# Rivière-Fabes Symposium 2022

## April 29-May 1, 2022

## Speakers

The symposium's program consists of two hour-long lectures from the following speakers:

**Henri Berestycki**— École des hautes études en sciences sociales (EHESS), Paris**Yaiza Canzani**— University of North Carolina**Juhi Jang**— University of Southern California**Charles Smart**— Yale University

## Sponsors

The 24th annual Rivière-Fabes Symposium is sponsored by the NSF, the Rivière-Fabes Fund at the University of Minnesota, and the IMA (Institute for Mathematics and its Applications) through its Participating Institution Conference Program. The Symposium was established in 1998 in memory of our colleagues Nestor Rivière and Eugene Fabes. These meetings are a continuation of the Rivière Lectures, which ran from 1979-1997.

## Organizers

Dmytro Bilyk, Max Engelstein, Hao Jia, Markus Keel, Svitlana Mayboroda, Peter Polacik, Mikhail Safonov, Daniel Spirn, and Vladimir Sverak.

## Schedule

#### Friday, April 29, 2022

**3–3:25pm — Coffee***Keller 3-176*

**3:25–3:30pm — Welcome and Introduction***Keller 3-180*

**3:30–4:30pm — Gravitational collapse of gaseous stars (Juhi Jang)***Keller 3-180*

In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. Self-gravitating Newtonian stars and relativistic stars are modeled by the gravitational Euler-Poisson system and the Einstein-Euler system respectively. In these lectures, Professor Juhi Jang will review some recent progress on the local and global dynamics of Newtonian stars, and discuss mathematical constructions of gravitational collapse that show the existence of smooth initial data leading to finite time collapse, characterized by the blow-up of the star density. For Newtonian stars, dust-like collapse and self-similar collapse will be presented, and the relativistic analogue and formation of naked singularities for the Einstein-Euler system will be discussed.

**4:30–5pm — Coffee***Keller 3-176*

**5–6pm — Eigenfunction concentration and Weyl Laws via geodesic beams (Yaiza Canzani)***Keller 3-180*

A vast array of physical phenomena, ranging from the propagation of waves to the location of quantum particles, is dictated by the behavior of Laplace eigenfunctions. Because of this, it is crucial to understand how various measures of eigenfunction concentration respond to the background dynamics of the geodesic flow. In collaboration with J. Galkowski, Professor Yaiza Canzani and team developed a framework to approach this problem that hinges on decomposing eigenfunctions into geodesic beams. During the two lectures, Prof. Canzani will present these techniques and explain how to use them to obtain quantitative improvements on the standard estimates for the eigenfunction's pointwise behavior, averages over submanifolds, and Weyl Laws. One consequence of this method is a quantitatively improved Weyl Law for the eigenvalue counting function that holds on most manifolds.

#### Saturday, April 30, 2022

**9–9:30am — Coffee***Keller 3-176*

**9:30–10:30am — The question of uniqueness of steady states for reaction–diffusion equations in general domains (Henri Berestycki)***Keller 3-180*

At long times, reaction–diffusion processes tend to settle into steady states. This begs the question: are these states unique? In this talk, Professor Henri Berestycki will report on ongoing work with Cole Graham on this topic. They study a variety of reaction types, boundary conditions, and domains, and they encounter an unexpected wealth of behavior. To frame these results, Prof. Berestycki will recall some earlier works on qualitative properties of semi-linear elliptic equations in unbounded domains as well as some elements of the theory of generalized principal eigenvalues. Prof. Berestycki will also mention a host of open problems.

**10:30–10:45am — Group Photo***Keller 3-176*

**10:45–11am — Coffee***Keller 3-176*

**11:00am–12pm — Gravitational collapse of gaseous stars (Juhi Jang)***Keller 3-180*

In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. Self-gravitating Newtonian stars and relativistic stars are modeled by the gravitational Euler-Poisson system and the Einstein-Euler system respectively. In these lectures, Professor Juhi Jang will review some recent progress on the local and global dynamics of Newtonian stars, and discuss mathematical constructions of gravitational collapse that show the existence of smooth initial data leading to finite time collapse, characterized by the blow-up of the star density. For Newtonian stars, dust-like collapse and self-similar collapse will be presented, and the relativistic analogue and formation of naked singularities for the Einstein-Euler system will be discussed.

**12–2pm — Lunch**

**2–3pm — Segregation in predator-prey models and the emergence of territoriality (Henri Berestycki)***Keller 3-180*

Professor Henri Berestycki report here on a series of joint works with Alessandro Zilio (Université de Paris) about systems of competing predators interacting with a single type of prey. They focus on the asymptotic behavior of steady states when the competition parameter becomes unbounded. In the limit, they see segregation between the various components of the system. The analysis rests on a series of a priori estimates (involving Liouville-type results) and properties of a free boundary problem. They classify solutions using spectral properties of the limiting system. Results shed light on the conditions under which predators segregate into packs, on whether there is an advantage to such hostile packs, and on the various territorial configurations that arise in this context. These questions lead to nonstandard optimization problems.

**3–3:30pm — Coffee***Keller 3-176*

**3:30pm–4:30pm — Eigenfunction concentration and Weyl Laws via geodesic beams (Yaiza Canzani)***Keller 3-180*

A vast array of physical phenomena, ranging from the propagation of waves to the location of quantum particles, is dictated by the behavior of Laplace eigenfunctions. Because of this, it is crucial to understand how various measures of eigenfunction concentration respond to the background dynamics of the geodesic flow. In collaboration with J. Galkowski, Professor Yaiza Canzani and team developed a framework to approach this problem that hinges on decomposing eigenfunctions into geodesic beams. During the two lectures, Prof. Canzani will present these techniques and explain how to use them to obtain quantitative improvements on the standard estimates for the eigenfunction's pointwise behavior, averages over submanifolds, and Weyl Laws. One consequence of this method is a quantitatively improved Weyl Law for the eigenvalue counting function that holds on most manifolds.

**6–9pm — Banquet***Graduate Hotel*

#### Sunday, May 1, 2022

**8:30–9am — Coffee***Keller 3-176*

**9–10am — Homogenization and Doubling Inequalities for Periodic Elliptic Equations (Charles Smart)***Keller 3-180*

Solutions of periodic elliptic equations behave on large scales like harmonic functions. The theory of homogenization can be used to make this precise. Professor Charles Smart will discuss optimal homogenization results for periodic elliptic equations with measurable coefficients. Prof. Smart will also discuss an application to large-scale doubling inequalities for solutions. This is joint work with Armstrong and Kuusi.

**10–10:30am — Coffee***Keller 3-176*

**10:30–11:30am** **— Unique Continuation for Lattice Schrodinger Operators (Charles Smart)***Keller 3-180*

Professor Charles Smart will discuss unique continuation principles for solutions of Schrodinger operators on lattices. This will include a discussion of joint work with Ding as well as works of Bukovsky-Logunov-Malinnokova-Sodin and Li-Zhang.

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History

This Symposium was established in memory of our colleagues Nestor M. Rivière and Eugene B. Fabes. Both of them were analysts and did their graduate work together at the University of Chicago. After finishing his Ph.D. under Alberto Calderón in 1966, Nestor joined the School of Mathematics the same year. Gene finished his Ph.D. under Antoni Zygmund in 1965 and spent two years at Rice University before coming to Minnesota in 1967. The two started a new era in classical analysis at Minnesota. Unfortunately for us, cancer claimed Nestor's life at the young age of 38 in 1978, ending a brilliant career. The department established the Nestor M. Rivière Lecture in his memory. Gene usually took care of the organizational work and the Rivière Lecture was supported by a fund established by donations from friends of Nestor. In 1997 another tragedy struck. Gene passed away just after he turned sixty and was still at the peak of his productive career. A list of his mathematical achievements can be found in his obituary in the Amer. Math. Soc. Notices, v. 45 (1998), pp. 706-708, and in the Journal of Fourier Analysis and Appl., v. 4, no. 4/5 (1998). Former colleagues, students and friends of Nestor and Gene from all over the world expressed the sentiment that we should establish an annual symposium in their memory. Families of Nestor and Gene fully endorsed the idea of turning the Nestor M. Rivière Lecture into the Rivière-Fabes Symposium. With financial support from interested people the symposium was formally established in 1998.

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Financial support

#### Request for Graduate Student or Postdoc Partial Travel Support

To apply for partial travel support please complete the application form.

Please have your advisor or a postdoctoral mentor write a brief letter that confirms your status, identifies your research area and addresses the possible value to you of your participation. The writer should email (rfs@umn.edu) their brief letter to the organizers of the symposium.

Applications will be reviewed on an ongoing basis until **March** 15, 2022.

Early applications are strongly encouraged.

**Partial Support Information**

The maximum travel support is $900 per person. In order to be reimbursed, we need receipts covering your airfare, hotel, and transportation. Please note that only domestic airfare expenses on U.S. flag carriers are allowed for reimbursement. Please let us know if you have any questions. (Email to rfs@umn.edu.)

Past symposia: Speakers, abstracts, and materials

*This conference is supported by the National Science Foundation under DMS-2000940.*