# Math Calendar

## The Mathematics of Three-Dimensional Structure Determination of Molecules by Cryo-Electron Microscopy

*Amit Singer, Princeton University*

Location: Stevenson 5211

Cryo-electron microscopy (EM) is used to acquire noisy 2D projection images of thousands of individual, identical frozen-hydrated macromolecules at random unknown orientations and positions. The goal is to reconstruct the 3D structure of the macromolecule with sufficiently high resolution. We will discuss algorithms for solving the cryo-EM problem and their relation to other branches of mathematics such as tomography, random matrix theory, representation theory, spectral geometry, convex optimization and semidefinite programming. Tea at 3:30 pm in SC 1425.

## Continuous Maximal Regularity on Manifolds with Singularities and Applications to Geometric Flows

*Yuanzhen Shao, Vanderbilt University*

Location: Stevenson 1307

In this talk, we study continuous maximal regularity theory for a class of degenerate or singular differential operators on manifolds with singularities. Based on this theory, we show local existence and uniqueness of solutions for several nonlinear geometric flows and diffusion equations on non-compact, or even incomplete, manifolds, including the Yamabe flow and parabolic p-Laplacian equations. In addition, we also establish regularity properties of solutions by means of a technique consisting of continuous maximal regularity theory, a parameter-dependent diffeomorphism and the implicit function theorem.

## Fuglede-Kadison Determinats and Sofic Entropy

*Ben Hayes, Vanderbilt University*

Location: Stevenson 1432

Let G be a countable discrete group. An algebraic action of G is an action of G by automorphisms on a compact, metrizable, abelian group X. We will typically ignore the algebraic structure of X and think of the action of G on X as either an action on a compact metrizable space by homeomorphisms, or as a probablity measure-preserving action (giving X the Haar measure). When G is sofic, (to be defined in the talk) Lewis Bowen, with an extension by David Kerr and Hanfeng Li, defined the entropy of an action of G on a compact metrizable space or probability space. We will discuss results on the entropy of an algebraic action in the case that X is the Pontryagin dual of a finitely presented Z(G)-module. It turns out the answer is related to Fuglede-Kadison determinants (defined via the group von Neumann algebra) of finite matrices over Z(G). Our work extends results many others: Li-Thom,Kerr-Li,Bowen-Li etc, and in particular is a generalization of recent results of Li-Thom from amenable groups to sofic groups. Moreover, the techniques are the first in the subject to avoid a nontrivial determinant approximation. Time permitting, I will sketch a proof of the main result.

## Departmental Welcome Event

Location: Stevenson 5211

Department Chair Dietmar Bisch will introduce newcomers to the department. Refreshments will follow in Stevenson 1425.

## Mathematical Properties of Effective Potentials in String Theory

*Marcelo Disconzi, Vanderbilt University*

Location: Stevenson 1307

We study effective potentials coming from compactifications of string theory. We show that, under mild assumptions, such potentials are bounded from below in four dimensions, giving an affirmative answer to a conjecture proposed by Michael Douglas. We also derive some sufficient conditions for the existence of critical points, and establish their positivity in the case of slowly varying warp factors. All proofs and mathematical hypotheses are discussed in the context of their relevance to the physics of the problem.

## A Center Construction for Braided Subfactors and Defects of Conformal Nets

*Marcel Bischoff, Vanderbilt University*

Location: Stevenson 1432

Braided subfactors are subfactors whose even part sits inside a given (non-degenerately) braided tensor category C. Using the braiding of C, there is a notion of commutativity for such subfactors. Maximal commutative subfactors in Z(C) can all be obtained by a subfactor in C via a double construction, the generalized Longo-Rehren construction. Given two commutative subfactors there is the notion of a defect between them, which is basically a commuting square containing both subfactors. We introduce a fusion product between defects and discuss how these defects describe topological defects in algebraic conformal quantum field theory. We give a classification of defects for subfactors coming from the double construction in terms of bimodules of the original subfactors. This classification will also show that the double construction is basically a special case of the "Functoriality of the center of an algebra" by Kong and Runkel.

## Talk Title TBA

*Martin Kassabov, Cornell University*

Location: Stevenson 5211

## Talk Title TBA

*Uffe Haagerup, Copenhagen University
*

Location: Stevenson 1432

## Talk Title TBA

*Uffe Haagerup, University of Copenhagen*

Location: Stevenson 5211

## Talk Title TBA

*François Le Maître, École Normale Supérieure de Lyon
*

Location: Stevenson 1432

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