# Math Calendar

### Upcoming Events

## Computational Analysis Seminar-Many-particle Interactions and Multivariate Geometric Potentials- SC 1310

Ryan Matzke- Vanderbilt University

A great deal of study has been devoted to energies involving two-particle interactions. Such systems have applications in discrete geometry, signal processing, and modeling various natural phenomena (e.g., electrostatic or gravitational energy), among others. Energies involving many-particle interactions are far less understood, which have been appearing recently to improve upon previous results in discrete geometry (such as bounds for optimal codes and kissing numbers) which made use of two-particle interactions. In this talk, we will discuss recent work in developing general theory for energy optimization with multivariate potentials (i.e. ones that model many-particle interactions), and some connections and applications. In particular, we will discuss geometric kernels, such powers of the volume of a simplex generated by k points (a multivariate generalization of Riesz kernels). The work in this talk has been in collaboration with Dmitriy Bilyk (University of Minnesota), Damir Ferizovic (KU Leuven), Alexey Glazyrin (University of Texas Rio Grande Valley), Josiah Park (Texas A&M University), and Oleksandr Vlasiuk (Vanderbilt University).

## Starting of the Point Configurations and Related Equilibria Seminar-Tangent-point energy minimization problem- SC 1404

Liudmyla Kryvonos- Vanderbilt University

## VandyGRAF Fall Seminar Series- Black Hole Shadows and Photon Rings: Experimental Relativity with Radio Interferometry- SC 6333- Physics Building

Michael Johnson- Harvard- Smithsonian- Daniel Kapec- Harvard- CMSA

By combining global networks of radio telescopes, very long baseline interferometry provides the sharpest images in astronomy. This technique has recently culminated in the first images of a black hole, produced using the Event Horizon Telescope (EHT). These images revealed dark “shadows” encircled by bright and unresolved rings for the nuclear supermassive black holes in M87 and the Milky Way. I will discuss these results, including their implications for astrophysical theories of black hole accretion and jet formation. I will then describe our efforts to develop the next-generation EHT (ngEHT) and a space-enhanced EHT over the coming decade, which will improve the dynamic range of current EHT images by two orders of magnitude and will enable studies of horizon-scale dynamics through black hole movies. These efforts can resolve the fractal substructure from unstable photon orbits near a black hole that is predicted to appear within the blurry EHT ring, and they will ultimately measure the masses of thousands of supermassive black holes across cosmic history.

## MTSU/Vanderbilt Graph Theory and Combinatorics Seminar- Recent problems in partitions and other combinatorial functions – MTSU: KOM 204; Vanderbilt: Buttrick 206

Larry Rolen, Vanderbilt University

In this talk, I will discuss recent work, joint with a number of collaborators, on analytic and combinatorial properties of the partition and related functions. This includes work on recent conjectures of Stanton, which aim to give a deeper understanding into the “rank” and “crank” functions which “explain” the famous partition congruences of Ramanujan. I will describe progress in producing such functions for other combinatorial functions using the theory of modular and Jacobi forms and recent connections with Lie-theoretic objects due to Gritsenko-Skoruppa-Zagier. I will also discuss how analytic questions about partitions can be used to study Stanton’s conjectures, as well as recent conjectures on partition inequalities due to Chern-Fu-Tang and Heim-Neuhauser, which are related to the Nekrasov-Okounkov formula.

## PDE Seminar

## Classical Developments of Compressible Fluid Flow- SC 1431

Leo Abbresica- Vanderbilt University

The flow of compressible fluids is governed by the Euler equations, and understanding the dynamics for large times is an outstanding open problem whose full resolution is unlikely to happen in our lifetimes. The main source of difficulty is that any global-in-time theory must incorporate singularities in the PDEs, a fact we have known even in one spatial dimension since Riemann’s 1860 work. In this 1D setting, mathematicians have successfully spent the past 160 years painting a nearly-full picture of fluid dynamics that incorporates singularities.There is a monumental gap in our understanding of compressible fluids in the physical 3D setting compared to the 1D case. This is due in large to the (provable) inaccessibility of the technical PDE tools used in 1D when quantifying the dynamics in 3D. Nevertheless, Christodoulou’s 2007 celebrated breakthrough on shock singularities for the Euler equation has sparked a dramatic wave of results and ideas in multiple space dimensions that have the potential to make the first meaningful dent in the global-in-time theory of compressible fluids. Roughly, shocks are a form of singularity where the fluid solution remains regular but certain first derivatives blow up.

In this talk I will discuss the recent culmination of the wave of results initiated by Christodoulou: my work on the maximal classical development (MCD) for compressible fluids, joint with J. Speck. Roughly speaking, the MCD describes the largest region of spacetime where the Euler equations admit a classical solution. For an open set of smooth data, my work reveals the intimate relationship between shock singularity formation and the full structure of the MCD. This fully solves the 162 year old open problem of extending Riemann’s historic 1D result to 3D without symmetry assumptions. In addition to the mathematical contribution, the geo-analytic information of the MCD is precisely the correct “initial data” needed to physically describe the fluid “past” the initial shock singularity in a weak sense. I will also briefly discuss the countless open problems in the field, all of which can be viewed as “building blocks” which will shine the first lights onto the outstanding global-in-time open problem of fluids.

## Subfactor Seminar

## Subfactor Seminar-Fracton models, defect networks, and enriched fusion categories- SC 1432

Peter Huston- Vanderbilt University

Fracton phases of matter are classes of physical models, related to topological phases of matter, in which quasiparticles exhibit subdimensional mobility. Recently, networks of defects between topological phases have been proposed as a way of understanding fracton order in (3+1)D. In this talk, I will introduce fracton order from the perspective of topological defect networks. I will then describe recent joint work with Fiona Burnell and Dave Penneys on understanding (2+1)D slices of such defect networks using 3-categories of enriched fusion categories.

## Topology & Group Theory Seminar

## Topology and Group Theory – Title: TBA- SC1312

Abstract: TBA

**Abstract: **TBA

## VandyGRAF Seminar Series- SC 6333- Physics Building- Title TBD

Daniel Kapec- Harvard University

Abstract- TBD

## PDE Seminar

## Characteristic initial value problem for 3D compressible Euler equations- SC 1431

Sifan Yu- Vanderbilt University

We consider the characteristic initial value problem for 3D compressible Euler equations (“CharIVP” for short), which is a Cauchy problem whose initial data set is given in a pair of transversally intersecting characteristic initial hypersurface. The resolution of this problem will provide the first solution to a multi-speed characteristic initial value problem in the context of compressible fluid mechanics in multiple spatial dimensions. The setup of “CharIVP” is expected to be useful for studying the long-time dynamics of solutions. We use the geometric framework from [Speck, 19], where the Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. We determine the initial data, which consists of a pair of transversally intersecting null hypersurfaces embedded in Cartesian spacetime and constrained fluid data prescribed on such hypersurfaces. Then we prove a well-posedness result in a “characteristic diamond” region bounded below by initial null hypersurfaces, and propagate regularity with the help of the integral identities derived in [Abbrescia-Speck, 20]. This is a joint work with Prof. Speck.

## VandyGRAF Seminar Series- SC 6333 Physics Building- Title- TBD

Jorge Noronha, UIUC

Abstract- TBD

## VandyGRAF Seminar Series- SC 6333 Physics Building- Title- TBD

Christopher Monahan- William and Mary University

Abstract- TBD

## VandyGRAF Seminar Series- SC 6333 Physics Building- Title- Exotic Matter in Neutron Stars

Veronica Dexheimer, Kent State University

The high densities achieved in neutron stars and the high densities and temperatures achieved in neutron-star mergers create ideal testing grounds in which to learn about exotic matter, namely hyperons and deconfined quarks. The presence of exotic matter can strongly affect the interior of neutron stars, but cannot be directly observed. New electromagnetic and gravitational-wave constraints have been slowly constraining the dense QCD equation of state, allowing us to learn important information about the strong interaction. Nevertheless, strong constraints on dense and hot matter depend on (a) the not yet observed post-merger period of gravitational-wave production from neutron-star mergers and (b) non-trivial comparisons with particle collision experimental data. In this talk, I discuss where we stand and what we expect to learn about dense matter in the near future.

## VandyGRAF Seminar Series- SC 6333 Physics Building- Title- TBD

Bill Press- UT Austin

Abstract- TBD

## VandyGRAF Seminar Series- SC 6333 Physics Building- Title- TBD

James Dent- ULL

Abstract- TBD

## https://as.vanderbilt.edu/math/events/conferences-workshops/

## 2023 Shanks Workshop on Advances in Mathematical and Theoretical Biology March 13-17, 2023

https://my.vanderbilt.edu/mathbio/

## VandyGRAF Seminar Series- SC 6333 Physics Building- Title- TBD

Andrew Strominger- Harvard University

Abstract- TBD

## VandyGRAF Seminar Series- SC 6333 Physics Building- Title- TBD

Edgar Shaghoulian- University of Pennsylvania

Abstract- TBD

## VandyGRAF Seminar Series- SC 6333 Physics Building- Title- TBD

Charles Gale- McGill University

Abstract- TBD