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William Chen, Rutgers University
Markoff numbers are positive integers which encode how resistant certain irrational numbers are to being approximated by rationals. In 1913, Frobenius asked for a description of all congruence conditions satisfied by Markoff numbers modulo primes p. In 1991 and 2016, Baragar, Bourgain, Gamburd, and Sarnak conjectured a refinement of Frobenius’s question, which amounts to showing that the Markoff equation x^2 + y^2 + z^2 - xyz = 0 satisfies “strong approximation”; that is to say: they conjecture that its integral points surject onto its mod p points for every prime p. In this talk we will show how to prove this conjecture for all but finitely many primes p, thus reducing the conjecture to a finite computation. A key step is to understand this problem in the context of describing the orbits of certain group actions. Primarily, we will consider the action of the mapping class group of a topological surface S on (a) the set of G-covers of S, where G is a finite group, and (b) on the character variety of local systems on S. Questions of this type have been related to many classical problems, from proving that the moduli space of curves of a given genus is connected, to Grothendieck’s ambitious plan to understand the structure of the absolute Galois group of the rationals by studying its action on "dessins d’enfant". We will explain some of this history and why such problems can be surprisingly difficult.
Erich Stachura, Kennesaw State University
Abstract: I will discuss a basic model of passive intermodulation (PIM). PIM occurs when multiple signals are active in a passive device that exhibits a nonlinear response. It is known that certain nonlinearities (e.g. the electro-thermal effect) which are fundamental to electromagnetic wave interaction with matter should be accounted for. In this talk, I will discuss existence, uniqueness, and regularity of solutions to a simple model for PIM. This in particular includes a temperature dependent conductivity in Maxwell's equations, which themselves are coupled to a nonlinear heat equation. I will also discuss challenges related to a similar problem when the permittivity $\varepsilon$ also depends on temperature. This is joint work with Niklas Wellander and Elena Cherkaev.
Sourav Chatterjee, Stanford/IAS
I will talk about the Higgs mechanism for mass generation and some recent progress on this topic. No background is necessary. I will start by introducing lattice gauge theories coupled to Higgs fields. After a survey of existing results, I will discuss what is needed to prove rigorously that the Higgs mechanism can indeed generate mass in the continuum limit of these theories. Finally, I will present a result which shows that in a certain scaling limit in any dimension three or higher, SU(2) Yang-Mills-Higgs theory converges to a continuum limit object which has an explicit description as a scale-invariant random distribution. This allows an exact computation of the mass generated by the Higgs mechanism in the continuum limit of this theory.
Yujin Kim, Courant Institute, NYU
The extremal process of branching Brownian motion (BBM) —i.e., the collection of particles furthest from the origin— has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a description of the extrema of BBM in $d > 1$ has been obtained. In this talk, we address the following geometrical question that can only be asked in $d > 1$. Generate a BBM at a large time, and draw the outer envelope of the cloud of particles: what is its shape? Macroscopically, the shape is known to be a sphere; however, we focus on the outer envelope around an extremal point— the "front" of the BBM. We describe the scaling limit for the front, with scaling exponent 3/2, as an explicit, rotationally-symmetric random surface. Based on joint works with Julien Berestycki, Bastien Mallein, Eyal Lubetzky, and Ofer Zeitouni.
Hongbin Sun, Rutgers University
Abstract: We show that any arithmetric lattice $\Gamma<\text{Isom}_+(\mathbb{H}^n)$ with $n\geq 4$ is not LERF (locally extended residually finite), including type III lattices in dimension 7. One key ingredient in the proof is the existence of totally geodesic 3-dim submanifolds, which follows from the definition if $\Gamma$ is in type I or II, but is much harder to prove if $\Gamma$ is in type III. This is a joint work with Bogachev and Slavich.
Christopher-Lloyd Simon, Pennsylvania State University
We study several arithmetic and topological structures on the set of conjugacy classes of the modular group PSL(2;Z), such as equivalence relations or bilinear functions.
A) The modular group PSL(2; Z) acts on the hyperbolic plane with quotient the modular orbifold M, whose oriented closed geodesics correspond to the hyperbolic conjugacy classes in PSL(2; Z). For a field K containing Q, two matrices of PSL(2; Z) are said to be K-equivalent if they are conjugated by an element of PSL(2;K). For K = C this amounts to grouping modular geodesics of the same length. For K = Q we obtain a refinement of this equivalence relation which we will relate to genus- equivalence of binary quadratic forms, and we will give a geometrical interpretation in terms of the modular geodesics (angles at the intersection points and lengths of the ortho-geodesics).
T) The unit tangent bundle U of the modular orbifold M is a 3-dimensional manifold homeomorphic to the complement of trefoil in the sphere. The modular knots in U are the periodic orbits for the geodesic flow, lifts of the closed oriented geodesics in M , and also correspond to the hyperbolic conjugacy classes in PSL(2; Z). Their linking number with the trefoil is well understood as it has been identified by E. Ghys with the Rademacher cocycle. We are interested in the linking numbers between two modular knots. We will show that the linking number with a modular knot minus that with its inverse yields a quasicharacter on the modular group, and how to extract a free basis out of these. For this we prove that the linking pairing is non degenerate. We will also associate to a pair of modular knots a function defined on the character variety of PSL(2;Z), whose limit at the boundary recovers their linking number.
Francois-Henry Rouet, Ansys
Element-by-element preconditioners were an active area of research in the 80s and 90s, and they found some success for problems arising from Finite Element discretizations, in particular in structural mechanics and fluid dynamics (e.g., the "EBE" preconditioner of Hughes, Levit, and Winget). Here we consider problems arising from Boundary Element Methods, in particular the discretization of Maxwell's equations in electromagnetism. The matrix comes from a collection of elemental matrices defined over all pairs of elements in the problem and is therefore dense. Inspired by the EBE idea, we select subsets of elemental matrices to define different sparse preconditioners that we can factor with a direct method. Furthermore, the input matrix is rank-structured ("data sparse") and is compressed to accelerate the matrix-vector products. We use the Block Low-Rank approach (BLR). In the BLR approach, a given dense matrix (or submatrix, in the sparse case) is partitioned into blocks following a simple, flat tiling; off-diagonal blocks are compressed into low-rank form using a rank-revealing factorization, which reduces storage and the cost of operating with the matrix. We demonstrate results for industrial problems coming from the LS-DYNA multiphysics software.
Joint work with Cleve Ashcraft and Pierre L'Eplattenier
There are no conferences this week.