# Past Events

## Tracking Topological Features Across Neural Stimulus Spaces

**Tuesday, Aug. 2, 2022, 1:30 p.m.** through **Tuesday, Aug. 2, 2022, 2:15 p.m.**

Keller 3-180

Chad Giusti (University of Delaware)

## Effective constructions in algebraic topology and topological data analysis

**Tuesday, Aug. 2, 2022, 10:30 a.m.** through **Tuesday, Aug. 2, 2022, 11:30 a.m.**

Keller 3-180

Anibal Medina-Mardones (Max Planck Institute for Mathematics)

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.

## Motivic Euler characteristics and the Motivic Segal-Becker theorem (Remotely)

**Tuesday, Aug. 2, 2022, 9 a.m.** through **Tuesday, Aug. 2, 2022, 10 a.m.**

Keller 3-180

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.

## Invertibility in Category Representations

**Monday, Aug. 1, 2022, 4 p.m.** through **Monday, Aug. 1, 2022, 4:45 p.m.**

Keller 3-180

Sanjeevi Krishnan (The Ohio State University)

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.

## Limits of Dense Simplicial Complexes

**Monday, Aug. 1, 2022, 3 p.m.** through **Monday, Aug. 1, 2022, 3:45 p.m.**

Keller 3-180

Santiago Segarra (Rice University)

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.

## Coarse coherence of metric spaces and groups

**Monday, Aug. 1, 2022, 2 p.m.** through **Monday, Aug. 1, 2022, 2:45 p.m.**

Keller 3-180

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.

## Homology crowding in configuration spaces of disks

**Monday, Aug. 1, 2022, 1 p.m.** through **Monday, Aug. 1, 2022, 1:45 p.m.**

Keller 3-180

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.

## Braids and Hopf algebras

**Monday, Aug. 1, 2022, 10:30 a.m.** through **Monday, Aug. 1, 2022, 11:30 a.m.**

Keller 3-180

Craig Westerland (University of Minnesota, Twin Cities)

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.

## Topological explorations of neuron morphology

**Monday, Aug. 1, 2022, 9 a.m.** through **Monday, Aug. 1, 2022, 10 a.m.**

Keller 3-180

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.

## Algebraic Topology and Topological Data Analysis: A Conference in Honor of Gunnar Carlsson

**Monday, Aug. 1, 2022, 8 a.m.** through **Friday, Aug. 5, 2022, 11:30 a.m.**

University of Minnesota, Twin Cities

### 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

Time | Activity | Location |
---|---|---|

8:00 am - 8:50 am | Coffee and Registration | Keller 3-176 |

8:50 am - 9:00 pm | Welcome and Introduction | Keller 3-180 |

9:00 am - 10:00 am | Topological explorations of neuron morphology
Kathryn Hess-Bellwald (École Polytechnique Fédérale de Lausanne (EPFL)) |
Keller 3-180 |

10:00 am - 10:30 am | Coffee Break | Keller 3-176 |

10:30 am - 11:30 am | Braids and Hopf algebras
Craig Westerland (University of Minnesota, Twin Cities) |
Keller 3-180 |

11:30 am - 1:00 pm | Lunch | |

1:00 pm - 1:45 pm | Homology crowding in configuration spaces of disks
Hannah Alpert (Auburn University) |
Keller 3-180 |

2:00 pm - 2:45 pm | Coarse coherence of metric spaces and groups
Boris Goldfarb (State University of New York - Albany) |
Keller 3-180 |

3:00 pm - 3:45 pm | Limits of Dense Simplicial Complexes
Santiago Segarra (Rice University) |
Keller 3-180 |

4:00 pm - 4:45 pm | Invertibility in Category Representations
Sanjeevi Krishnan (The Ohio State University) |
Keller 3-180 |

#### Tuesday, August 2, 2022

Time | Activity | Location |
---|---|---|

8:30 am - 9:00 am | Coffee | Keller 3-176 |

9:00 am - 10:00 am | Motivic Euler characteristics and the Motivic Segal-Becker theorem (Remotely)
Roy Joshua (The Ohio State University) |
Keller 3-180 |

10:00 am - 10:15 am | Group Photo | |

10:15 am - 10:30 am | Coffee Break | Keller 3-176 |

10:30 am - 11:30 am | Effective constructions in algebraic topology and topological data analysis
Anibal Medina-Mardones (Max Planck Institute for Mathematics) |
Keller 3-180 |

11:30 am - 1:00 pm | Lunch | |

1:30 pm - 2:15 pm | Tracking Topological Features Across Neural Stimulus Spaces
Chad Giusti (University of Delaware) |
Keller 3-180 |

2:30 pm - 3:15 pm | Persistent cup-length
Ling Zhou (The Ohio State University) |
Keller 3-180 |

3:30 pm - 4:15 pm | Witness complexes and Lagrangian duality
Erik Carlsson (University of California, Davis) |
Keller 3-180 |

4:30 pm - 5:15 pm | Ramification in Higher Algebra
John Berman (University of Massachusetts) |
Keller 3-180 |

#### Wednesday, August 3, 2022

Time | Activity | Location |

8:30 am - 9:00 am | Coffee | Keller 3-176 |

9:00 am - 10:00 am | Toward conjectures of Rognes and Church--Farb--Putman (Lecture Remotely)
Jenny Wilson (University of Michigan) |
Keller 3-180 |

10:00 am - 10:30 am | Coffee Break | Keller 3-176 |

10:30 am - 11:30 am | Path induction and the indiscernibility of identicals
Emily Riehl (Johns Hopkins University) |
Keller 3-180 |

11:30 am - 1:00 pm | Lunch | |

1:00 pm - 1:45 pm | Decomposition of topological Azumaya algebras in the stable range
Niny Arcila-Maya (Duke University) |
Keller 3-180 |

2:00 pm - 2:45 pm | Persistent homology and its fibre
Ulrike Tillmann (University of Oxford) |
Keller 3-180 |

#### Thursday, August 4, 2022

Time | Activity | Location |
---|---|---|

8:30 am - 9:00 am | Coffee | Keller 3-176 |

9:00 am - 10:00 am | Alpha Magnitude (Remotely)
Sara Kalisnik (ETH Zürich) |
Keller 3-180 |

10:00 am - 10:30 am | Coffee Break | Keller 3-176 |

10:30 am - 11:30 am | Vector bundles for data alignment and dimensionality reduction
Jose Perea (Northeastern University) |
Keller 3-180 |

11:30 am - 1:00 pm | Lunch | |

1:00 pm - 1:45 pm | Equivariant K-Theory of G-Manifolds
Mona Merling (University of Pennsylvania) |
Keller 3-180 |

2:00 pm - 2:45 pm | Equivariant methods in chromatic homotopy theory
XiaoLin (Danny) Shi (University of Chicago) |
Keller 3-180 |

3:00 pm - 3:45 pm | Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips Complexes
Henry Adams (Colorado State University) |
Keller 3-180 |

4:00 pm - 4:45 pm | Gratitude
Vin de Silva (Pomona College) |

#### Friday, August 5, 2022

Time | Activity | Location |
---|---|---|

8:30 am - 9:00 am | Coffee | Keller 3-176 |

9:00 am - 10:00 am | Speculations
Gunnar Carlsson (Stanford University) |
Keller 3-180 |

10:00 am - 10:30 am | Coffee Break | Keller 3-176 |

10:30 am - 11:30 am | Approximations to Classifying Spaces from Algebras
Ben Williams (University of British Columbia) |
Keller 3-180 |

### Participants

Name | Department | Affiliation |
---|---|---|

Henry Adams | Department of Mathematics | Colorado State University |

Hannah Alpert | Department of Applied Mathematics and Statistics | Auburn University |

Niny Arcila-Maya | Duke University | |

John Berman | Department of Mathematics | University of Massachusetts |

Robyn Brooks | Department of Mathematics | Boston College |

Johnathan Bush | Department of Mathematics | University of Florida |

Marco Campos | Department of Mathematics | University of Houston |

Erik Carlsson | Department of Computational and Applied Mathematics | University of California, Davis |

Gunnar Carlsson | Department of Mathematics | Stanford University |

Christopher Chia | Department of Mathematical Sciences | Binghamton University (SUNY) |

Jacob Cleveland | Department of Mathematics | Colorado State University |

Mathieu De Langis | Department of Mathematics | University of Minnesota, Twin Cities |

Vin de Silva | Department of Mathematics | Pomona College |

Alex Elchesen | Department of Mathematics | Colorado State University |

Russell Funk | Strategic Management and Entrepreneurship | University of Minnesota, Twin Cities |

Thomas Gebhart | Department of Computer Science and Engineering | University of Minnesota, Twin Cities |

Chad Giusti | Department of Mathematics | University of Delaware |

Boris Goldfarb | Department of Mathematics and Statistics | State University of New York - Albany |

Iryna Hartsock | Department of Mathematics | University of Florida |

Kathryn Hess-Bellwald | Department of Mathematics | École Polytechnique Fédérale de Lausanne (EPFL) |

Anh Hoang | Department of Mathematics | University of Minnesota, Twin Cities |

Roy Joshua | Department of Mathematics | The Ohio State University |

Matthew Kahle | Department of Mathematics | The Ohio State University |

Sara Kalisnik | Department of Computational and Applied Mathematics | ETH Zürich |

Jennifer Kloke | Data | LinkedIn Corporation |

Miroslav Kramar | Department of Mathematics | University of Oklahoma |

Sanjeevi Krishnan | Department of Mathematics | The Ohio State University |

Chung-Ping Lai | Department of Mathematics | Oregon State University |

Kang-Ju Lee | Department of Mathematical Sciences | Seoul National University |

Guchuan Li | Department of Mathematical Sciences | University of Michigan |

Wenwen Li | Department of Mathematics | University of Oklahoma |

Miguel Lopez | Department of Mathematics | University of Pennsylvania |

Anibal Medina-Mardones | Max Planck Institute for Mathematics | |

Facundo Mémoli | Department of Mathematics | The Ohio State University |

Mona Merling | University of Pennsylvania | |

Elias Nino-Ruiz | Department of Computer Science | Universidad del Norte |

Jose Perea | Department of Mathematics and Computer Science | Northeastern University |

Emily Riehl | Department of Mathematics & Statistics | Johns Hopkins University |

Thomas Roddenberry | Department of Electrical and Computer Engineering | Rice University |

Jerome Roehm | Department of Mathematical Sciences | University of Delaware |

Benjamin Ruppik | Institute for Informatics & Institute for Mathematics | Heinrich-Heine-Universität Düsseldorf |

Eli Schlossberg | Department of Mathematics | University of Minnesota, Twin Cities |

Nikolas Schonsheck | Department of Mathematical Sciences | University of Delaware |

Santiago Segarra | Department of Electrical and Computer Engineering | Rice University |

XiaoLin (Danny) Shi | Department of Mathematics | University of Chicago |

Alexander Smith | Chemical and Biological Engineering | University of Wisconsin, Madison |

Andrew Thomas | Center for Applied Mathematics | Cornell University |

Ulrike Tillmann | Mathematical Institute | University of Oxford |

Mikael Vejdemo-Johansson | Department of Mathematics | College of Staten Island, CUNY |

Elena Wang | Department of Computational Mathematics, Science, and Engineering | Michigan State University |

Craig Westerland | School of Mathematics | University of Minnesota, Twin Cities |

Kirsten Wickelgren | Department of Mathematics | Duke University |

Ben Williams | Department of Computational and Applied Mathematics | University of British Columbia |

Jenny Wilson | University of Michigan | |

Iris Yoon | Mathematical Institute | University of Oxford |

Ningchuan Zhang | Department of Mathematics | University of Pennsylvania |

Ling Zhou | Department of Mathematics | The Ohio State University |

Shaopeng Zhu | Department of Computer Science | University of Maryland |

Lori Ziegelmeier | Department of Mathematics | Macalester College |

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