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**CANCELLED**

Alex Cowen, Harvard University

"A twisted additive divisor problem"

What correlation is there between the number of divisors of N and the number of divisors of N+1? This is known as the classical additive divisor problem. This talk will be about a generalized form of this question: I'll give asymptotics for a shifted convolution of sum-of-divisors functions with nonzero powers and twisted by Dirichlet characters. The spectral methods of automorphic forms used to prove the main result are quite general, and I'll present a conceptual overview. One step of the proof uses a less well-known technique called "automorphic regularization" for obtaining the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable.

MC 5417

Kaleb D. Ruscitti, Department of Pure Mathematics, University of Waterloo

"Singular elements & bundles of principal parts"

We will cover sections 7.1-7.3 of the text, which tells us how to compute the singular elements in our enumeration problems. To do this, we will introduce and study bundles of principal parts.

This seminar will be held both online and in person:

• Room: MC 5501
• Zoom link: https://uwaterloo.zoom.us/j/97297197868?pwd=aEhPNHJyWkJqQ20zUkpiL1NUL2FnQT09

Jacques van Wyk, Department of Pure Mathematics, University of Waterloo

"(Some) Essentials of Schemes"

We continue with chapter 1 of Eisenbud and Harris, starting by introducing morphisms of schemes.

MC 5417

Ross Willard, University of Waterloo

"Residually finite equational theories"

An equational theory T is said to be residually finite if every model of the theory can be embedded in a product of finite models of the theory.  Equivalently, T is residually finite if and only if its irreducible models (those that cannot be embedded in products of “simpler” models) are all finite.  In practice, it seems that whenever a theory is both “interesting” and residually finite, then there is a finite upper bound to the sizes of its irreducible models.  In other words, we see a sort of compactness principle for “interesting” equational theories: if such a theory has arbitrarily large finite irreducible models, then it must have an infinite irreducible model.  Whether or not this observation holds generally has been open for almost 50 years.  In this talk I will discuss some recent progress with collaborators Keith Kearnes and Agnes Szendrei.

MC 5479

Michael Francis, Western University

"Local normal forms in complex b^k geometry"

The b-tangent bundle (terminology due to Melrose) is defined so that its sections are smooth vector fields on the base manifold tangent along a given hypersurface. Complex b-manifolds, studied by Mendoza, are defined just like ordinary complex manifolds, replacing the usual tangent bundle by the b-tangent bundle. Recently, a Newlander-Nirenberg theorem for b-manifolds was obtained by Francis-Barron, building on Mendoza's work. This talk will discuss the extension of the latter result to the setting of b^k-geometry for k>1. The original approach to b^k-geometry is due to Scott. A slightly different approach that allows for global holonomy phenomena not present in Scott's framework was introduced by Francis and, independently, by Bischoff-del Pino-Witte.

This seminar will be held both online and in person:

• Room: MC 5417
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Yvon Verberne, Western University

"Pseudo-Anosov Homeomorphisms"

The mapping class group is the group of orientation preserving homeomorphisms of a surface up to isotopy. In particular, the mapping class group encodes information about the symmetries of a surface. The Nielsen-Thurston classification states that elements of the mapping class group are of one of three types: periodic, reducible, and pseudo-Anosov. In this talk, we will focus our attention on the pseudo-Anosov elements, which are the elements of the mapping class group which mix the underlying surface in a complicated way. In this talk, we will discuss both classical and new results related to pseudo-Anosov mapping classes, as well as the connections to other areas of mathematics.

MC 5501

Kunjakanan Nath, University of Illinois, Urbana-Champaign

"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 link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Jiahui Huang, Department of Pure Mathematics, University of Waterloo

"Deformations and Lines on Cubics"

We continue studying the problem of lines on cubic surfaces. By considering first order deformations on the Hilbert scheme, we show that there are 27 distinct lines on any smooth surfaces.

MC 5501

Faisal Romshoo, Department of Pure Mathematics, University of Waterloo

"Some computations with gauge transformations on a $G_2$ manifold"

Given a (torsion-free) $G_2$-manifold $(M, \varphi, g)$ and a gauge transformation $P: TM \rightarrow TM$, we want to look at the $G_2$ structures $\Tilde{\varphi} = P^*g$ and explore the conditions for it to be torsion-free. In this talk, we will start in a more general setting with a Riemannian manifold $(M, g)$ and obtain an expression for the tensor $B(X, Y) = \tilde{\nabla}_X Y -\nabla_X Y$ before moving on to computing the full torsion tensor $\tilde{T}_{pq}$ in the case when $M$ is a $G_2$ manifold.

MC 4058

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory III"

We will continue our discussion on forcing and take a look at nontrivial structures. 

MC 5479