Geometry and Topology

Monday 23rd June

15:10 – 15:50 Chunyi Li (Warwick)
15:50 – 16:30 Cristina Manolache (Sheffield)
16:30 – 17:10 Anne-Sophie Kaloghiros (Brunel)

Tuesday 24th June

15:40 – 16:20 Emanuele Dotto (Warwick)
16:20 – 17:00 Joao Faria Martins (Leeds)
17:00 – 17:40 Luciana Basualdo Bonatto (Oxford)

Wednesday 25th June

15:40 – 16:04 Oli Gregory (Imperial)
16:04 – 16:28 Katerina Santicola (Warwick)
16:28 – 16:52 Dmytro Karvatskyi (NAS Ukraine)
16:52 – 17:16 Marija Jelic Milutinovic (Belgrade)
17:16 – 17:40 Ka Man Yim (Cardiff)

Titles and Abstracts

Oli Gregory – Griffiths groups of supersingular varieties
The 2nd Griffiths group ${\rm Griff}^2(X)$ of a variety $X$ is an invariant which measures the difference between homological equivalence and algebraic equivalence for codimension 2-cycles on $X$ . It plays an important role in the theory of motives, but is still mysterious. Over the complex numbers ${\rm Griff}^2(X)$ interacts with Hodge theory, and over global fields it is related to special values of $L$ -functions. I will investigate the situation in characteristic $p>0$ . Here, a consequence of the Beilinson-Bloch conjecture says that ${\rm Griff}^2(X)$ should be ‘large’ if $X$ is ordinary, and ‘small’ if $X$ is supersingular. I will show that ${\rm Griff}^2(X)$ actually vanishes for supersingular varieties over the algebraic closure of a finite field.

Marija Jelic Milutinovic – Matching Complexes
Simplicial complexes constructed from graphs form a rich and diverse research area, establishing numerous interesting connections to a variety of mathematical fields. These complexes provide a natural combinatorial framework, bridging graph-theoretic properties with concepts from algebraic topology. Among them, matching complexes represent a particularly significant class. For a simple graph $G$ , the matching complex $M(G)$ is the simplicial complex on the set of edges, whose faces correspond to matchings in the graph, where a matching is a set of edges no two of which share a vertex. The topology of matching complexes has been extensively studied over the years, leading to many applications in topological combinatorics. However, for most classes of graphs, much about this topology remains mysterious. This talk will offer a brief overview and then present our recent results on matching complexes of outerplanar and related graphs. We prove that the matching complex of an outerplanar graph is either contractible or homotopy equivalent to a wedge of spheres. Furthermore, we introduce a subcomplex of the matching complex, called the perfect matching complex, whose facets correspond to perfect matchings in a graph. We then discuss several results concerning the homotopy type of perfect matching complexes. The talk is based on papers co-authored with Margaret Bayer (University of Kansas, USA) and Julianne Vega (Maret School, Washington, USA), as part of a project supported by the NSF Grant No. DMS-1440140 (MSRI, Berkeley).

Dmytro Karvatskyi – Topological and fractal properties of homogeneous self-similar sets
We study the topological and fractal properties of attractors of homogeneous IFSs on the real line with a rational contraction factor and translations. In simple cases, the generated self-similar set is either a closed interval or homeomorphic to the classical Cantor set. The most intriguing scenario arises when the attractor has a non-empty interior and a fractal boundary, which will be the main focus of the talk. We further compute the Lebesgue measure of the interior and the Hausdorff dimension of the boundary for these complex self-similar sets within a specific family.

Chunyi Li – Bridgeland Stability Conditions and Their Applications in Algebraic Geometry
Inspired by work in mathematical physics, Bridgeland introduced the concept of stability conditions on triangulated categories. Over the past two decades, Bridgeland stability conditions have developed into a powerful tool for addressing problems across multiple mathematical fields. The existence of such stability conditions forms the foundation for several important theories in this area. In this talk, I will discuss recent progress in constructing stability conditions and their applications in algebraic geometry.

Katerina Santicola – Sharp interpolation of rational objects
Interpolation is the art of building an object that passes through a given set of points. For instance, if we are given two points, we can always pass a line through them, and if we are given five points, we can always pass a conic through them. A recent breakthrough in algebraic geometry relates the minimal degree of the curve to the number of points we want to interpolate. What if we wanted to interpolate a set of points belonging in a set, and ensure the object passes through no other points in the set? The aim of this talk is to give an overview of how this question is answered when we ask for sharp interpolation instead.

Ka Man Yim – An algebraic derivation of Morse complexes for poset-graded chain complexes
The Morse-Conley complex is a central object in the application of homological algebra to analysing dynamical systems, as well as information compression in topological data analysis. Given a poset-graded chain complex, its Morse-Conley complex is the optimal chain-homotopic reduction of the initial complex that respects the poset grading. In this work, we give a purely algebraic derivation of the Conley complex using homological perturbation theory. Our algebraic formulation presents an alternative to combinatorial approaches in Forman’s discrete Morse theory. We show how this algebraic perspective also yields efficient algorithms for computing the Conley complex. This talk features joint work with Álvaro Torras Casas and Ulrich Pennig.