Past Events

Chad Giusti (University of Delaware)

Anibal Medina-Mardones (Max Planck Institute for Mathematics)

Slides

In order to incorporate ideas from algebraic topology in concrete contexts such as topological data analysis and topological lattice field theories, one needs effective constructions of concepts defined only abstractly or axiomatically. In this talk, I will discuss such constructions for certain invariants derived from the cup product on the cohomology of spaces or, more specifically, from an E∞-structure on their cochains. Together with allowing for the concrete computation of finer cohomological invariants in persistent homology -Steenrod barcodes- these effective constructions also reveal combinatorial information connected to convex geometry and higher category theory.

Roy Joshua (The Ohio State University)

A well-known and very useful result in algebraic topology is the statement that the Euler characteristic of G/N(T) in singular cohomology is 1, where G is a compact Lie group and N(T) is the normalizer of a maximal torus. In the presence of a transfer map as constructed by Becker and Gottlieb the above result shows that in any generalized cohomology theory the classifying space of G is a split summand of the classifying space of N(T).

Based on this, Fabien Morel made a conjecture that an analogous motivic Euler characteristic for a split reductive group G over a field k and N(T) the normalizer of a split maximal torus is 1. We will sketch a proof of this conjecture in the first part of the talk under the assumption the base field has a square root of -1. In the second part of the talk we will apply this result to prove what we call a motivic Segal-Becker theorem for Algebraic K-Theory.

All of this is based on joint work with Gunnar Carlsson and Pablo Pelaez.

Sanjeevi Krishnan (The Ohio State University)

Slides

It is often desirable to equip a representation of a poset or more general small category with inner products on the relevant vector spaces so that the linear maps are partial isometries, maps which restrict to isometries on orthogonal complements of kernels. This sort of inner product structure can be used, for example, to simply representations of interest in multidimensional persistence, circuit design, and network coding. The existence of suitable inner product structure is much more difficult to ascertain in the general categorical setting than in the group setting. However, we can characterize the existence of a slightly weaker inner product structure as factorizability of the representation through a special dagger category called an inverse category. This factorizability admits a coordinate-free, numerical characterization that is decidable for finite categories. We give some concrete applications in circuit design. Time-permitting, we will discuss some connections between this work and a nascent theory (by others) of principle S-bundles for S an inverse semigroup. This talk is joint work with Crichton Ogle.

Santiago Segarra (Rice University)

Slides

We develop a theory of limits for sequences of dense abstract simplicial complexes, where a sequence is considered convergent if its homomorphism densities converge. The limiting objects are represented by stacks of measurable [0,1]-valued functions on unit cubes of increasing dimension, each corresponding to a dimension of the abstract simplicial complex. We show that convergence in homomorphism density implies convergence in a cut-metric, and vice versa, as well as showing that simplicial complexes sampled from the limit objects closely resemble its structure. Applying this framework, we also partially characterize the convergence of nonuniform hypergraphs.

Boris Goldfarb (State University of New York - Albany)

I will introduce properties of metric spaces and, specifically, finitely generated groups with word metrics which are called “coarse coherence” and “coarse regular coherence”. They are geometric counterparts of the classical notion of coherence in homological algebra and the regular coherence property of groups defined and studied by Waldhausen. The properties make sense in the general context of coarse metric geometry and are coarse invariants of spaces and groups. They are in fact a weakening of Waldhausen's regular coherence. In a joint project with Gunnar Carlsson we show they can be used as effectively in K-theory computations. The family of all coarsely regular coherent groups is a very large class of groups containing all groups with straight finite decomposition complexity. This includes almost all known fundamental groups of aspherical manifolds. The new framework allows to prove structural results for the family by developing permanence properties of coarse coherence, a joint work with Jonathan Grossman.

Hannah Alpert (Auburn University)

Configuration spaces of disks in a region of the plane vary according to the radius of the disks, and their topological invariants such as homology also vary. Realizing a given homology class means coordinating the motion of several disks, and if there is not enough space for the disks to move, the homology class vanishes. We explore how clusters of orbiting disks can get too crowded, some topological conjectures that describe this behavior, and some progress toward those conjectures.

Craig Westerland (University of Minnesota, Twin Cities)

Slides

The Milnor–Moore theorem identifies a large class of Hopf algebras as enveloping algebras of the Lie algebras of their primitives. If we broaden our definition of a Hopf algebra to that of a braided Hopf algebra, much of this structure theory falls apart. The most obvious reason is that the primitives in a braided Hopf algebra no longer form a Lie algebra. In this talk, we will discuss recent work to understand what precisely is the algebraic structure of the primitives in a braided Hopf algebra in order to “repair” the Milnor–Moore theorem in this setting. It turns out that this structure is closely related to the dualizing module for the braid groups, which implements dualities in the (co)homology of the braid groups.

Kathryn Hess-Bellwald (École Polytechnique Fédérale de Lausanne (EPFL))

To understand the function of neurons, as well as other types of cells in the brain, it is essential to analyze their shape. Perhaps unsurprisingly, topology provides us with tools ideally suited to performing such an analysis. In this talk I will present a selection of the results of a long-standing collaboration with Lida Kanari of the Blue Brain Project on applying topology to the study of neuron shape and function, emphasizing that even simple topogical tools can prove remarkably powerful for analyzing biological data. I will also illustrate how work on applications can feed back into the development of new mathematical ideas.

Organizers

• Matthew Kahle, The Ohio State University
• Facundo Mémoli, The Ohio State University
• Kirsten Wickelgren, Duke University

The conference brings together researchers from both traditional aspects within Algebraic Topology (such as homotopy theory, knot theory, K-theory, etc.) with more recently developed techniques such as those from Topological Data Analysis and Applied Algebraic Topology (such as persistent homology, applied category theory, quantitative topology, dimension reduction, etc.). 

Having mentored and collaborated with many mathematicians and applied scientists, Gunnar Carlsson has been a central figure in the recent development of both currents.  This week-long conference will therefore explore a wide range of topics at the confluence between Algebraic Topology and Topology Data Analysis. As such it has a strong potential to seed new research directions which will not only widen the landscape of topological techniques in data analysis, but could also suggest new possible directions within algebraic topology. 

Schedule

Subscribe to this event's calendar

Monday, August 1, 2022

Tuesday, August 2, 2022

Wednesday, August 3, 2022

Thursday, August 4, 2022

Friday, August 5, 2022

Participants

The conference is supported by the National Science Foundation under DMS-2223905.