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for events the day of Thursday, February 15, 2018.

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Thursday, February 15, 2018

11:00 am in 241 Altgeld Hall,Thursday, February 15, 2018

Potential automorphy and applications

Patrick Allen (University of Illinois)

Abstract: The philosophy of Langlands reciprocity predicts that many L-functions studied by number theorists should be equal to L-functions coming from automorphic forms. This leads to Langlands's functoriality conjecture, which very roughly states L-functions naturally created from a given automorphic L-function should also be automorphic. I'll describe this in the case of symmetric power L-functions and how the Langlands program in this special case has applications to the Sato-Tate conjecture and the Ramanujan conjecture. The former concerns the distribution of points on elliptic curves modulo various primes, while the latter concerns the size of the Fourier coefficients of modular forms. I'll then discuss joint work in progress with Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne establishing a weak form of Langlands reciprocity and functoriality for symmetric powers of certain rank 2 L-functions over CM fields.

12:00 pm in 243 Altgeld Hall,Thursday, February 15, 2018

Coding geodesic flows and various continued fractions

Merriman Claire (Illinois Math)

Abstract: Continued fractions are frequently studied in number theory, but they can also be described geometrically. I will give both pictorial and algebraic descriptions of the flows that describe continued fraction expansions. This talk will focus on continued fractions of the form $a_1\pm\frac{1}{a_2\pm\frac{1}{a_3\pm\ddots}}$, where the $a_i$ are odd. I will show how to describe these continued fractions as geodesic flows on a modular surface, and compare it to the modular surface needed when $a_i$ are even.

2:00 pm in 241 Altgeld Hall,Thursday, February 15, 2018

p-adic families of modular forms

Ravi Donepudi   [email] (UIUC)

Abstract: This talk is an introduction to the theme of p-adic variation in number theory, especially concerning modular forms. We will first give a general overview of the p-adic Galois representation attached to a classical Hecke eigenform. Then we will closely study the example of Eisenstein series which are classically parametrized by integer weights and see how they can be naturally interpolated p-adically to give a family of p-adic modular forms. As a corollary, this yields a very simple construction of p-adic L-functions. Finally we will tie these two themes together to study the structure of p-adic Galois representations arising from modular forms in the larger space of all p-adic Galois representations (of a fixed Galois group), highlighting Gouvêa-Mazur’s construction of the “infinite fern of Galois representations" and Coleman-Mazur’s Eigencurve. Have I mentioned the word “p-adic” yet?

4:00 pm in 245 Altgeld Hall,Thursday, February 15, 2018

Conformal Tilings of the Plane: Theory and Practice

Kenneth Stephenson (University of Tennessee)

Abstract: The Penrose Tiling of the plane is the most famous traditional "tiling" of the type which motivated this work. With just two tile types, a "kite" shape and a "dart" shape, and a corresponding subdivision rule, it provides fascinating tilings, ones in which the eye finds endless repetitions, but which in fact have no periodicities. Here we will discuss a new, even richer class of tilings, one in which conformal shapes replace the traditional euclidean shapes. The talk will intermingle the discrete world of circle packing with the continuous world of analytic maps and will convey --- I hope --- the pleasing blend of theory, computation, experimentation, and visualization that I have so enjoyed in this new topic. It will be a largely visual tour, so no background in tiling, circle packing, or analytic function theory is needed. (Much of this is joint work with Phil Bowers of Florida State University.)