Speaker: Howen Chuah
Title: The free Boundary in a higher dimensional Long-Range Segregation Model
Abstract: We consider a system of elliptic equations, depending on a small parameter epsilon, that models long-range segregation of populations. The diffusion is governed by the Laplacian. This system was previously investigated by Caffarelli, Patrizi, and Quitalo as a model in population dynamics, and they established the regularity of the free boundary in two dimensions. In this paper we study the free boundary in the higher dimensional case. We extend the concept of angles and asymptotic cones to higher dimensions, and give a characterization of regular and singular points in terms of their densities and angles. We obtain a structure result of the free boundary and show that, if the angles at the singular points are away from n\omega_n/2, the regular set is open in the free boundary and locally a C^1 manifold of dimension n-1. We also show that, if the supports of the populations are convex, they are convex polytopes. A weak form of the equality of angles for the convex configuration is also derived.
Speaker: General Ozochiawaeze
Title: A Factorization Method for Clamped Obstacles from Near-Field Data
Abstract: This talk will address the inverse problem of recovering a clamped obstacle in a thin elastic plate from multi-static near-field measurements. Unlike the classical acoustic setting, the biharmonic plate model with clamped boundary conditions leads to a more complicated inverse problem: unique recovery requires measurements generated by both point-source and dipole incident fields. We develop a qualitative, non-iterative factorization method based on a new factorization of the associated near-field operator and a far-field transformation that converts the near-field operator into one amenable to the factorization framework. This transformation can be precomputed independently of the unknown obstacle. The resulting method provides a direct imaging procedure for reconstructing the unknown obstacle, with some numerical experiments provided to demonstrate its effectiveness.
Speaker: Patrick Carper
Title: If You Give A Random Walk A Cookie...
Abstract: Cookie random walks are a type of self-interacting random walk. In the most basic model, an (M,p)-cookie random walk, M cookies of strength p are placed at each site on Z. When a random walker reaches a site with a cookie, he steps right with probability p>1/2 and eats the cookie there. At sites with no cookies, the walker steps left and right with equal probability. Pinsky proposed a variation in which only one cookie is placed at each site, but the random walker only eats it after stepping left from that site. He was able to compute the speed of this walk for small values of p and sufficiently large values of p using martingale arguments. To close the gap he left open, we study a certain branching process. We calculate the mean of this process's stationary distribution, obtaining formulae for the speed of both Pinsky's random walk and other related walks as a consequence. For a further application of our methods, we obtain an explicit (but not particularly attractive) formula for the speed of (3,p)-cookie random walk. We discuss how beautifying this formula is equivalent to establishing an asymptotic expansion of several orders for certain generalized Hypergeometric-like functions.
No Talk
Speaker: Nick Gismondi
Title: A Sharp Flexibility/Rigidity Result for the Stationary Isotropic Landau Equation
Abstract: In this talk I will discuss a sharp rigidity/flexibility threshold for stationary solutions of the Krieger--Strain equation, an isotropic model of the Landau--Coulomb equation. More precisely, there are nontrivial wild solutions lying in L^p(R^3), p < 6/5 , while solutions lying in L^{6/5}(R^3) must be identically zero. These results are the first of their kind in kinetic theory. This talk is based on joint work with Will Golding and Matt Novack.
Fall Break - No Talk
No Talk
Speaker: Shantanu Deodhar
Title: L^p Integrability for manifolds of higher codimension and application to uniqueness of homogeneous PDEs
Abstract: Agranovsky and Narayanan proved that if a function f ∈ L^p (R^n) for p ≤ 2n/d has its Fourier transform supported on a d-dimensional submanifold, then f = 0. We show that the exponent p is governed by a quantitative spectral complexity parameter, the Fourier ratio, in addition to the geometric size of the Fourier support. We explore manifolds of higher codimension; In particular, we estimate p for quadratic manifolds depending upon the singularities of the joint Hessian. We further establish a general improvement for manifolds of higher codimension using certain incidence tube arguments. As an application, we present an L^p uniqueness theorem for a class of homogeneous PDEs, incorporating curvature information with the aid of the associated Fourier ratio
Speaker: Dominic Cordova
Title: When Matrices Don’t Commute: Quadratic Pseudospectra and the Spectral Localizer
Abstract: Commuting Hermitian matrices can be diagonalized simultaneously, and their joint spectrum consists of the eigenvalue tuples shared by common eigenvectors. When commutativity fails, no common eigenvectors exist, but one can still ask how close a single vector can come to being an approximate eigenvector for all of the matrices at once. This expository talk introduces two answers to that question. The quadratic pseudospectrum, built from \sum_j (A_j-\lambda_j)^2, measures the smallest possible combined eigen-error and is Lipschitz and stable under distant perturbations. The spectral localizer, a Dirac-type operator built from Clifford matrices, can have a sphere or a higher-genus surface as its spectrum even for small matrices. We will see that the two agree up to the size of the commutators, but only the localizer carries an index, its signature, which can certify that its spectrum is not a finite set.
No talk
Speaker: Ataleshvara Bhargava
Title:
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Thanksgiving Week - No Talk
Speaker: Hudson Bouw
Title:
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Speaker: Howen Chuah
Title: The Regularity of The Free Boundary of a Nonlinear Long Range Segregation Model in Dimension 2
Abstract: We study a system of fully nonlinear elliptic equations, depending on a small parameter \eps, that models long-range segregation of populations.
The diffusion is governed by the negative Pucci operator. This system was previously investigated by the authors as a model in population dynamics. In this work, we continue the study of the regularity of the free boundary in the two-dimensional case. We prove the sharp characterization of the interface by a viscosity argument. Moreover, we also prove that the regular set is open and C^1, while the singular set is discrete. Moreover, it is also shown whenever there are two free boundary points that are of distance \sizeofboundary from each other, the corresponding angles are equal. The talk is based on the joint work with Prof. Monica Torres and Prof. Stefania Patrizi.