Mathematics

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.

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.

Geometric Statistics of Ford Circles

Jayadev Athreya (UIUC Math)

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.

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.

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.

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.

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.

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.