Past Events

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.

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

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.

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.

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

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.

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.

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.