Geometry and Topology

Monday 23rd June, Newman Purple (WSGT1)

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, Newman Purple (WSGT2)

15:40 – 16:20 Emanuele Dotto (Warwick)
16:20 – 17:00 Joao Faria Martins (Leeds)
17:00 – 17:40 Ka Man Yim (Cardiff)

Wednesday 25th June, Newman Purple (WSGT3)

15:40 – 16:10 Oli Gregory (Imperial)
16:10 – 16:40 Katerina Santicola (Warwick)
16:40 – 17:10 Dmytro Karvatskyi (NAS Ukraine)
17:10 – 17:40 Marija Jelic Milutinovic (Belgrade)

Titles and Abstracts

Emanuele Dotto – Characteristic polynomials in homotopy theory
The properties of the characteristic polynomial of matrices can be encoded into a ring homomorphism from the cyclic K-group to the ring of Witt vectors. I will give an overview of this characterisation, due to Almkvist and Grayson, and explain how to obtain this homomorphism formally by extending the Witt vectors construction to bimodules. In particular, this construction makes sense also when R is not commutative, and can be refined for matrices self-adjoint with respect to a symmetric form. Most of this work is joint with Achim Krause, Thomas Nikolaus and Irakli Patchkoria.

Joao Faria Martins – A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from discrete higher gauge theory
Quinn’s Finite Total Homotopy TQFT is a topological quantum field theory defined for any dimension n of space, depending on the choice of a homotopy finite space B. For instance, B can be the classifying space of a finite group or a finite 2-group. In this talk, I will report on recent joint work with Tim Porter on once-extended versions of Quinn’s Finite Total Homotopy TQFT, taking values in the symmetric monoidal bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, yielding (0,1,2)-, (1,2,3)-, and (2,3,4)-extended TQFTs, given a homotopy finite space B. I will shown how to compute these once-extended TQFTs when B is the classifying space of a homotopy 2-type, represented by a crossed module of groups.

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.

Anne-Sophie Kaloghiros – Explicit K-stability and moduli construction of Fano 3-folds
The Calabi problem asks which compact complex manifolds are Kähler-Einstein (KE)- i.e. can be endowed with a canonical metric that satisfies both an algebraic condition (being Kähler) and the Einstein (partial differential) equation. Such manifolds always have a canonical class of definite sign. The existence of such a metric on manifolds with positive and trivial canonical class (general type and Calabi Yau) was proved in the 70s. In the case of Fano manifolds the situation is more subtle – Fano manifolds may or may not have a KE metric. We now know that a Fano manifold admits a KE metric precisely when it satisfies a sophisticated algebro-geometric condition called K-polystability. Surprisingly, K-polystability also sheds some light on another important problem: while Fano varieties do not behave well in families, K-polystable Fano varieties do, and form K-moduli spaces. Our explicit understanding of K-polystability is still partial, and few examples of K-moduli spaces are known. In dimension 3, Fano manifolds were classified into 105 deformation families by Mori-Mukai and Iskovskikh. In this talk, I will present an overview of the Calabi problem in dimension 3. Knowing – as we do – which families in the classification of Fano 3-folds have K-polystable members is a starting point to investigate the corresponding K-moduli spaces. I will describe explicitly some K-moduli spaces of Fano 3-folds of small dimension.

Cristina Manolache – Reduced Gromov–Witten invariants in any genus
Gromov–Witten (GW) invariants ideally give counts of curves of genus g in a given variety. However, GW invariants with g greater than one, have a more subtle enumerative meaning: curves of lower genus also contribute to GW invariants. In genus one this problem was corrected by Vakil and Zinger, who defined more enumerative numbers called “reduced GW invariants”. More recently Hu, Li and Niu gave a construction of reduced GW invariants in genus two. I will define reduced Gromov–Witten invariants in all genera. This is work with A. Cobos-Rabano, E. Mann and R. Picciotto.

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.