When is a triangle a triangle?

Tin Yau Tsang

October 27th, 2021 - 2:00pm - 3:00pm - RH 510M

Abstract

As we know, the sum of interior angles for triangles on spheres differs from those on the plane. We know that this differences arises from differences in curvature by the Gauss-Bonnet Theorem. A natural extension of this to ask "what happens in higher dimensional 'triangles'?" In this talk, we will discuss recent developments regarding the relationship between curvature and angles between faces.

About the Speaker

Tin Yau is a 4th year graduate student studying differential geometry.

What is... resolvent degree?

Alex Sutherland

May 19th, 2021 - 10:30am - 11:30am - Virtual via Zoom

Abstract

Given a polynomial f(z), how can we describe a root of f(z) in terms of its coefficients? Trying to answer this question leads to an invariant known as resolvent degree. In this talk, we will address the original question, discuss what we know about resolvent degree, and use resolvent degree to connect questions about solving polynomials to other geometric and representation theoretic problems.

About the Speaker

Alex is a sixth year graduate student working in algebraic geometry. Outside of math, Alex likes to go to concerts and play board games with friends.

Advisor and Collaborators

Alex's advisor is Jesse Wolfson.

On the Crepant Resolutions and Quiver Moduli Spaces

Fei Xiang

May 19th, 2021 - 4:00pm - 5:00pm - Virtual via Zoom

Abstract

In order to resolve a simple singularity of a surface, a possible method is repeatedly blowing it up until the canonical class becomes nontrivial, which raises the definition of crepant resolution. It is first observed by McKay that there exists a one-to-one correspondence between the components of the exceptional locus of a resolution and the nontrivial irreducible representations of the group, which is known as McKay Correspondence. The existence of a crepant resolution and the equivalence between the derived categories have been proved in some particular cases. In this talk, we will be interested in the conjecture that the existence of such a resolution, with some additional restriction, should result in one of the quiver moduli spaces. We will then give the main idea of the recovering process of the resolution from the derived category using Hochschild cohomology.

About the Speaker

Fei is a third year graduate student working in algebraic geometry and representation theory.

Advisor and Collaborators

Fei's advisor is Isaac Vladimir Baranovsky.

Tame Expansions of the Group of Integers

Michael Hehmann

May 13th, 2021 - 4:00pm - 5:00pm - Virtual via Zoom

Abstract

A standard research program in model theory is the study of which expansions of a given mathematical structure preserve model-theoretic tameness properties present in the original structure. Among the most studied tameness properties are those arising from the classification theory of Shelah, the purpose of which is to find interesting dividing lines on the class of first-order theories which separate theories with tame models (where one may be able to classify models up to isomorphism with a small list of invariants) from those with wild models (where no such classification is possible). To this end, Shelah isolated certain combinatorial configurations whose presence indicate that one is on the wild side of a dividing line. In this talk, we will be interested in expansions of the group of integers which omit these combinatorial configurations. We will survey some recent work and explore some open questions and directions for future research.

About the Speaker

Michael is a third year graduate student working in model theory and its applications to group theory and combinatorics.

Advisor and Collaborators

Jessica's advisor is Isaac Goldbring.

Existentially Closed Structures and Subgroups of Ultrapowers of Free Groups

Jessica Schirle

April 19th, 2021 - 4:00pm - 5:00pm - Virtual via Zoom

Abstract

As one may be interested in moving from the real numbers to the complex numbers in order to ensure a solution to every non-constant polynomial, the model theorist wishes to move from her base structure to one which is existentially closed (ec). In short, ec-structures contain solutions to every consistent system of equations. In this talk, we'll explore the class of limit groups, which were introduced in Z. Sela's first paper on his solution to the Tarski Problem. These groups coincide exactly with the class of finitely generated fully residually free groups and finitely generated subgroups of an ultrapower of a free group. We'll outline some of the historical results on ec-groups in order to motivate meaningful questions about ec-limit groups. In particular, we'll analyze a few of the proof techniques in approaching the case of ec-groups and discuss some of the hopes and challenges in applying these techniques to the case of ec-limit groups.

About the Speaker

Jessica is a third year graduate student researching topics in model theory and group theory. She's currently the Science Communication Fellow for the Math Department. In her spare time, she enjoys crosswords, puzzle games, and petting all of the dogs.

Advisor and Collaborators

Jessica's advisor is Isaac Goldbring.