Department of Mathematics

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Abstracts

Week of January 22, 2023

January 23, 2023
Colloquium	Projection theorems and Fourier restriction theory	4:00pm - LOM 206	Given a fractal set E on the plane and a set F of directions, can we find one direction L in F such that the orthogonal projection of E onto L is large? Suppose f is a function whose Fourier transform is supported on a curved manifold (for instance, a sphere), what can we say about this function? It turns out that these two questions are related. We will survey some classical and recent projection theorems and discuss their relation to Fourier restriction theory.

January 24, 2023
Geometry & Topology	Hitchin representations and minimal surfaces	4:15pm - LOM 206	Abstract: Labourie proved that every Hitchin representation into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3) and explained that if true, then the Hitchin component admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmüller space. After giving the relevant background, we will explain that Labourie’s conjecture fails for n at least 4, and point to some future questions.

January 25, 2023
Applied Mathematics	An integral equation method for the advection-diffusion equation on time-dependent domains in the plane	1:00pm - AKW 200	Boundary integral methods are attractive for solving homogeneous elliptic partial differential equations on complicated geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy.
Colloquium	Growth of unimodular random graphs	4:15pm - LOM 214	Abstract: Unimodular random graphs are probabilistic objects arising, for example, as the limits of sequences of finite graphs or as the connected components of a percolation on a transitive graph. In general, a unimodular random graph might fail to have an exponential growth rate but for unimodular random trees there is a curious dichotomy where the growth can be shown to exist once the “upper growth” passes certain threshold. Based on a joint work with Miklos Abert and Ben Hayes.

January 26, 2023
Group Actions, Geometry and Dynamics	Co-spectral radii and subgroup intersections,	4:00pm -	A subgroup H of a countable group G is co-amenable if the left regular representation on the coset space G/H admits almost invariant vectors. Co-amenablility is a notion of largeness of a subgroup, but it is not the best behaved one. For example, the intersections of co-amenable subgroups can fail to be co-amenable. I will talk about a joint work with Wouter van Limbeek in which we prove that the class of co-amenable invariant random subgroups is closed under taking finite intersection. This follows from more general results on the co-spectral radii of intersections of invariant random subgroups.
Analysis	Exploring the power of nonlinearity in complex function theory (in 1 d and higher dimensions)	4:15pm -	We describe recent remarkable nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It is known that all harmonic functions in higher dimensions are combinations of holomorphic functions on 2 dimensional planes, extended as, constant in normal directions. We derive representation theorems, with corresponding isometries, opening the door for applications in higher dimensions, to the processing of highly oscillatory multidimensional signals. This is joint work with Guido Weiss Stefan Steinerberger , Jacques Peyriere , Hau -tieng Wu and many others

January 27, 2023
Friday Morning Seminar	Friday Morning Seminar	9:30am - LOM 215	A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!
Graduate Student Seminar	Symplectic representations of mapping class groups restricted on (orientable) pseudo-Anosovs	12:00pm -	It is an open question whether mapping class groups of a closed orientable surface is linear. One natural representation of a mapping class group into a linear group is its symplectic representation, which is induced from the action of the mapping class group on the first homology of the surface. It is well-known that the symplectic representation is surjective onto the integral lattice. It is a natural question that what we can observe when we restrict the symplectic representation on the set of pseudo-Anosov mapping classes. Thurston proved that there are plenty of pseudo-Anosovs in the kernel of the symplectic representation, employing his construction of pseudo-Anosovs from filling multicurves. In this talk, we show that the surjectivity still holds after restricting the symplectic representation on the set of pseudo-Anosovs. On the other hand, we also show that the surjectivity does not hold on the set of pseudo-Anosovs with orientable invariant measured foliations. This is the joint work with Hyungryul Baik and Inhyeok Choi, answering the question of Ursula Hamenstädt.
Geometric Analysis and Application	Hawking mass monotonicity for initial data sets	2:00pm - LOM 215	An interesting feature of General Relativity is the presence of singularities which can happen in even the simplest examples such as the Schwarzschild spacetime. However, in this case the singularity is cloaked behind the event horizon of the black hole which has been conjectured to be generically the case. To analyze this so-called Cosmic Censorship Conjecture Penrose proposed in 1973 a test which involves Hawking’s area theorem, the final state conjecture and a geometric inequality on initial data sets (M,g,k). For k=0 this Penrose inequality has been proven by Huisken-Ilmanen and by Bray using different methods, but in general the question is wide open. Huisken-Ilmanen’s proof relies on the Hawking mass monotonicity formula under inverse mean curvature flow (IMCF), and the purpose of this talk is to generalize the Hawking mass monotonicity formula to initial data sets. For this purpose, we start with recalling spacetime harmonic functions and their applications which have been introduced together with Demetre Kazaras and Marcus Khuri in the context of the spacetime positive mass theorem.