# Math Calendar

## An Unexpected Application of Topological Graph Theory II

*Mark Ellingham, Vanderbilt University*

Location: Stevenson 1432

Last week I introduced the idea of a k-link graph, which generalizes the idea of the line graph, which is the 1-link graph. A natural question is whether H = L_k(G) uniquely determines the graph G. For minimum degree at least three, it turns out that it does. Somewhat surprisingly, part of the proof uses the classification of quadrangular embeddings of 4-regular graphs, which are always on the torus or Klein bottle. This week we show how this classification from topological graph theory can be used to deal with the case where the edges of H have more than one partition into 4-cycles. This is joint work with Bin Jia.

## Tarski Numbers of Group Actions

*Gili Golan, Bar Ilan, Israel*

Location: Stevenson 1310

The Tarski number of an action of a group G on a set X is the minimal number of pieces in a paradoxical decomposition of it. For any k > 3 we construct a faithful transitive group action with Tarski number k. Since every k<4 is not a Tarski number, this provides a complete characterization of Tarski numbers of group actions. Using similar techniques we construct a group action of a free group F with Tarski number 6 such that the Tarski numbers of restrictions of this action to finite index subgroups of F are arbitrarily large.

## 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

*Andrei Martinez-Finkelshtein, Vanderbilt University*

Location: Stevenson 1432

## Talk Title TBA

*Uffe Haagerup, University of Copenhagen*

Location: Stevenson 5211

## Talk Title TBA

*François Le Maître, École Normale SupĂ©rieure de Lyon
*

Location: Stevenson 1432

## Talk Title TBA

*Dustin Mixon, Air Force Institute of Technology*

Location: Stevenson 1432

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