Titles and Abstracts

Title: Using Homology to Explore Hypergraph Structure in Biological Systems

Abstract: Glucocorticoids, a class of steroid hormones, play a key role in immune regulation, metabolism, and the stress response. In adrenal insufficiency, a condition where the adrenal glands do not produce enough glucocorticoids, patients rely on glucocorticoid replacement therapy. However, this is often over or under-replaced in the clinic, causing complications.  My project aims to find markers of adequate glucocorticoid replacement in order to personalise treatment. 

Traditional approaches to study biological networks often use pairwise associations, but this may not capture the complexity of the system. Hypergraphs, which allow for higher-order associations, are useful for integrating data from different tissues and omic layers and better modelling the complex system. Homology and persistent homology present powerful tools to analyse these hypergraphs, comparing the network structure in states of glucocorticoid exposure and withdrawal, and identifying biologically relevant changes.

Title: From String Theory to Quantum Cohomology and Quantum K-Theory: Insights into Gromov-Witten Invariants

Abstract: A method to calculate instanton corrections to the non-perturbative effects of the topological string is by finding the Gromov-Witten (GW) invariants. They are rational numbers that appear in the enumerative geometry of Calabi-Yau 3-folds and have applications in string theory. They arise in the topological A-model as counts of worldsheet instantons. GW invariants are also present in quantum K-theory; however, in this context these numbers are integers. In this work, we use the Atiyah-Bott localization method to compute the GW invariants in both contexts: quantum cohomology and quantum K-theory, for local Calabi-Yau manifolds. The final goal is to compare the results in both theories and provide an interpretation within string theory.

Title: Some Remarks on the Special Linear and Metalinear Algebraic Cobordism

Abstract: In classical topology, different cobordism theories can be thought of as universal cohomology theories with certain orientations. For example, complex cobordism $(\text{MU})$ is the universal complex oriented cohomology theory; that is cohomology theories with Thom isomorphism for every complex vector bundle. The analogous idea of different notions of orientations, and corresponding algebraic cobordism theories are well studied in $\mathds{A}^1$-homotopy theory. I will mostly talk about special linear, and \say{metalinear} orientations, and their corresponding universal algebraic theories in the context of $\mathds{A}^1$-homotopy theory. I would like to report on some ongoing computations of stable homotopy groups of special and metalinear algebraic cobordism. This is joint work in progress with Egor Zolotarev.

Title: Positivity in cohomology for algebraic varieties

Abstract: This talk will explain a special feature of the (co)homology of algebraic varieties, namely “positivity”.  We’ll see where this positivity comes from, and what kind of information it can give us about a variety.

Title: Deforming Derived Categories 

Abstract: It is well-known that the Hochschild cohomology of an algebraic object (such as an algebra or a linear category) should govern its deformations. In the case of a dg-category however, the Hochschild cohomology has been shown to parametrize not the dg-deformations, but the curved A-infinity deformations. This phenomenon is known as the curvature problem. We therefore seek a different deformation-theoretic interpretation of the Hochschild cohomology of a dg-category. Motivated by the example of the bounded derived category of an abelian category — which can be approached via the deformation theory of abelian categories in the sense of Lowen-Van den Bergh — we develop the deformation theory of other dg-categories with similar extra data: pretriangulated dg-categories with a t-structure, abbreviated to t-dg-categories.

In this talk, I will establish the deformation theory of t-dg-categories following the prototypical example of the bounded derived category of an abelian category, thus allowing for novel interpretations of the higher Hochschild cohomology groups also in this case. In particular, we will discuss a deformation equivalence between the bounded t-deformations of a bounded t-dg-category on the one hand, and dg-deformations of the dg-category of derived injective ind-dg-objects on the other hand. Since this latter dg-category is cohomologically concentrated in nonpositive degrees, we do not encounter curvature.

This is joint work with Francesco Genovese, Wendy Lowen and Michel Van den Bergh.

Title: Persistent Hochschild homology of directed graphs: From connectivity functors to reachability category

Abstract: There is currently an active interest in homotopy and homology theories in the world of graphs and directed graphs; discrete homotopy theory, magnitude homology and path homology are few prominent themes. Many of these also have incarnations in topological data analysis and persistent homology. In a joint work with Luigi Caputi we extend the use of Hochschild homology of directed graphs in persistence setting. We "lift" Hochschild homology to higher degrees via so called connectivity structures, of which I will present functorial examples. Unfortunately, to get an efficiently computable pipeline we have to resort to non-functorial constructions. To remedy this leads to defining the reachability category of a directed graph, and associated persistent reachability homology. In addition to application to persistence, basic equivalence of categories allows us to give a very direct proof that recently introduced commuting algebras are Morita equivalent to incidence algebras.

Title: Why Condensed Mathematics is Better than Topology

Abstract: The category PAb of profinite abelian groups is an abelian category with many nice properties, which allows us to do most of standard homological algebra. The category PAb naturally embeds into the category TAb of topological abelian groups, but TAb is not abelian, nor does it have a satisfactory theory of tensor products. On the other hand, PAb also naturally embeds into the category CondAb of "condensed abelian groups", which is an abelian category with nice properties. We will show that the embedding of profinite modules into condensed modules (actually, into "solid modules") preserves usual homological notions such Ext and Tor, so that the condensed world might be a better place to study profinite modules than the topological world.

Title: Homotopy transfer over rings and minimal models

Abstract: A classical theorem by Kadeishvili states that the information of an associative dg algebra over a field can be transferred without loss of (homotopical) information to a minimal A_\infty structure on its homology. In my talk, I will discuss this result and generalizations that simultaneously allow transferring commutative multiplications and working over a generic ground ring. One main tool is to work with diagrams of chain complexes indexed by finite sets and injections.