Published on Department of Mathematics Calendar (https://calendar.math.yale.edu)

Week of October 1, 2023

October 4, 2023
Colloquium [3]	Limit measures for topologically and geometrically random surfaces	Jeremy Kahn - Brown University	4:00pm - KT219

October 5, 2023
Analysis [4]	Nodal count and Morse theory of magnetic operators on graphs.	Lior Alon - M.I.T	4:00pm - KT 219
Group Actions, Geometry and Dynamics [5]	Patterson--Sullivan theory and equidistribution in Hilbert geometry	Pierre-Louis Blayac - University of Michigan	4:00pm - KT801
Graduate Learning Seminar [6]	Learning seminar on D-modules	Soumik Ghosh - Yale	4:00pm - Prospect 204, B-02

October 6, 2023
Friday Morning Seminar [7]	Friday Morning Seminar		10:00am - 809 Commons
Geometric Analysis and Application [8]	Recent Developments in Constant Mean Curvature Hypersurfaces	Liam Mazurowski - Cornell University	2:00pm - KT 906

October 11, 2023
Applied Mathematics [9]	The mean first passage time as a natural diffusion distance	Maxim Goldberg - Ramapo College	3:00pm - LOM 214

October 12, 2023
Graduate Learning Seminar [6]	Learning seminar on D-modules	David Bai - Yale	4:00pm - KT 801
Analysis [4]	The nonlinear stochastic heat equation in the critical dimension	Alexander Dunlap - Duke University	4:00pm -

October 13, 2023
Friday Morning Seminar [7]	Friday Morning Seminar		10:00am - KT 801
Geometric Analysis and Application [8]	Geometric Flows and Nearly-Parallel G2-Structures	Aaron Kennon - UC Santa Barbara	2:00pm - KT 906

October 16, 2023
Group Actions, Geometry and Dynamics [5]	The space of traces of certain discrete groups	Raz Slutsky - Weizmann Institute of Science	4:00pm - KT 801

October 20, 2023
Friday Morning Seminar [7]	Friday Morning Seminar		10:00am - KT 801

October 25, 2023
Applied Mathematics [9]	Solving partial differential equations exactly over polynomials	Keaton Burns - MIT	3:00pm - LOM 214

October 26, 2023
	Learning seminar on D-modules	Mengwei Hu - Yale	4:00pm - KT 801
Group Actions, Geometry and Dynamics [5]	Classification of Horocycle Orbit Closures in Z-covers	Or Landesberg - Yale University	4:00pm - KT205
Analysis [4]	Quantitative stability of traveling waves	Chris Henderson -	4:00pm - KT 219

October 27, 2023
Friday Morning Seminar [7]	The optimal paper Moebius band	Richard Schwartz - Brown University	10:00am - KT 801
Geometric Analysis and Application [8]	Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds	Longzhi Lin - UC Santa Cruz	2:00pm - KT 906

Abstracts

Week of October 1, 2023

October 4, 2023
Colloquium [3]	Limit measures for topologically and geometrically random surfaces	4:00pm - KT219	An immersed surface S in a Riemannian manifold M induces a probability measure on the space G_2(M) of two-planes in the tangent bundle of M. If M is a hyperbolic 3-manifold and S_n is a sequence of surfaces with principal curvatures going to zero, then any weak* limit of their induced measures is a convex combination of the Liouville (equidistributed) measure on G_2(M), and measures that come from immersed totally geodesic surfaces in M. We consider two ways of generating a “random” nearly geodesic surface in M, one by bounding the genus, and the other by bounding the area. We show that limits of the measures in the former case must come exclusively from the totally geodesic surfaces (if there are any in M), while limits in the latter case must have some portion that is equidistributed. This is joint work with V. Markovic and I. Smilga.

October 5, 2023
Analysis [4]	Nodal count and Morse theory of magnetic operators on graphs.	4:00pm - KT 219	Our intuition regarding waves suggests that the k-th eigenvector of a graph Laplacian L (or any Discrete Schrodinger operator) should exhibit greater fluctuations as k increases. In this context, the "nodal count" is the number of edges on which the eigenvector changes sign. The works of Fiedler (1975) and Berkolaiko (2007) show that the nodal count is bounded between k-1 and k-1+b, where b is the first Betti number of the graph. We establish that these bounds hold for signed graphs as well when considering sign changes accordingly. The “nodal surplus”, the deviation from k-1, is expected to concentrate around b/2. Numerical observations indicate that the distribution of the nodal surplus, across all eigenvectors and different signings of the graph, resembles a Gaussian distribution centered at b/2, regardless of the graph's characteristics. We prove that it is precisely binomial with mean b/2 in the case of operators on complete graphs with sufficiently high potential. This outcome, among others, stems from a noteworthy relationship. The magnetic perturbations of L are achieved by multiplying the off-diagonal entries of L by phases (in a Hermitian manner), modulo gauge invariance. The eigenvalues of L extend to piece-wise analytic functions of the phases. At non-degenerate critical points, the Morse index is equal to the associated nodal surplus. If time permits, I will draw the line connecting this work to spectral gaps of periodic operators. This talk is based on joint works with Mark Goresky and John Urschel
Group Actions, Geometry and Dynamics [5]	Patterson--Sullivan theory and equidistribution in Hilbert geometry	4:00pm - KT801	In the theory of discrete subgroups of Lie groups, given a length function on the Lie group G, one popular object of study is the asymptotic when R goes to infinity of the number of loxodromic elements with length less R in a discrete subgroup H, and the distribution of the fixed points of these loxodromic elements. When G is the group of (real) projective transformations and H acts properly discontinuously and cocompactly on a strictly convex domain O of the projective space, Yves Benoist noticed that one can use the Hilbert metric of O, and the associated Hilbert geodesic flow, to estimate the above counting function, for a special length function called the Hilbert length. An important ingredient in Benoist’s result is the uniform hyperbolicity of the Hilbert geodesic flow, which does not hold when O is not strictly convex. In this talk we will develop and use instead the theory of Patterson–Sullivan measures for Hilbert geometries O/H that satisfy a mild rank-one assumption, and are not necessarily strictly convex nor compact. This will yield counting results for H in the cases where the induced Bowen–Margulis measure is finite. (This is joint work with Feng Zhu).
Graduate Learning Seminar [6]	Learning seminar on D-modules	4:00pm - Prospect 204, B-02	This is the fourth lecture in the seminar.

October 6, 2023
Friday Morning Seminar [7]	Friday Morning Seminar	10:00am - 809 Commons	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!
Geometric Analysis and Application [8]	Recent Developments in Constant Mean Curvature Hypersurfaces	2:00pm - KT 906	A constant mean curvature surface is a critical point of the area functional subject to a volume constraint. Min-max theory is a powerful method for finding saddle type critical points of functionals. Recently, Xin Zhou and Jonathan Zhu developed a min-max theory for finding constant mean curvature surfaces in closed manifolds. In this talk, I will discuss some recent results in the min-max theory of constant mean curvature hypersurfaces. In particular, I will discuss an extension of the CMC min-max theory to certain non-compact manifolds. I will also discuss joint work with Xin Zhou on min-max theory with a volume constraint.

October 11, 2023
Applied Mathematics [9]	The mean first passage time as a natural diffusion distance	3:00pm - LOM 214	Given an irreducible finite Markov chain, we propose the mean first passage time (mfpt) as a diffusion distance. We motivate this definition by considering a compact Riemannian manifold, and the submanifold resulting from removing the closure of a small ball. The steady-state solution to an associated inhomogeneous heat flow problem on the submanifold is non-negative and can be viewed as having large values at locations which are far away from the removed ball. The same function is shown to give the expected value of the first hitting time of the removed ball from any location in the submanifold. The latter interpretation leads to our proposing the mfpt as a diffusion distance for a given finite set of states (samples) and an associated transition matrix. Even if the transition matrix does not arise from heat flow and may in fact be non-symmetric and non-bistochastic, we note that the mfpt satisfies the triangle inequality. Moreover, various efficient ways to compute the mfpt have been proposed in the literature. Zoom link: https://yale.zoom.us/j/99114648888 [12]

October 12, 2023
Graduate Learning Seminar [6]	Learning seminar on D-modules	4:00pm - KT 801	This is a fifth meeting of the seminar.
Analysis [4]	The nonlinear stochastic heat equation in the critical dimension	4:00pm -	I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on joint work with Cole Graham and Yu Gu.

October 13, 2023
Friday Morning Seminar [7]	Friday Morning Seminar	10:00am - KT 801	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!
Geometric Analysis and Application [8]	Geometric Flows and Nearly-Parallel G2-Structures	2:00pm - KT 906	A 3-Sasakian structure on a 7-manifold may be used to define two distinct Einstein metrics: the 3-Sasakian metric and the squashed Einstein metric. Both metrics are induced by nearly parallel G2-structures which may also be expressed in terms of the 3-Sasakian structure. Just as Einstein metrics are critical points for the Ricci flow up to rescaling, nearly parallel G2-structures provide natural critical points of the (rescaled) geometric flows of G2-structures known as the Laplacian flow and Laplacian coflow. We study each of these flows in the 3-Sasakian setting and see that their behaviour is markedly different, particularly regarding the stability of the nearly parallel G2-structures. We also compare the behaviour of the flows of G2-structures with the (rescaled) Ricci flow. This was joint work with Jason Lotay

October 16, 2023
Group Actions, Geometry and Dynamics [5]	The space of traces of certain discrete groups	4:00pm - KT 801	A trace on a group is a positive-definite conjugation-invariant function on it. These functions play an important role in harmonic analysis of discrete groups, and their study has found many exciting connections to rigidity, stability, and dynamics in the past couple of decades. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will then see the different behaviours of these spaces for free groups vs. higher-rank lattices. Finally, some open questions about free products and surface groups will be presented. This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.

October 20, 2023
Friday Morning Seminar [7]	Friday Morning Seminar	10:00am - KT 801	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!

October 25, 2023
Applied Mathematics [9]	Solving partial differential equations exactly over polynomials	3:00pm - LOM 214	Numerical simulations of partial differential equations (PDEs) are indispensable across science and engineering. For simple geometries, spectral methods are a powerful class of techniques that produce exceptionally accurate solutions for wide ranges of equations. But many variations of these methods exist, each with distinct properties and performance, and developing the best method for a complex nonlinear problem is often quite challenging. In this context, we present a framework that unifies all polynomial and trigonometric spectral methods, from classical “collocation” to the more recent “ultraspherical” schemes. In particular, we examine the exact discrete equations solved by each method and characterize their deviation from the original PDE in terms of perturbations called “tau corrections”. By analyzing these corrections, we can precisely categorize existing methods and design new solvers that robustly accommodate new boundary conditions, eliminate spurious numerical modes, and satisfy exact conservation laws. This approach conceptually separates what discrete model a spectral scheme solves from how it solves it. This separation provides much more freedom when building and optimizing new numerical models. We will illustrate these advantages with some examples from fluid dynamics using Dedalus, an open-source package for solving PDEs with modern spectral methods.

October 26, 2023
	Learning seminar on D-modules	4:00pm - KT 801	This is the sixth talk in the seminar.
Group Actions, Geometry and Dynamics [5]	Classification of Horocycle Orbit Closures in Z-covers	4:00pm - KT205	Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We study Z-covers of compact hyperbolic surfaces and provide the first description of all possible horocycle orbit closures in this category. Surprisingly, the topology and Hausdorff dimension of these non-homogeneous orbit closures delicately and discontinuously depends on the choice of a hyperbolic metric on the covered compact surface. Nevertheless, some rigidity is preserved in the form of integer Hausdorff dimension of all orbit closures. Based on an ongoing series of works together with James Farre and Yair Minsky.
Analysis [4]	Quantitative stability of traveling waves	4:00pm - KT 219	In their original paper, Kolmogorov, Petrovsky, and Piskunov demonstrated stability of the minimal speed traveling wave with an ingenious compactness argument based on, roughly, the decreasing "steepness" of the profile. This proof is extremely flexible, yet entirely not quantitative. On the other hand, more modern PDE proofs of this fact for general reaction-diffusion equations are highly tailored to the particular equation, fairly complicated, and often not sharp in the rate of convergence. In this talk, which will be elementary and self-contained, I will introduce a natural if "hidden" quantity, the shape defect function, that allows a simple approach to quantifying convergence to the traveling wave for a large class of reaction-diffusion equations. This is a joint work with Jing An and Lenya Ryzhik.

October 27, 2023
Friday Morning Seminar [7]	The optimal paper Moebius band	10:00am - KT 801	In this talk I will explain my recent solution of the Halpern-Weaver Conjecture. The result is that a strip of paper that is 1 unit wide must be more than sqrt(3) units long in order for it to be smoothly folded into a paper Moebius band, and this bound is sharp. The proof is elemetary enough that I can explain the whole thing during the talk. I’ll also talk about lots of related unsolved problems.
Geometric Analysis and Application [8]	Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds	2:00pm - KT 906	Abstract: For a long time, there has been a folklore conjecture, often attributed to Thurston, which asserts that every almost Fuchsian manifold is foliated by closed incompressible constant mean curvature (CMC) surfaces. In this talk I will discuss our recent work using the modified mean curvature flow to prove the existence of closed incompressible surfaces of constant mean curvature in a certain class of almost Fuchsian manifolds. As an application, we confirm this CMC foliation conjecture for such a class of almost Fuchsian manifolds. This is joint work with Zheng Huang and Zhou Zhang.

Links
[1] https://calendar.math.yale.edu/list/calendar/grid/week/2023-W39 [2] https://calendar.math.yale.edu/list/calendar/grid/week/2023-W41 [3] https://calendar.math.yale.edu/seminars/colloquium [4] https://calendar.math.yale.edu/seminars/analysis [5] https://calendar.math.yale.edu/seminars/group-actions-geometry-and-dynamics [6] https://calendar.math.yale.edu/seminars/graduate-learning-seminar [7] https://calendar.math.yale.edu/seminars/friday-morning-seminar [8] https://calendar.math.yale.edu/seminars/geometric-analysis-and-application [9] https://calendar.math.yale.edu/seminars/applied-mathematics [10] https://calendar.math.yale.edu/list/calendar/grid/week/abstract/2023-W39 [11] https://calendar.math.yale.edu/list/calendar/grid/week/abstract/2023-W41 [12] https://yale.zoom.us/j/99114648888