Events

Colin Jahel (TU Dresden)

When invariance implies exchangeability (and what it means for invariant Keisler measures)

Let M be a countable model-theoretic structure. We study the actions of Aut(M) on spaces of expansions of M and more precisely, the invariant probability measures under this action. In particular we are interested in understanding when Aut(M)-invariance actually implies S_\infty invariance. We obtain a nice classification for many classical structures. Finally we connect this to invariant Keisler measures, showing how for many structures, they must be S_\infty-invariant. We use this fact to illustrate the difference between two notion of smallness for formulas, forking and universal measure zero.

MC 5403

Faisal Romshoo

Perspectives on the moduli space of torsion-free $\textrm{G}_2$-structures

Joyce showed that the moduli space of torsion-free $\textrm{G}_2$-structures for a compact $7$-manifold forms a non-singular smooth manifold. In this talk, we consider the action of gauge transformations of the form $e^{tA}$ where $A$ is a $2$-tensor, on the space of torsion-free $\textrm{G}_2$-structures. This gives us a new framework to study the moduli space. 

We will see that a $\textrm{G}_2$-structure $\widetilde{\varphi} = P^*\varphi$ acted upon by a gauge transformation $P = e^{tA}$ is infinitesimally torsion-free if and only if  $A \diamond \varphi$ is harmonic and if $A$ satisfies a ``gauge-fixing" condition, where $A \diamond \varphi$ is a special type of $3$-form. This may be the first step in giving an alternate proof of the fact that the moduli space forms a manifold in our framework of gauge transformations.

MC 5501

Spencer Whitehead (Duke University)

An introduction to the Nahm transform and construction of instantons on tori

A Nahm transform recognizes the moduli space of instantons in some setting as an isometric 'dual space'. In this sense the Nahm transform is a 'nonlinear Fourier transform'. In this talk, I will give an introduction to Nahm transforms, sketching from two different points of view the classical Nahm transform of hermitian bundles over 4-tori. Along the way, we will develop a zoo of instanton examples in all ranks using constructions from differential and complex geometry.

MC 5417

Lucia Martin Merchan

Closed G2 manifolds with finite fundamental group

In this talk, we construct a compact closed G2 manifold with b1=0 using orbifold resolution techniques. Then, we study some of its topological properties: fundamental group, cohomology algebra, and formality.

MC 5417