Events

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

Jacques Van Wyk

An Introduction to Generalised Geometry

Generalised geometry is a field in differential and complex geometry in which one views the direct sum TM ⊕ T*M instead of TM as the bundle associated to a manifold M. Generalised geometry has seen great success in acting as a unifying framework in which structures defined on TM and T*M can be viewed as specific instances of structures defined on TM ⊕ T*M. For example, almost complex structures and pre-symplectic structures can both be viewed as generalised almost complex structures, a certain kind of automorphism of TM ⊕ T*M.

In this talk, I will give an introduction to generalised geometry. I will show TM ⊕ T*M comes with an intrinsic non-degenerate bilinear form. I will introduce the Dorfman bracket on Γ(TM ⊕ T*M), an analogue of the Lie bracket, which together with the aforementioned bilinear form gives TM ⊕ T*M the structure of a Courant algebroid. I will define generalised almost complex structures in this setting, and show how almost complex structures and pre-symplectic structures can be viewed as generalised almost complex structures. I will introduce generalised metrics and generalised connections, and if time permits, I will discuss integrability of generalised almost complex structures in terms of generalised connections, and/or discuss the analogue of the Levi-Civita connection and what complications it comes with.

MC 5417

Mathias Sonnleitner, University of Passau

Covering completely symmetric convex bodies

A completely symmetric convex body is invariant under reflections or permutations of coordinates. We can bound its metric entropy numbers and consequently its mean width using sparse approximation. We provide an extension to quasi-convex bodies and present an application to unit balls of Lorentz spaces, where we can provide a complete picture of the rich behavior of entropy numbers. These spaces are compatible with sparse approximation and arise from interpolation of Lebesgue sequence spaces, for which a similar result is by now classical. Based on joint work with J. Prochno and J. Vybiral.

MC5403

Andy Zucker

Topological groups with tractable minimal dynamics

For Polish groups, there are several interesting dividing lines in how complicated their minimal flows can be. While metrizability of the universal minimal flow is the most obvious, a theorem of Ben Yaacov, Melleray, and Tsankov suggests the broader class of Polish groups whose universal minimal flows have a comeager orbit. In joint work with Gianluca Basso, we find natural extensions of these classes to general topological groups, obtaining the classes of topological groups with ``concrete minimal dynamics'' or ``tractable minimal dynamics,'' respectively. Both classes admit a wide variety of non-trivial characterizations. In particular, the class of groups with tractable minimal dynamics is the largest class of topological groups admitting any form of KPT correspondence, allowing us to show that this class is absolute between models of set theory.

MC 5479

Spiro Karigiannis

Organizational Meeting

We will meet to plan out the Differential Geometry Working Seminar for the Fall 2024 term.

MC 5479

Brent Nelson, Michigan State University

Uniqueness of almost periodic states on hyperfinite factors

Murray and von Neumann initiated the study of "rings of operators" in the 1930's. These rings, now known as von Neumann algebras, are unital *-algebras of operators acting on a Hilbert space that are closed under the topology of pointwise convergence. Elementary examples include square complex matrices and essentially bounded measurable functions, but the smallest honest examples come from infinite tensor products of matrix algebras. These latter examples are factors—they have trivial center—and are hyperfinite—they contain a dense union of finite dimensional subalgebras. Highly celebrated work of Alain Connes from 1976 and Uffe Haagerup from 1987 showed that these infinite tensor products are in fact the unique hyperfinite factors. Haagerup eventually provided several proofs of this uniqueness, and one from 1989 included as a corollary a uniqueness result for so-called periodic states. This result only holds for some infinite tensor products of matrix algebras and is known to fail for certain other examples, but in recent joint work with Mike Hartglass we show that it can be extended to the remaining examples when periodicity is generalized to almost periodicity. In this talk, I will discuss these results beginning with an introduction to von Neumann algebras that assumes no prior knowledge of the field.

MC 5501

John Yin, Ohio State University

A Chebotarev Density Theorem over Local Fields

I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.

MC 5479

Kaleb D Ruscitti, University of Waterloo

ntroducing the Log Canonical Threshold of a Singularity

Given a variety X, an ideal sheaf a, and a point p in X, the log canonical threshold of a at p is a birational invariant which generalizes the order of a. It appears in asymptotic expansions of certain intergrals, in the minimal model program for log-canonical pairs, and in many other algebraic geometry contexts.   In this seminar, I will give an introduction to this invariant, following the IMPANGA Lecture Notes on Log Canonical Thresholds by Mircea Mustață.

MC 5403