# Past Events

## 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.*

## On the long-term regularity of water waves

**Friday, July 29, 2022, 10:30 a.m.** through **Friday, July 29, 2022, 11:30 a.m.**

Vincent 570

Alexandru Ionescu (Princeton University)

I will discuss the Euler equations and water waves systems.

The main topics will include

- local regularity theory
- quartic and quintic energy inequalities
- Strichartz estimates, dispersion, and decay
- long-term regularity of solutions

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

**Friday, July 29, 2022, 9 a.m.** through **Friday, July 29, 2022, 10 a.m.**

Vincent 570

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.

## Singularity formation in incompressible fluids and related models

**Thursday, July 28, 2022, 3 p.m.** through **Thursday, July 28, 2022, 4 p.m.**

Vincent 570

Jiajie Chen (California Institute of Technology)

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. In these lectures, we will describe some recent progress on singularity formation in incompressible fluids and related models. We will begin with some properties of the 3D Euler equations useful for studying singularity formation and the dynamic rescaling formulation of the 3D Euler equations. Then we will discuss some ideas to overcome some difficulties in singularity formation and study finite time blowup based on the stability of an approximate blowup profile. We will compare this stability and another notion of stability of blowup. Lastly, we will discuss some ideas for constructing finite time blowup from smooth initial data, particularly in 1D models of the Euler equations, which can be helpful in studying the singularity formation of 3D Euler with smooth data.

## On the long-term regularity of water waves

**Thursday, July 28, 2022, 1:30 p.m.** through **Thursday, July 28, 2022, 2:30 p.m.**

Vincent 570

Alexandru Ionescu (Princeton University)

I will discuss the Euler equations and water waves systems.

The main topics will include

- local regularity theory
- quartic and quintic energy inequalities
- Strichartz estimates, dispersion, and decay
- long-term regularity of solutions

## On the long-term regularity of water waves

**Thursday, July 28, 2022, 11 a.m.** through **Thursday, July 28, 2022, Noon**

Vincent 570

Alexandru Ionescu (Princeton University)

I will discuss the Euler equations and water waves systems.

The main topics will include

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

## Instability and non-uniqueness in the Navier-Stokes equations

**Thursday, July 28, 2022, 9:30 a.m.** through **Thursday, July 28, 2022, 10:30 a.m.**

Vincent 570

Dallas Albritton (Princeton University)

It is not yet known whether Navier-Stokes solutions develop singularities in finite time. If they do, then the solutions can be continued beyond the singularities as Leray-Hopf solutions. It is therefore a fundamental question, Are Leray-Hopf solutions unique? The goal of this course is to present very recent developments in our understanding of this question.

We will begin by quickly reviewing the Navier-Stokes basics: dimensional analysis, the energy balance, pressure, weak solutions, perturbation theory, and weak-strong uniqueness. To save time, we will not present the complete proofs.

Next, we will explain aspects of the Jia, Sverak, and Guillod program (Jia-Sverak, Inventiones 2014, JFA 2015; Guillod-Sverak, arXiv 2017) and, in particular, how instability in self-similarity variables can generate non-uniqueness. We will briefly discuss bifurcations, stable and unstable manifolds, and, time permitting, a short proof of the existence of large self-similar solutions.

Finally, we will present the recent work (A.-Brue-Colombo, Ann. Math. 2022) which rigorously established non-uniqueness of Leray-Hopf solutions with forcing. We will present the main idea but focus on the spectral perturbation arguments.

No prior knowledge of the Navier-Stokes equations is required, though it might be beneficial to preview background on weak and mild solutions in, for example, Chapters 4 and 11 in Robinson-Rodrigo-Sadowski or Chapters 3 and 5 in Tsai.