Events

Viktor Majewski, Humboldt University Berlin

Resolutions of Spin(7)-Orbifolds

In Joyce’s seminal work, he constructed the first examples of compact manifolds with exceptional holonomy by resolving flat orbifolds. Recently, Joyce and Karigiannis generalised these ideas in the G2 setting to orbifolds with Z2-singular strata. In this talk I will present a generalisation of these ideas to Spin(7) orbifolds and more general isotropy types. I will highlight the main aspects of the construction and the analytical difficulties.

MC 5479

Anand Pillay, University of Notre Dame

Quasirandomness of definable subsets of algebraic groups over finite fields

We give an arithmetic version of Tao’s algebraic regularity lemma (which was itself an improved Szemerédi regularity lemma for graphs uniformly definable in finite fields). In the arithmetic regime the objects of study are pairs (G,D) where G is a group and D an arbitrary subset. We obtain optimal results, namely that the algebraic regularity lemma holds for the associated bipartite graph (G,G,E) where E(x,y) is xy−1 ∈ D, witnessed by a the decomposition of G into cosets of a (uniformly definable) small index normal subgroup H of G. We compare to results of Green and Gowers. (This is joint work with Atticus Stonestrom.)

MC 5501

Kunjakanan Nath, IECL Nancy, France

Circle method and binary correlation problems

One of the key problems in number theory is to understand the correlation between two arithmetic functions. In general, it is an extremely difficult question and often leads to famous open problems like the Twin Prime Conjecture, the Goldbach Conjecture, and the Chowla Conjecture, to name a few. In this talk, we will discuss a few binary correlation problems involving primes, square-free integers, and integers with restricted digits. The objective is to demonstrate the application of Fourier analysis (aka the circle method) in conjunction with the arithmetic structure of the given sequence and the bilinear form method to solve these problems.

Zoom: https://uwaterloo.zoom.us/j/94276302733?pwd=stZaTKvufL02c5UlpyubhpXYkTSDoN.1

Meeting ID: 942 7630 2733 Passcode: 144512

Mark Hamilton, Mount Allison University

Toric degenerations and independence of polarization

In the theory of geometric quantization, one essential ingredient is the choice of a "polarization"; a natural question is then whether the resulting quantization depends on this choice.  One recent approach to the question of "independence of polarization" is using a deformation of complex structure to "deform" one polarization into another.  Originally applied to smooth toric varieties, this has also been applied to a broader class of examples, such as flag varieties, by using a toric degeneration. 

In this talk I will present an overview of this program (including a short introduction to the key ideas of geometric quantization), and mention several examples of its application, including flag manifolds, more general varieties, and moduli spaces of flat connections (work in progress).

MC 5403

Anand Pillay, University of Notre Dame du Lac

On theories of "nice" fields equipped with a generic derivation

There is a growing body of work on differential fields which are NOT differentially closed but nevertheless have a tractable model theory. I will discuss various results, including a description of definable groups and analogues of algebraic D-groups.

MC 5479

Joey Lakerdas-Gayle, University of Waterloo

Fundamentals of Computability Theory 4

We will continue working through some examples of injury arguments, following Robert Soare's textbook.

MC 5403

Faisal Romshoo, University of Waterloo

Special Lagrangian Geometry

I will talk about special Lagrangian submanifolds, which have garnered considerable interest in several areas in differential geometry and theoretical physics. In particular, I will describe some examples of special Lagrangian submanifolds explicitly.

MC 5479

Matthew Wiersma, University of Waterloo

Entropies and Poisson boundaries of random walks on groups with rapid decay

Let $G$ be a countable group and $\mu$ a probability measure on $G$. The Avez entropy of $\mu$ provides a way of quantifying the randomness of the random walk on $G$ associated with $\mu$. We build a new framework to compute asymptotic quantities associated with the $\mu$-random walk on $G$, using constructions that arise from harmonic analysis on groups. We introduce the notion of \emph{convolution entropy} and show that, under mild assumptions on $\mu$, it coincides with the Avez entropy of $\mu$ when $G$ has the rapid decay property. Subsequently, we apply our results to stationary dynamical systems consisting of an action of a group with the rapid decay property on a probability space, and give several characterizations for when the Avez entropy coincides with the Furstenberg entropy of the stationary space. This leads to a characterization of Zimmer amenability for stationary dynamical systems whenever the acting group has the property of rapid decay.

This talk is based on joint work with B. Anderson-Sackaney, T. de Laat and E. Samei.

MC 5417 or Zoom link below

https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Candace Bethea, Duke University

The local equivariant degree and equivariant rational curve counting

I will talk about joint work with Kirsten Wickelgren on defining a global and local degree in stable equivariant homotopy theory. We construct the degree of a proper G-map between smooth G-manifolds and show a local to global property holds. This allows one to use the degree to compute topological invariants, such as the equivariant Euler characteristic and Euler number. I will discuss the construction of the equivariant degree and local degree, and I will give an application to counting orbits of rational plane cubics through 8 general points invariant under a finite group action on CP^2. This gives the first equivariantly enriched rational curve count, valued in the representation ring and Burnside ring. I will also show this equivariant enrichment recovers a Welchinger invariant in the case when Z/2 acts on CP^2 by conjugation.

MC 5417

Valeriya Kovaleva, University of Montreal

Correlations of the Riemann Zeta on the critical line

In this talk we will discuss the behaviour of the Riemann zeta on the critical line, and in particular, its correlations in various ranges. We will prove a new result for correlations of squares, where shifts may be up to size T^{3/2-\epsilon}. We will also explain how this result relates to Motohashi’s formula for the fourth moment, as well as the moments of moments of the Riemann Zeta and its maximum in short intervals.

MC 5479