• Ulrich Bunke, Motivic ideas in large-scale geometry

Coarse homology theories provide interesting large-scale invariants of metric and more general coarse spaces which are often computable using tools from homotopy theory. Constructions or assertions which work uniformly for all coarse homology theories are called "motivic". In this talk I will give an overview on the axiomatics of coarse homology theories and construct the universal coarse homology in analogy to constructions in algebraic geometry or the theory of non-commutative motives. I will then present examples of motivic constructions and assertions. I will in particular discuss the coarse assembly map and transgressions to the Higson corona.

• Emanuele Dotto, Poincaré structures on the Fukaya category

A Poincaré structure on a stable category is an efficient way of encoding a notion of "form" (as in quadratic, symmetric,...) on the objects of the category. We will sketch how, given a symplectic manifold M with an anti-symplectomorphism, the Fukaya category of M inherits a suitably linear Poincaré structure. Its definition is analogous to the definition of the so called "visible" Poincaré structure on the category of compact parametrised spectra over a space. We will then relate the real Hochschild homology of this category to the symplectic cohomology of M, in a way which is analogous (and in fact extends) the relation between the Hochschild homology of the chains on the based loop space with the homology of the free loop space. This is all joint work with Cheuk Yu Mak.

• Anthea Monod, Topological Graph Kernels from Tropical Geometry

We introduce a new class of graph kernels for machine learning based on tropical geometry and the topology of metric graphs.  Unlike traditional graph kernels that are defined by graph combinatorics (nodes, edges, subgraphs), our approach considers only the geometry and topology of the underlying metric space.  A key property of our construction is its invariance under edge subdivision, making the kernels intrinsically well-suited for comparing graphs that represent different underlying spaces.  Our kernels are efficient to compute and depend only on the graph genus rather than the size.  In label-free settings, our kernels outperforms existing methods, which we showcase on synthetic, benchmarking, and real-world road network data.  Joint work with Yueqi Cao (KTH Sweden).

• Oscar Randal-Williams, A chromatic approach to homological stability

I will explain a new point of view on the subject of homological stability inspired by aspects of chromatic stable homotopy theory. In this worldview, the usual stabilisation map plays the role of multiplication by p on the p-local sphere spectrum S_(p), and describing the stable homology (i.e., inverting this map) therefore plays the role of calculating the rational stable homotopy groups of spheres. The "secondary stabilisation maps" described by Galatius, Kupers, and I play the role of a v_1-self map on S_(p)/p, and describing the "secondary stable homology" plays the role of calculating the v_1-periodic stable homotopy groups of spheres. Adopting this point of view, several ideas from stable homotopy theory (finite localisations, Smith-Toda complexes, Adams periodicity, ...) can be applied to obtain a good qualitative understanding of what "higher order homological stability" should be about.

• Sarah Whitehouse, Discs and décalage

The talk will give an overview of joint work with Muriel Livernet offering a homotopy perspective on spectral sequences. A certain category of discs plays an important role and gives insight into décalage functors. This generalises earlier work with Cirici-Egas Santander-Livernet providing model structures on filtered chain complexes where the weak equivalences are those maps inducing an isomorphism from the r-page of the associated spectral sequence onwards. Two situations where these ideas can be used will be discussed. One is recent work of Boavida de Brito-Cirici-Horel establishing rational formality of the little n-discs operad. The second is work in progress with Livernet and Roff, using the magnitude-path spectral sequence to give homotopy theories of directed graphs.

• Mohammad Alattar, Deforming Lipschitz Homeomorphisms

In this talk, I will discuss my research on deformations of homeomorphisms on spaces more singular than manifolds, yielding Lipschitz isotopy extension results, a Lipschitz fibration theorem, and a Lipschitz gluing theorem.  These results extend the work of Siebenmann, Sullivan and Perelman. The topological theory of deforming homeomorphisms plays an important role in the proofs of several results in geometry and topology, such as the Grove-Petersen Wu finiteness theorem, and Perelman’s stability theorem for Alexandrov spaces.

• Andrew Baker, Hecke algebras for finite subHopf algebras of the Steenrod algebra

Hecke algebras for pairs of finite groups are well known. I will discuss a generalisation to certain pairs of finite dimensional Hopf algebras, then discuss some examples of the form $\mathcal{E}\subseteq\mathcal{A}(n)$ where the latter is the subHopf of the mod~$2$ Steenrod $\mathcal{A}$ generated by $\mathrm{Sq}^1,\mathrm{Sq}^2,\ldots,\mathrm{Sq}^{2^n}$.

• Severin Bunk, Connections on ∞-bundles

Principal bundles with higher-categorical structure groups are becoming increasingly important in mathematics and its applications, including in the differential geometry of supergravity and higher symmetries in quantum field theory. I will survey a general theory of connections on such bundles, based on derived geometry, which has been missing from the literature so far. This is joint work with Lukas Müller (Perimeter Institute), Joost Nuiten (Toulouse), and Richard Szabo (Heriot-Watt).

• Xavier Crean, Uncovering the phase structure of quantum chromodynamics with topological data analysis 

Recent numerical results have provided evidence for a new conjectured phase in finite temperature quantum chromodynamics (QCD) – the theory underpinning the elementary strong interaction of quarks and gluons. Standard observables used to probe the deconfinement phase transition in pure SU(3) Yang-Mills theory are not suitable in QCD. There is a growing body of evidence that observables constructed from non-trivial topological features of the Yang-Mills vacuum, namely monopoles and vortices, can be highly sensitive to the deconfinement transition. In this talk, we apply methods from topological data analysis to extract topological features from finite temperature pure gauge and QCD lattice configurations. We demonstrate that our TDA-inspired observables precisely capture the quantitative features of the deconfinement phase transition.

• Jack Davidson, Operads and extensions of reflexive homology

An oriented group is a discrete group G with a homomorphism from G to the cyclic group of order two. Koam and Pirashvili studied cohomology theories for oriented algebras, i.e. associative algebras equipped with an action of the oriented group G (where elements of G act via (anti)-automorphisms, depending on their image in the cyclic group of order two). I will describe an extension of their theory using the framework of functor (co)homology and crossed simplicial groups, viewing it as an extension of reflexive homology (which is a homology theory for algebras with an anti-involution). We will demonstrate how this framework allows us to identify these homology theories as "homotopical objects", i.e. as the homology of algebras over an operad. As a special case, we can describe reflexive homology as operadic homology. This is all joint work with Dan Graves and Sarah Whitehouse.

• Kamen Pavlov, On the homotopy classification of 4-manifolds with fundamental group ZxZ/p 

The quadratic 2-type introduced by Hambleton and Kreck in 1988 has been shown to determine the homotopy type of closed 4-manifolds for certain classes of fundamental groups, including ZxZ/2, though it fails in general. A counterexample is given by products of 3-dimensional lens spaces with the circle. In this talk we will sketch the proof of the result that, for general manifolds with fundamental group ZxZ/p, fixing the quadratic 2-type imposes the upper bound of p on the number of possible homotopy types. Semidirect products of Z and Z/p will also be covered.

• Mariam Pirashvili, An isometry theorem for persistent homology of circle-valued functions

This talk is based on the preprint arXiv:2506.02999 and is a collaboration with Nathan Broomhead. Circle-valued functions provide a natural extension of real-valued functions, where instead of measuring values along a linear scale the values lie on a circle. This opens up new possibilities for analysing data in settings where the underlying structure is periodic or has a direction associated to it. There has been significant work on circle-valued maps in the context of persistent homology. Zig-zag persistence generalises to circle-valued functions, leading to persistence modules which are representations of a zig-zag cyclic quiver of type $\tilde{A_n}$. This approach was first introduced in the work of Burghelea and Dey, who classified the resulting indecomposable represenatitons of the $\tilde{A_n}$ quiver as barcodes and Jordan blocks and proposed an algorithm for computing these. The stability of the numerical invariants of persistent homology with respect to the interleaving distance is the fundamental result in this area that gives this method its strong theoretical foundation. Over the years, this distance has been generalised to the zig-zag setting and to general poset representations, using tools from representation theory. Most notably, the involvement of the Auslander-Reiten translate in the definition of the interleaving distance has meant that the robust machinery of representation theory could be employed to derive algebraic stability theorems in more general settings. Our main result is defining an interleaving distance on circle-valued persistence modules using the Auslander-Reiten translate. Moreover, we propose a novel, computer-friendly way to encode the invariants of circle-valued functions via the so-called geometric model, a relatively new tool from representation theory. We also propose a matching distance based on the geometric model, and show that this matching metric coincides with the interleaving distance.

• Lewis Stanton, Anick's conjecture for polyhedral products

Anick conjectured the following after localisation at any sufficiently large prime - the pointed loop space of any finite, simply connected CW complex is homotopy equivalent to a finite type product of spheres, loops on spheres, and a list of well studied torsion spaces defined by Cohen, Moore and Neisendorfer. We study this question in the context of moment-angle complexes, a central object in toric topology which are indexed by simplicial complexes. Recently, much work has been done to find families of simplicial complexes for which Anick's conjecture holds integrally.  In this talk, I will survey what is known, and show that the loop space of any moment-angle complex is homotopy equivalent to a product of looped spheres after localisation away from a finite set of primes. This is then used to show Anick’s conjecture holds for a much wider family of spaces known as polyhedral products. This talk is based on joint work with Fedor Vylegzhanin.

• Jan Steinebrunner, Integral characteristic classes for Poincaré duality spaces 

A surprising result of Berglund and Madsen says that for d>2 and as g -> infinity the rational cohomology of BhAut( (S^d x S^d)^{#g} ) is a free graded commutative algebra on classes indexed by the cohomology of Out(F_n). Using joint work in progress with Shaul Barkan, I will explain how to arrive at these classes from a categorical perspective. This in particular yields integral refinements in BhAut(M) for any oriented Poincaré duality space M. We use that C^*(M) is a (twisted) E-infinity-Frobenius algebra: it has both an E-infinity-algebra structure and a ""trace"" [M]: C^*(M) --> Z[n], which together exhibit a (shifted) self-duality of C^*(M). Via the theory of cyclic and modular infinity-operads we show that the graph bordism category (as studied by Galatius) contains the universal example of an E-infinity-Frobenius algebra. The mapping spaces of this category contain a copy BOut(F_n) and so this space yields universal operations associated to any family of Poincaré duality spaces.

• Robin Stoll, A rational model for the fiberwise THH-transfer 

I will report on work joint with Florian Naef and Nils Prigge in which we produce, for a map $f$ of spaces over a space $B$ such that $f$ has compact fibers, a rational model for the fiberwise transfer of fiberwise topological Hochschild homology, considered as a map of parametrized spectra over $B$. It is constructed in terms of an $A_\infty$-model for the map $f$. This is motivated by applications to moduli spaces of high-dimensional manifolds: in particular we can detect the vanishing of certain cohomology classes originating from the classifying space of block diffeomorphisms.