Past Events

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

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

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.

Limits of Dense Simplicial Complexes

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.

Coarse coherence of metric spaces and groups

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

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

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.

Topological explorations of neuron morphology

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

Advisory

A block of rooms have been reserved for the conference:

For the Graduate Hotel, please use this link to make your reservation and receive the preferred rate.

For the Days Inn, please call 612-254-6418 (front desk is staffed 24/7), ask for Topology Workshop - Gunnar Carlsson Birthday and receive the preferred rate.

Organizers

Group of people posing for a group photo

This conference will bring 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 Lecture

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 Lecture (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 Lecture

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 Lecture

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

On the long-term regularity of water waves

Alexandru Ionescu (Princeton University)

I will discuss the Euler equations and water waves systems.
The main topics will include

  1. local regularity theory
  2. quartic and quintic energy inequalities
  3. Strichartz estimates, dispersion, and decay
  4. long-term regularity of solutions

Inviscid damping, enhanced dissipation, and dynamical stability for Euler and Navier Stokes equations

Hao Jia (University of Minnesota, Twin Cities)

In this sequence of lectures, we will introduce the classical stability problem for incompressible Euler and Navier Stokes equations. We will focus on the specific setting of the perturbative regime near a spectrally stable monotonic shear flow, and explain various dynamical phenomena, such as inviscid damping for the Euler equation and enhanced dissipation for the Navier Stokes equations in the high Reynolds number regime. Recent advances and further open problems will also be discussed if time permits.