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for events the day of Thursday, September 5, 2019.

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Thursday, September 5, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 5, 2019

On the modularity of elliptic curves over imaginary quadratic fields

Patrick Allen (Illinois)

Abstract: Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point, implying that the mod 3 Galois representation attached to the elliptic curve arises from a modular form of weight one. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch" that gives a criterion for when a given mod 6 representation arises from an elliptic curve. As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.

12:00 pm in 243 Altgeld Hall ,Thursday, September 5, 2019

Counting incompressible surfaces in 3-manifolds

Nathan Dunfield   [email] (Illinois)

Abstract: Counting embedded curves on a hyperbolic surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting incompressible surfaces in a hyperbolic 3-manifold, with the key difference that now the surfaces themselves have intrinsic topology. As are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Hakenís normal surface theory and facts about branched surfaces, we can characterize not just the rate of growth but show it is (essentially) a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples. This is joint work with Stavros Garoufalidis and Hyam Rubinstein.

1:00 pm in 464 Loomis,Thursday, September 5, 2019


Ignacio A. Reyes (Max Planck Institute, Potsdam)

Abstract: We uncover various novel aspects of the entanglement of free fermions at finite temperature on the circle. The modular flow involves a bi-local coupling between a discrete but infinite set of points, even for a single interval. The modular Hamiltonian transitions from locality to complete non-locality as a function of temperature. We derive the entanglement and relative entropies, and comment on the applications to bulk reconstruction in higher spin holography.

2:00 pm in 347 Altgeld Hall,Thursday, September 5, 2019

Introduction to Random Planar Maps

Grigory Terlov (UIUC Math)

Abstract: A "typical" continuous curve on a plane looks like a path of Brownian motion. A natural next question we might ask is "what does a "typical" continuous 2d-surface looks like?" One of the ways to construct such a model is to find a discrete object and consider a scaling limit of it (analogous to considering a scaling limit of a random walk to construct Brownian motion). Such objects are called random planar maps - planar multi-graphs embedded in a sphere or a plane. Of course, similarly to random walks, there are many other reasons why these discrete objects are interesting. In these two talks we will consider several ways of defining random planar maps and a measure on them, connections with random walks and random trees. Finally, in the remaining time I will try to mention several highlights of the field in connection with combinatorics, percolation theory, scaling limits, and Ergodic theory.

3:00 pm in 347 Altgeld Hall,Thursday, September 5, 2019

Polytopes, polynomials and recent results in 1989 mathematics

Bruce Reznick   [email] (University of Illinois at Urbana-Champaign)

Abstract: Hilbertís 17th Problem discusses the possibility of writing polynomials in several variables which only take non-negative values as a sum of squares of polynomials. One approach is to substitute squared monomials into the arithmetic-geometric inequality. Sometimes this is a sum of squares, sometimes it isnít, and I proved 30 years ago that this depends on a property of the polytope whose vertices are the exponents of the monomials in the substitution. Whatís new here is an additional then-unproved claim in that paper and its elementary, but non-obvious proof. This talk lies somewhere in the intersection of combinatorics, computational algebraic geometry and number theory and is designed to be accessible to first year graduate students.