physical, and Engineering Sciences many researcher motivated towards the study of Bessel functions. In the present talk an attempt is made to discuss Bessel function series, its connection with Ramanujan's Mathematics and its interplay with Number Theory.

2:00 pm in 243 Altgeld Hall,Thursday, April 3, 2014

#### Rigidity and maps in Carnot groups

###### Alessandro Ottazzi (CIRM, Trento)

Abstract: In this seminar I am going to discuss some classes of maps in Carnot groups: isometric, conformal, quasiconformal, contact maps. Assuming smoothness, I am interested into their geometric characterization. In particular, I shall present some new results in the study of rigidity of these maps, which I obtained in different collaborations with M. Cowling, E. Le Donne and B. Warhurst.

2:30 pm in 140 Henry Administration Building,Thursday, April 3, 2014

#### On the parity of broken $k$-diamond partitions

###### Amita Malik (UIUC Math)

Abstract: We discuss several new parity results for broken $k$-diamond partitions on certain types of arithmetic progressions. We also discuss bounds for the parity of broken k-diamond partitions and more general colored partitions.

3:00 pm in 347 Altgeld Hall,Thursday, April 3, 2014

#### Integrability of Lie Algebroids

###### Joel Villatoro (UIUC Math)

Abstract: We will discuss the basic definitions and theory of Lie Groupoids and Lie Algebroids. Then we will give a summary of the main results regarding the integrability of general algebroids. More specifically, we will discuss the precise obstructions to fully generalizing Lie's 3rd Theorem to the Groupoid case.

4:00 pm in 245 Altgeld Hall,Thursday, April 3, 2014

#### Arnold Diffusion via Invariant Cylinders and Mather Variational Method

###### Vadim Kaloshin (University of Maryland)

Abstract: The famous ergodic hypothesis claims that a typical Hamiltonian dynamics on a typical energy surface is ergodic. However, KAM theory disproves this. It establishes a persistent set of positive measure of invariant KAM tori. The (weaker) quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian dynamics on a typical energy surface has a dense orbit. This question is wide open. In early 60th Arnold constructed an example of instabilities for a nearly integrable Hamiltonian of dimension n>2 and conjectured that this is a generic phenomenon, nowadays, called Arnold diffusion. In the last two decades a variety of powerful techniques to attack this problem were developed. In particular, Mather discovered a large class of invariant sets and a delicate variational technique to shadow them. In a series of preprints: one joint with P. Bernard, K. Zhang and one with K. Zhang and one with M. Guardia we prove strong form of Arnold's conjecture in dimension n=3.