Events

Michael Chapman, NYU (Courant Institute)

Subgroup Tests and the Aldous-Lyons conjecture

The Aldous-Lyons conjecture from probability theory states that every (unimodular) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.

In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.

This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.

MC 5501

2:30pm - 3:30pm

Jesse Huang, University of Waterloo

Organizational Meeting

This is an organizational meeting for the mirror symmetry learning seminar. We will skim through the reading list and topics to cover and assign talks. All are welcome!

MC 2017

Akash Singha Roy, University of Georgia

Residue-class distribution and mean values of multiplicative functions

The distribution of values of arithmetic functions in residue classes has been a problem of great interest in elementary and analytic number theory. The analogous question commonly studied for multiplicative functions is the distribution of their values in coprime residue classes. In work studying this problem for large classes of multiplicative functions, Narkiewicz obtained criteria deciding when a family of such functions is jointly uniformly distributed among the coprime residue classes to a fixed modulus. In the first part of this talk, we shall extend Narkiewicz's criteria to moduli that are allowed to vary in a wide range. Our results are essentially the best possible analogues of the Siegel-Walfisz theorem in this setting. One of the primary themes behind our arguments is the quantitative detection of a certain "mixing" (or ergodicity) phenomenon in multiplicative groups via methods belonging to the "anatomy of integers", but we also rely heavily on more classical analytic arguments, tools from arithmetic and algebraic geometry, and from linear algebra over rings.

In the second part of this talk, we shall gain a finer understanding of these distributions, such as the second-order behavior. This shall rely on extending some of the most powerful known estimates on mean values of multiplicative functions (precisely, the Landau-Selberg-Delange method) to a result that is much more uniform in certain important parameters. We will see several applications of this extended result in other interesting settings as well.

This talk is partially based on joint work with Prof. Paul Pollack.

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Jesse Huang, University of Waterloo

Mirror symmetry for complex tori

We discuss various forms of mirror symmetry using the example of a complex torus and its compactifications.

MC 5479

Owen Sharpe, University of Waterloo

The Selberg-Delange Method

For complex w and z, the expression w^z is ambiguous, requiring a choice of branch of log(w). In particular, there is no way to make w^z an entire function of w; a branch cut will always be present. In turn, this makes it difficult to perform contour integration and calculate residues with functions of the form f(w)^z, which are fundamental operations in number theory. We describe Selberg's method for performing such computations and some of its applications, such as those by Selberg and Delange. Incidentally, we will also discuss Hankel's formula for the Gamma function and Perron's formula for partial sums of Dirichlet series.

MC 5403

Amanda Maria Petcu, University of Waterloo

Cohomogeneity one solitons of the hypersymplectic flow

Given a manifold X^4 x T^3 where X^4 is hypersymplectic, one can give a flow of hypersymplectic structures that evolve according to the equation dt(w) = d(Q d^*(Q^{-1} w)), where w is the triple that gives the hypersymplectic structure and Q is a 3x3 symmetric matrix that relates the symplectic forms w_i to one another. We will let X^4 be R^4 with a cohomogeneity one action and explain what it means to be a soliton for the hypersymplectic flow and examine a (potentially hyperkahler) metric that comes from this set-up.

MC 5479

Rahim Moosa, University of Waterloo

Curve excluding fields III

We continue to read the paper by Johnson and Ye.

MC 5403

Kaleb Ruscitti, University of WaterlooA category theory joke

A category theory joke

In this talk I will tell one joke. To ensure that all participants find the joke funny, I will spend the first 50 minutes explaining the background material (applied category theory) required for the joke.

MC 5501

Joey Lakerdas-Gayle, University of Waterloo

Compactness and connectives in continuous logic

We will look at the compactness theorem and systems of connectives following "Model Theory for Metric Structures" by Ben Yaacov, Berenstein, Henson, and Usvyatsov.

MC 5403

Ben Webster, University of Waterloo

Intro to 3-d mirror symmetry

This will be an overview talk, aiming to get people hyped up for the 3-d mirror symmetry seminar.

MC 2017