M597 is a student-led seminar for graduate students supported by the Department of Mathematics at the University of Tennessee. Each week, one graduate student presents a talk, often expository and informal. Attendees are encouraged to ask questions and engage in discussion, with the level of interaction guided by the speaker.

In Fall 2025 we are meeting on Mondays 4:10PM - 5PM EDT at AYR 111.

If you’re interested to give a talk, fill out the sign-up form here. If you want to be included on the mailing list please include your name in the following link.

August 25 - Caden Farley

Title: Recent developments in the isoperimetric inequality

Abstract:

The isoperimetric inequality is one of the oldest problems in geometry, dating back to a legend from Virgil's Aeneid. In its original form, it asks the question: among all domains in the plane with a given area, which has the smallest boundary length? The inequality can be generalized to Euclidean spaces of arbitrary dimension and proved using a sharp Sobolev inequality.

September 15 - Jefferson Cannon

Title: Grothendieck Topologies: A Brief Introduction to Topos Theory

Abstract:

The notion of a topos has its origins with Grothendieck in his reformulation of algebraic geometry via sheaves and schemes (as well as with Lawvere in his work to axiomatize the category of sets). On one hand, a topos can be thought of as a "generalized space," or, on the other hand, as a "generalized universe of sets." In this sense, topos theory provides a remarkable union between algebraic geometry and topology, and logic and set theory. The category of sheaves on a topological space is one such example of a topos. But the need quickly arose for sheaves to be defined on categories more general than the lattice of open sets, and in particular, on categories where not every morphism is necessarily a monomorphism. This is what brought about the formulation of a Grothendieck topology, and consequently a (Grothendieck) topos. I will elaborate on Grothendieck's motivation via an analogy between the Galois theory of fields and the Galois theory of covering spaces, then formally describe the construction of Grothendieck topologies, list some basic properties, and discuss their connection to topoi and topos theory at large. If time allows, I will also briefly discuss Lawvere's axiomatic, category-theoretic formulation of an (elementary) topos, which is a slight generalization of a Grothendieck topos.

September 22 - Jefferson Cannon

Title: Localic Reflections of Grothendieck Topoi

Abstract:

In my previous talk, I introduced the notion of a Grothendieck topos as a generalization of the category of sheaves on a topological space, and in particular, on its lattice of open sets, which form a complete Heyting algebra. Now I will introduce another special type of Grothendieck topos via locales, which can be thought of as generalized topological spaces, that is, spaces which may or may not "have enough points," or may have no points at all. A localic topos is one that is equivalent to the category of sheaves on a locale. These topoi have many valuable properties, and among them is the fact that any Grothendieck topos has a so-called "localic reflection." The localic reflection is characterized by a powerful adjunction that gives a (bijective) correspondence between geometric morphisms of topoi and maps of locales. I will prove this fact, then state several consequences, such as the fact that given any locale X, the sublocales of X correspond exactly to the subtopoi of Sh(X).

September 29 - Caden Farley

Title: This Talk Will Almost Schur-ly Be Good

Abstract: Schur's lemma in Riemannian geometry says that Einstein manifolds in dimensions at least 3 have constant scalar curvature. De Lellis and Topping investigated the stability of Schur's lemma, asking instead how close the scalar curvature must be to a constant value if the trace-free Ricci tensor is allowed to be small, rather than identically zero as for Einstein manifolds. They prove a quantitative version of Schur's lemma that they call the Almost-Schur lemma, and they demonstrate that the constants and assumptions of their lemma are sharp using a second variation argument and results of Thurston on the hyperbolic geometry of 3-manifolds which fiber over the circle. This talk will sketch the major ideas of their proof of this Almost-Schur inequality, assuming nothing but basic manifold theory and in the process introducing the major curvature tensors of Riemannian geometry, their behaviors in different dimensions, and foundational tools and techniques in geometric analysis such as the Bochner formula, second variation arguments, and eigenfunctions of the Laplacian.

Monday, October 06, 2025

FALL BREAK

October 20-  Benjamin Meyer

Title: (Title TBA)

October 27, 2025

Speaker: (Open)

Title: (Title TBA)

November 03, 2025- Madison Pratt

Title: (Title TBA)

November 10, 2025

Speaker: (Open)

Title: (Title TBA)

November 17, 2025

Speaker: (Open)

Title: (Title TBA)

November 24, 2025

Speaker: (Open)

Title: (Title TBA)

December 01, 2025

Speaker: (Open)

Title: (Title TBA)