We will consider the category of polynomial functors from finitely generated free groups to abelian groups. Favorite examples include exterior powers of abelianization, and the so called Passi functors, whose definition we will recall. Our main result says that the derived category of this category is equivalent to the category of excisive functors from pointed spaces to chain complexes. Under this equivalence, exterior powers of abelianization correspond to chains on reduced symmetric powers of spaces. Using this correspondence, one can calculate Ext groups in the functor category in many cases that were previously unknown. This is joint work with Marco Nervo

I will outline how to describe the category of motivic spaces (over a field \(k\)) which have rational homotopy sheaves, are simply connected, and are also "simply connected in the \(\mathbb{G}_m\)-direction" in terms of stable motivic homotopy theory. In particular, for unorderable fields, this category embeds fully faithfully into rational coalgebras in Voevodsky's derived category of motives. In the orderable case, real points and norms have to be taken into account.

Topological methods have recently gotten more traction in the machine learning community and are actively applied and developed. In this talk I will first give a short primer on machine learning and topological data analysis. The main part of the talk will then be devoted to showcase how topological methods can be used in machine learning and give my perspective on important open problems that need to be addressed to allow for more wide-spread use of topological methods in machine learning.

Massey products are an algebraic tool most widely known for detecting the knotting of the Borromean rings. In previous work, Grbić-Linton created systematic constructions of non-trivial Massey products in moment-angle complexes that generalised all former examples of Massey products in toric topology. We take inspiration from this work to construct non-trivial Massey products in partial quotients of moment-angle complexes. Moment-angle complexes \(Z_K\) are built by products of discs and circles glued together according to an associated simplicial complex \(K\). Much of the topology of these spaces is encoded in the combinatorics of the \(K\), which is how it was possible to build general constructions of Massey products. Since moment-angle complexes also admit the action of a torus T, a partial quotient of a moment-angle complex is a quotient \(Z_K/H\) for a closed subgroup \(H\leq T\). Partial quotients are topological analogues of smooth toric varieties, whose cohomology is no longer completely determined by the simplicial complex \(K\) but also an integral matrix associated to the quotient \(T \to T/H\). So far there is no nice model of the cohomology of partial quotients, but nonetheless we show that it is possible to have systematic constructions of Massey products also in partial quotients. This is joint work with Xin Fu and Jelena Grbić.

The framework of extended topological field theories, where these are defined as symmetric monoidal functors from \((\infty,n)\)-categories of cobordisms to some target, can be used to describe classical (as opposed to quantum) topological field theories by taking the target to be various \((\infty,n)\)-categories of iterated spans. I will try to give some motivation for this claim and then discuss some general constructions of such field theories where the target has as objects derived algebraic stacks equipped with symplectic or Poisson structures. This is joint work with Damien Calaque and Claudia Scheimbauer (in the symplectic case) and Valerio Melani and Pavel Safronov (in the Poisson case).

Décalage was first introduced by Deligne in his work on Hodge theory and provides us with a way to construct a new filtered chain complex from an old one, in a certain way. Thinking of spectral sequences as a way to process the data available in a filtration, one can roughly think of the décalage machine as providing us with a way to encode “turning the page of a spectral sequence” on the level of filtrations. Although not originally phrased in this way, décalage can be made sense in terms taking connective covers of a filtration in a certain t-structure on the category of filtered complexes called the Beilinson t-structure. This allows one to generalise the construction also to filtered objects in other stable \(\infty\)-categories, such as spectra. In this talk, we show that the language of the Beilinson \(t\)-structure and décalage provides access to highly structured results on filtered spectra and their associated spectral sequences. This is joint work with Achim Krause and Thomas Nikolaus.

We show that so-called "wedge theorems" hold for all properly immersed, not necessarily compact, ancient solutions to the mean curvature flow in \(R^{n+1}\). Such nonlinear parabolic Liouville-type results add to a long story, generalizing our recent results for self-translating solitons, which in turn imply the minimal surface case (Hoffman-Meeks, '90) that contains the classical cases of cones (Omori '67) and graphs (Nitsche, '65). As an application, we classify the convex hulls of the spacetime tracks of all proper ancient flows, without any of the usual curvature or topology assumptions. The proofs make use of a parabolic Omori-Yau maximum principle for (non-compact) ancient flows. This is joint work with F. Chini.

Discrete Morse Theory is a versatile tool to analyze regular CW complexes and, in particular, simplicial complexes in many contexts. We focus on the sublevel filtration induced by a discrete Morse function and on the persistent connectivity of said filtration. The combinatorial data of persistent connectivity can be tracked by the induced merge tree. In this talk, I will give an introduction to merge trees and discuss the case of discrete Morse functions on trees. For that case, I will discuss sublevel-symmetry equivalences and component-merge equivalences under which the induced merge tree turns out to be invariant. These notions of equivalence lead to a classification result for discrete Morse functions on trees: Any discrete Morse function on a tree is up to matched cells, symmetry equivalences, and component-merge equivalences uniquely determined by the isomorphism class of its induced Morse-labeled merge tree. If time permits, we will continue with a discussion on the difference between Morse-labeled merge trees and unlabeled merge trees.

Koszul duality is a phenomenon appearing in several areas of algebra and topology such as rational homotopy theory and deformation theory. Commonly it is formulated as a Quillen equivalence between the model categories of non-unital differential graded (dg) algebras and non-unital conilpotent dg coalgebras. In this talk we consider how Koszul duality can be extended to an enriched Quillen equivalence over non-unital conilpotent dg coalgebras. We will further note that the same construction goes through for another case of Koszul duality, between dg Lie algebras and non-unital cocommutative dg coalgebras, and discuss why a more general operadic version runs into difficulties.

The Grothendieck construction is a fundamental construction in higher category theory, as it allows one not only to deal efficiently with the higher coherences that appear when dealing with infinity-categories, but it is also a way to encode some notions of \((\infty,2)\)-category theory with \((\infty,1)\)-categories, which are by far more manageable. The goal of this talk will be to outline a multiplicative version of this construction - that is, a version which takes into account monoidal structures on the infinity-categories involved. We will see that there are several possible things one could mean by this, and explain how they are related.

The space of persistence diagrams \(D_p(\mathbb{R}^2,\Delta)\) is the set of all countable multiset of points \(x\in \mathbb R^2/\Delta\) endowed with the \(p\)-Wasserstein metric, \(p\in[1,\infty]\). Persistence diagrams have received considerable attention since they are complete invariants of the persistent homology of filtered spaces. In recent work, Bubenik and Elchesen generalised the notion of a persistence diagram to include multisets of points \(x\in X/A\), where \(X\) is a metric space and \(A\subset X\). The construction of spaces of (generalised) persistence diagrams, assigning a pointed metric space \(D_p(X,A)\) to a metric pair \((X,A)\), is functorial. Thus, it is natural to ask what topological and geometric properties of the space \(X\) the functor \(D_p\) preserve. In this talk, we will give an introduction to spaces of persistence diagrams over metric pairs. We will examine the functoriality of \(D_p\) with respect to separability, completeness, compactness, geodesicity and curvature bound conditions.