webnotice

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

"Essentials of Schemes"

We continue reading Eisenbud and Harris, actually starting with morphisms of schemes this time.

MC 5417

Dylan McGinley, McMaster University

"Cohomogeneity One Ricci Solitons"

Here we study the Ricci Soliton equation in the steady case, utilising symmetry to reduce the problem to an ordinary differential equation. I will focus in particular on the case where the principal orbits of the group action are line bundles over Fano Kahler manifolds.

MC 5403

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

"An exercise in Riemannian geometry (or how to make a Riemannian geometric omelet without breaking any eggs)"

 I will describe a particular class of Riemannian metrics on the total space of a vector bundle, depending only on one natural coordinate $r$, and which are thus of cohomogeneity one. Such metrics arise frequently in the study of special holonomy, By carefully thinking before diving in, one can extract many useful formulas for such metrics without needing to explicitly compute all of the Christoffel symbols and the curvature. For example, these include the rough Laplacian of a function or of a vector field which are invariant under the symmetry group. If time permits, I will explain why I care about such formulas, as they are ingredients in the study of cohomogeneity one solitons for the isometric flow of $\mathrm{G}_2$-structures.

MC 5403

Wanchun Rosie Shen, Harvard University

"Du Bois singularities, rational singularities, and beyond"

We survey some extensions of the classical notions of Du Bois and rational singularities, known as the k-Du Bois and k-rational singularities. By now, these notions are well-understood for local complete intersections (lci). We explain the difficulties beyond the lci case, and propose new definitions in general to make further progress in the theory. This is joint work (in progress) with Matthew Satriano, Sridhar Venkatesh and Anh Duc Vo.

MC 5417

**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