# Mathematical Physics and Harmonic Analysis Seminar

## Fall 2017

Date: | September 1, 2017 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Selim Sukhtaiev, Rice University |

Title: | The Maslov index and the spectra of second order elliptic operators |

Abstract: | In this talk I will discuss a formula relating the spectral flow of the one-parameter families of second order elliptic operators to the Maslov index, the topological invariant counting the signed number of conjugate points of certain paths of Lagrangian planes. In addition, I will present formulas expressing the Morse index, the number of negative eigenvalues, in terms of the Maslov index for several classes of second order differential operators. The talk is based on joint work with Yuri Latushkin. |

Date: | September 8, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Boris Hanin, Texas A&M University |

Title: | Deep Learning: Approximation Theory, Convexity, and the Expressivity of Depth in Neural Networks |

Abstract: | Deep learning (DL) is the analysis and application of a class of algorithms that in the past few years have become state of the art in a huge number of machine learning problems: playing Go, image recognition/segmentation, machine transcription/translation, to name a few. While DL works incredibly well in practice, a robust mathematical theory of why it works is still in its infancy. The purpose of this talk is to introduce one aspect of this subject. Namely, the expressive power - the ability of a approximate a rich class of functions - of deep neural nets. Some of the first theorems about this were proved in the late 80's and early 90's. I will review them, talk about some more recent work, and point to a few open questions. This talk is based in part on ongoing joint work with Mark Sellke. |

Date: | September 15, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | In-Jee Jeong, Princeton University |

Title: | Evolution of singular vortex patches |

Abstract: | A vortex patch is a solution to the 2D Euler equations whose vorticity is given by the characteristic function of a domain in the plane which evolves in time. In the 90s it was shown by Chemin, Bertozzi-Constantin, and Serfati that if the boundary of the domain is initially smooth (at least C^{1,\alpha} for \alpha > 0), then this smoothness propagates for all time. Much less is known for patches supported on domains with not so smooth boundaries, for example when the domain is initially a polygon. In this work, we show global well-posedness for vortex patches with corners when there is a certain rotational symmetry. We also prove some ill-posedness results in the absense of symmetries. This is joint work with Tarek M. Elgindi. |

Date: | September 22, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Thomas Beck, MIT |

Title: | Ground state eigenfunctions on convex domains of high eccentricity |

Abstract: | In this talk, I will discuss the ground state eigenfunction of a class of Schrödinger operators on a convex planar domain. We will see how to construct two length scales and an orientation of the domain defined in terms of eigenvalues of associated differential operators. These length scales will determine the shape of the intermediate level sets of the eigenfunction, and as an application allow us to deduce properties of the first Dirichlet eigenfunction of the Laplacian for a class of three dimensional convex domains. In the two dimensional case, with constant potential, we will see that the eigenfunction satisfies a quantitative concavity property in a level set around its maximum, consistent with the shape of its intermediate level sets. |

Date: | September 29, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Gerardo Mendoza, Temple University |

Title: | Free real actions, invariant CR structures, hypoellipticity, and Kodaira's vanishing theorem (joint with Several Complex Variables Seminar) |

Abstract: | Suppose M is a compact CR manifold with a nowhere vanishing real transverse vector field T that preserves the structure and admits an invariant metric which is Hermitian on the CR structure. Then -iT commutes with the Laplacians of the dee-bar complex and defines a selfadjoint operator on the kernel, H^q, of the Laplacian in each degree q with discrete spectrum without finite points of accumulation. Assuming non-degeneracy of the Levi form, we prove that only finitely many eigenvalues of -iT lie on the positive (or negative, depending on q and the signature of the Levi form) real axis. Finiteness of spectrum on one side or the other of 0 is strongly related to Kodaira's vanishing theorem. |

Date: | October 6, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Tom Vogel, Texas A&M University |

Title: | Stability of Delaunay Surface Solutions to Capillary Problems |

Abstract: | Capillary surfaces arise from minimizing surface energy subject to a volume constraint. Capillary problems with rotational symmetry often have rotationally symmetric surfaces of constant mean curvature (Delaunay surfaces) as stationary solutions. Using eigenvalue methods, stability of Delaunay surfaces solutions can be compared across different capillary problems. In particular, the problem of a liquid bridge between parallel planes has stability implications for bridges between solid balls and toroidal drops inside circular cylinders. |

Date: | October 13, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Dmitri Pelinovsky, McMaster University |

Title: | Nonlinear Schrodinger equation on the periodic graph |

Abstract: | With a multiple scaling expansion, an effective amplitude equation can be derived for an oscillating wave packet. Using Bloch wave analysis and energy methods, we estimate the distance between the macroscopic approximation which is obtained via the amplitude equation and true solutions of the NLS equation on the periodic metric graph. These approximations are discussed in the context of bifurcations of standing localized waves on the periodic metric graphs. This work is joint with Guido Schneider (Stuttgart). |

Date: | October 20, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Wen Feng, University of Kansas |

Title: | Stability of Vortex solitons for n-dimensional focusing NLS |

Date: | October 27, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Yulia Meshkova, Chebyshev Laboratory, St.Petersburg State University |

Title: | Homogenization of periodic hyperbolic systems with the corrector |

Date: | November 10, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Luan T. Hoang, Texas Tech University |

Title: | Large-time asymptotic expansions for solutions of Navier-Stokes equations (Joint with Nonlinear PDEs Seminar) |

Abstract: | We study the long-time behavior of solutions to the three-dimensional Navier-Stokes equations of viscous, incompressible fluids with periodic boundary conditions. The body forces decay in time either exponentially or algebraically. We establish the asymptotic expansions of Foias-Saut-type for all Leray-Hopf weak solutions. If the force has an asymptotic expansion, as time tends to infinity, in terms of exponential functions or negative-power functions, then any weak solution admits an asymptotic expansion of the same type. Moreover, when the force's expansion holds in Gevrey spaces, which have much stronger norms than the Sobolev spaces, then so does the solution's expansion. This extends the previous results of Foias and Saut in Sobolev spaces for the case of potential forces. This talk is based on joint research projects with Dat Cao (Texas Tech University) and Vincent Martinez (Tulane University). |

Date: | November 17, 2017 |

Time: | 1:50pm |

Location: | BLOC 220 |

Speaker: | Fritz Gesztesy, Baylor University |

Title: | On factorizations of differential operators and Hardy-Rellich-type inequalities |

Abstract: | We will illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, using this factorization method, we will derive a general (and, apparently, new) inequality and demonstrate how particular choices of the parameters contained in this inequality yield well-known inequalities, such as the classical Hardy and Rellich inequalities as special cases. Actually, other special cases yield additional and apparently less well-known inequalities. We will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators. This is based on various joint work with Lance Littlejohn, I. Michael, M. Pang, and R. Wellman. |

Date: | November 17, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Maxim Zinchenko, University of New Mexico |

Title: | Chebyshev Polynomials on Subsets of the Real Line |

Abstract: | Chebyshev polynomials are the unique monic polynomials that minimize the sup-norm on a given compact set. These polynomials have important applications in approximation theory and numerical analysis. H. Widom in his 1969 influential work initiated a study of Szego-type asymptotics of Chebyshev polynomials on compact sets given by finite unions of disjoint arcs in the complex plane. He obtained several partial results on the norm and pointwise asymptotics of the polynomials and made several long lasting conjectures. In this talk I will present some of the classical results on Chebyshev polynomials as well as recent progress on Widom's conjecture on the large n asymptotics of Chebyshev polynomials on finite and infinite gap subsets of the real line. Based on Asymptotics of Chebyshev Polynomials, I. Subsets of R with J. Christiansen and B. Simon. Invent. Math. 208 (2017), 217-245, and Asymptotics of Chebyshev Polynomials, II. DCT Subsets of R with J. Christiansen, B. Simon, and P. Yuditskii (preprint arXiv:1709.06707). |

Date: | November 28, 2017 |

Time: | 4:00pm |

Location: | BLOC 624 |

Speaker: | Semyon Dyatlov, MIT/Berkeley |

Title: | Lower bounds on eigenfunctions on hyperbolic surfaces |

Abstract: | I show that on a compact hyperbolic surface, the mass of an $L^2$-normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassical version, has many applications including control for the Schr\"odinger equation and the full support property for semiclassical defect measures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain. |

Date: | December 1, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Mahmood Ettehad, Texas A&M University |

Title: | Network graph reconstruction from path correlations (joint with Inverse Problems Seminar) |

Abstract: | Routing network can be modeled by a connected graph with a set of boundary vertices where the measurements can be taken. Among the possible measurements are the transmission times between any pair of boundary vertices. Furthermore, it has been observed that the correlation between transmission times along two paths can be used as a proxy for the length of the intersection of the paths. The aim is to use this information to solve the problem of network tomography (reconstruct the structure of the entire network together with the length of all links). Mathematically, given an edge-weighted graph we can measure the weight of the segment common to any two paths A-Z and B-Z, where A, B and Z are boundary vertices. We will present a necessary and sufficient condition for the graph to be reconstructable from this information, and will describe the reconstruction algorithm. Based on a joint work with Gregory Berkolaiko and Nick Duffield (ECE TAMU). |

Date: | December 8, 2017 |

Time: | 1:50pm |

Location: | BLOC 628 |

Speaker: | Naser Talebi Zadeh, University of Wisconsin-Madison |

Title: | QUANTUM CHAOS ON RANDOM CAYLEY GRAPHS OF SL2 [Z/pZ] |

Abstract: | We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL2[Z/pZ] as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs i.e. the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of SL2[Z/pZ] and the explicit LPS Ramanujan projective graphs of P1(Z/pZ) have optimal spectral gap and diameter as the prime number p goes to infinity. |