Differential Geometry Seminars 2025

Tuesday
4 March 2025
11:10 am
Engineering & Math EMG06

Stephan Tillmann
University of Sydney

Computing shortest curves on surfaces

The task at hand is to compute a shortest loop that cannot be contracted to
a point on a surface. This is a classical problem that has been studied in
different contexts, and which enjoys practical applications.
In the setting of hyperbolic and convex projective geometry, I will
describe a simple algorithm to compute the three shortest curves on a
once-punctured torus and explain why it works. The algorithm uses an
attractive interplay between algebra and geometry -- it is what Jane Gilman
coined a "non-Euclidean Euclidean algorithm". This is joint work with
Sepher Saryazdi.

Topology dictates the algebra: an intro to TQFT

Topological quantum field theories originate in interactions between physics and mathematics that began in the 1980s. We give a gentle introduction to them, show how low dimensional examples give rise to natural algebraic structures, and what more extended classification results look like.

Friday
21 March 2025
12:10 pm
Napier 108

John Huerta
University of Lisbon/ANU

Poincaré duality for families of supermanifolds

It is well known to experts, but seldom discussed explicitly, that smooth supergeometry is best done in families. This is also called the relative setting, and it implies that we need relative versions of standard supergeometric constructions. Such constructions include the de Rham complex familiar from ordinary differential geometry, but in the supergeometric setting, they also include more exotic objects, such as the Berezinian line bundle (whose sections are the correct objects to integrate over supermanifolds) and the related complex of integral forms, where the super version of Stokes' theorem lives. To work in families, we introduce relative versions of the de Rham complex and the integral form complex, and we prove that they satisfy a relative version of Poincaré duality. No background in supergeometry will be assumed for this talk. 

This is joint work with Konstantin Eder and Simone Noja.

Holomorphic and algebraic immersed curves directed by a flexible cone

I will describe recent joint work with Antonio Alarcon (Crelle 2025) and Alarcon and Franc Forstneric (arXiv 2024). We investigate immersed complex curves in complex affine space, directed by a cone A satisfying one of the flexibility properties that are studied in Oka theory. When A is the so-called null quadric, such curves play a fundamental role in the theory of minimal surfaces. There are other important examples. We are interested in approximation and interpolation theory for such curves, as well as the "rough shape" of the space of all curves. I will review results from 5-10 years ago in the holomorphic case and then describe our recent results in the algebraic setting, where obstacles not present in the holomorphic case arise.

Friday
2 May 2025
12:10 pm
Napier 108

G2 Moduli Spaces May Have Holes

In his two seminal articles, Dominic Joyce not only constructed the first examples of closed manifolds with G2-holonomy metrics, but also proved that the moduli space of all G2-metrics on a closed manifold is itself a finite-dimensional manifold. The statement is, however, only a local one, and the global topological properties of these moduli spaces have remained quite mysterious ever since. Indeed, up to now, we only know that they may be disconnected by the work of Crowley, Goette, and Nordström; the question whether all path components are contractible or not has not been answered yet.

In this talk, I will outline a construction of an element in the second homotopy group of the moduli space of G2 metrics on Joyce’s first example and present along the way a homotopy-theoretical result that guarantees its non-triviality. This talk is based on joint work with Diarmuid Crowley and Sebastian Goette.

Note: Thorsten will also give an introductory talk on Thursday 1 May 2025, 11:10 am in Lower Napier LG28