Past work has shown that discretizing dynamical systems in a structure-preserving way can improve upon the performance of numerical methods constructed using more traditional approaches.  We aim to show that the Maxwell-Vlasov system of equations, which models plasma dynamics, is amenable to such a structure-preserving approach.  In particular, we will appeal to results obtained for compressible fluids and electromagnetic fields in our treatment of Maxwell-Vlasov, while discussing obstacles unique to that system.

That ellipsoidal shells do not exert gravitational force inside the cavity of the shell was known to Newton, Laplace, and Ivory.

In early 30’s P. Dive proved the inverse of this theorem. In this talk, I shall recall the (partially geometric) proof of this fact and then extend this result to unbounded domains.

Since ellipsoids, and any limit of a sequence of ellipsoids, are the so-called coincidence sets for the obstacle problem, there is a close link between the ellipsoidal potential theory and global solutions to the obstacle problem.

In this talk we present a complete classification (in terms of limit domains of ellipsoids) for global solutions to the obstacle problem in dimensions greater than five. The interesting ramification of this result is a new interpretation of the structure of the regular free boundary close to singular points.

This is a joint work with S. Eberle, and G.S. Weiss.

For further details and references see: https://www.scilag.net/problem/P-200218.1

The Newell-Littlewood numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood-Richardson coefficients form a special case. A. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone: an eigenvalue interpretation and a previously conjectured description by Extended Horn inequalities. This is joint work with S. Gao, N. Ressayre, and A. Yong.

Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on the specific example of pattern formation in zebrafish, which are named for the dark and light stripes that appear on their bodies and fins. Mutant zebrafish, on the other hand, feature different skin patterns, including spots and labyrinth curves. All of these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells. The longterm motivation for my work is to better link genes, cell behavior, and visible animal characteristics — I seek to identify the specific alterations to cell interactions that lead to different mutant patterns. Toward this goal, I develop agent-based models to simulate pattern formation and make experimentally testable predictions. In this talk, I will overview my models and highlight several future directions. Because agent-based models are not analytically tractable using traditional techniques, I will also discuss the topological methods that we have developed to quantitatively describe cell-based patterns, as well as the associated nonlocal continuum limits of my models.

We use addition of two two-digit numbers as a motivating example to introduce addition with carrying. Prerequisites for the talk include familiarity with the place value system and single-digit addition in base ten. Some knowledge of multi-digit addition, in particular addition with carrying, is recommended but not required.

The Knaster continuum, also known as the buckethandle, or the Brouwer–Janiszewski–Knaster continuum can be viewed as an inverse limit of 2-tent maps on the interval. However, there is a whole class (with continuum many non-homeomorphic members) of Knaster continua, each viewed as an inverse limit of p-tent maps, where p is a sequence of primes. In this talk, for each Knaster continuum K, we will give a projective Fraïssé class of finite objects that approximate K (up to homeomorphism) and examine the combinatorial properties of that the class (namely whether the class is Ramsey or if it has a Ramsey extension). We will give an extremely amenable subgroup of the homeomorphism group of the universal Knaster continuum.

The Tower variant of the Number Field Sieve (TNFS) is known to be asymptotically the most efficient algorithm to solve the discrete logarithm problem in finite fields of medium characteristics, when the extension degree is composite. A major obstacle to an efficient implementation of TNFS is the collection of algebraic relations, as it happens in dimensions greater than 2. This requires the construction of new sieving algorithms which remain efficient as the dimension grows.

In this talk,  I will present how we overcome this difficulty by considering a lattice enumeration algorithm which we adapt to this specific context. We also consider a new sieving area, a high-dimensional sphere, whereas previous sieving algorithms for the classical NFS considered an orthotope. Our new sieving technique leads to a much smaller running time, despite the larger dimension of the search space, and even when considering a larger target, as demonstrated by a record computation we performed in a 521-bit finite field GF($p^6$). The target finite field is of the same form as finite fields used in recent zero-knowledge proofs in some blockchains. This is the first reported implementation of TNFS.

In the pre-talk, I will briefly present the core ideas of the quadratic sieve algorithm and its evolution to the Number Field Sieve algorithm.

Evolutionary dynamics permeates life and life-like systems. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to make sense of many phenomena in life sciences. I will present two very general types of evolutionary patterns, loss-of-function and gain-of-function mutations, and discuss scenarios of population dynamics  -- including stochastic tunneling and calculating the rate of evolution. I will also talk about evolution in random environments.  The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. Applications to biomedical problems will be discussed.