Department of

Mathematics

Seminar Calendar
for events the day of Thursday, September 19, 2019.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, September 19, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 19, 2019

Indivisibility and divisibility of class numbers of imaginary quadratic fields

Olivia Beckwith (Illinois)

Abstract: For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down to -X for which the class group has trivial (resp. non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss refinements of these classic results in which we consider the imaginary quadratic fields for which the class number is indivisible (divisible) by p and which satisfy the property that a given finite set of rational primes split in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as in Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups which satisfy a finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.

1:00 pm in 464 Loomis,Thursday, September 19, 2019

QUANTUM GRAVITY AND THE SWAMPLAND

Gary Shiu (University of Wisconsin)

Abstract: String theory seems to offer an enormous number of possibilities for low energy physics. The huge set of solutions is often known as the String Theory Landscape. In recent years, however, it has become clear that not all quantum field theories can be consistently coupled to gravity. Theories that cannot be ultraviolet completed in quantum gravity are said to be in the Swampland. In this talk, I’ll discuss some conjectured properties of quantum gravity, evidences for them, and their applications to cosmology.

2:00 pm in 243 Altgeld Hall,Thursday, September 19, 2019

Inductive limits of C*-algebras and compact quantum metric spaces

Abstract: In this talk, we will place quantum metrics, in the sense of Rieffel, on certain unital inductive limits of C*-algebras built from quantum metrics on the terms of the given inductive sequence with certain compatibility conditions. One of these conditions is that the inductive sequence forms a Cauchy sequence of quantum metric spaces in the dual Gromov-Hausdorff propinquity of Latremoliere. Since the dual propinquity is complete, this will produce a limit quantum metric space. Based on our assumptions, we then show that the C*-algebra of this limit quantum metric space is isomorphic to the given inductive limit, which finally places a quantum metric on the inductive limit. This then immediately allows us to establish a metric convergence of the inductive sequence to the inductive limit. Another consequence to our construction is that we place new quantum metrics on all unital AF algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state.

2:00 pm in 347 Altgeld Hall,Thursday, September 19, 2019

The Yang Mills Problem for Probabilists

Kesav Krishnan (UIUC math)

Abstract: We aim to introduce the problem of rigorously defining the Yang-Mills field from the probability perspective. In this first talk, we will introduce lattice guage theory, and some geometric preliminaries

4:00 pm in 245 Altgeld Hall,Thursday, September 19, 2019

On the container method

Jozsef Balogh   [email] (University of Illinois at Urbana-Champaign)

Abstract: We will give a gentle introduction to a recently-developed technique, `The Container Method’, for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. The container method is very useful counting discrete structures with certain properties; transferring theorems into random environment; and proving the existence discrete structures satisfying some important properties. In the first half of the talk we will attempt to convey a general high-level overview of the method, in particular how independent sets in hypergraphs could be used to model various problems in combinatorics; in the second, we will describe a few illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry.