## Montreal Analysis Seminar

Seminars are usually held on Mondays or Fridays at Concordia, McGill or Universite de Montreal

## SUMMER 2016

Thursday, June 2, 13:30-14:30, Concordia, Room LB 921-04
Stability results for sections of convex bodies
Abstract: (pdf) Let $K$ be a convex body in $\mathbb R^n$. The parallel section function of $K$ in the direction $\xi\in S^{n-1}$ is defined by $$A_{K,\xi}(t)=\mathrm{vol}_{n-1}(K\cap \{\xi^{\perp}+t\xi\}), \quad t\in \mathbb R.$$ If $K$ is origin-symmetric (i.e. $K=-K$), then Brunn's theorem implies $$A_{K,\xi}(0) =\max_{t\in \mathbb R} A_{K,\xi}(t)$$ for all $\xi\in S^{n-1}$. The converse statement was proved by Makai, Martini and \'Odor. Namely, if $A_{K,\xi}(0) =\max_{t\in \mathbb R} A_{K,\xi}(t)$ for all $\xi\in S^{n-1}$, then $K$ is origin-symmetric. We provide a stability version of this result. If $A_{K,\xi}(0)$ is close to $\max_{t\in \mathbb R} A_{K,\xi}(t)$ for all $\xi\in S^{n-1}$, then $K$ is close to $-K$. Joint work with Matthew Stephen.
Monday, August 1, 13:30-14:30, McGill, Burnside 920 (room to be confirmed)
Fedor Pakovich (Ben Gurion University)
On semiconjugate Rational Functions
Abstract: pdf

## WINTER 2016

Friday, January 15, 13:30-14:30, Burnside 920
Yannick Bonthonneau (CRM and UQAM)
Obtaining Weyl estimates on manifolds with cusps
Abstract: I will explain how several Weyl estimates for compact manifolds can be extended to the case of manifolds with hyperbolic cusps. I will focus on the case of negative curvature to give a glimpse of the proof.
Friday, January 22, 13:30-14:30, Burnside 920
Dmitry Khavinson (University of South Florida)
Isoperimetric "sandwiches", free boundary boundary problems and approximation by analytic and harmonic functions
Abstract: The isoperimetric problem, posed by the Greeks, proposes to find among all simple closed curves the one that surrounds the largest area. The isoperimetric theorem then states that the curve is a circle. It is frst mentioned in the writings of Pappus in the third century A.D. and is attributed there to Zenodorus. Steiner in 1838 was the first to attempt a "rigorous" proof. However, first truly rigorous proofs were only achieved in the beginning of the 20th century (e.g., Caratheodory, Hurwitz, Carleman,...). We shall discuss a variety of isoperimetric inequalities, as , e.g., in Polya and Szego 1949 classics, but deal with them via a relatively novel approach based on approximation theory. Roughly speaking, this approach can be characterized by a recently coined term "isoperimetric sandwiches". A certain quantity is introduced, usually as a degree of approximation to a given simple function, e.g., \overline{z} , |x|^2, by either analytic or harmonic functions in some norm. Then, the estimates from below and above of the approximate distance are obtained in terms of simple geometric characteristics of the set, e.g., area, perimeter, capacity, torsional rigidity, etc. The resulting "sandwich" yields the relevant isoperimetric inequality. Several of the classical isoperimetric problems approached in this way lead to natural free boundary problems for PDE, many of which remain unsolved today. (An example of such free boundary problem is the problem of a shape of an electrifed droplet, or a small air bubble in fluid flow. Another example is identifying a cross-section of laminary flow of viscous fluid that exhibits constant pressure on the pipe walls, J. Serrin's problem.) I will make every effort to make most of the talk accessible to the first year graduate students, or advanced undergraduates majoring in mathematics and physics who have had a semester course in complex analysis and a routine course in advanced calculus.
Friday, January 29, 13:30-14:30, Burnside 920
Igor Khavkine (Trento, Italy)
Topology, rigid cosymmetries and linearization instability in higher gauge theories
Abstract: It is well known that some solutions of non-linear partial differential equations (PDEs), like Einstein or Yang-Mills equations, exhibit linearization instability: some linearized solutions do not extend to families of near-by non-linear solutions. Often, linearized solution fail to extend when some non-linear functional, which we refer to as a linearization obstruction, is non-zero on it. In the case of Einstein and Yang-Mills equations, such linearization obstructions are precisely related to spacetime topology, charges of linearized conservation laws and rigid symmetries of the background solution. I will describe a significant generalization this classic result. It is applicable to both elliptic and hyperbolic equations, to variational and non-variational equations, to determined systems and gauge theories, and to ordinary as well as higher gauge theories.
The talk has been cancelled due to illness; the talk has been rescheduled for Monday, February 22
Friday, February 12, 13:30-14:30, Burnside 920
Catherine Beneteau (University South Florida)
Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants
Abstract: In this talk, I will discuss some polynomials that solve a particular optimization problem in Dirichlet-type spaces. Examples of Dirichlet-type spaces include the Hardy space (analytic functions in the disk whose coefficients are square summable), the Bergman space (analytic functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (analytic functions in the disk whose image has finite area, counting multiplicity). The optimal approximants p(z) in question minimize the Dirichlet- type norm of p(z) f(z) - 1, for a given function f(z). I will examine the connections between these optimal approximants, orthogonal polynomials and reproducing kernels, and exploit these connections to describe what is currently known about the zeros. This work is joint with D. Khavinson, C. Liaw, D. Seco, and A. Sola.

Friday, February 19, 13:30-14:30, Burnside 920
Paul Hagelstein (Baylor)
Solyanik Estimates in Harmonic Analysis
Abstract: pdf. Let $\mathcal{B}$ be a collection of open sets in $\mathbb{R}^n$. Associated to $\mathcal{B}$ is the geometric maximal operator $M_{\mathcal{B}}$ defined by $$M_{\mathcal{B}}f(x) = \sup_{x \in R \in \mathcal{B}}\int_R|f|\;.$$ For $0 < \alpha < 1$, the associated \emph{Tauberian constant} $C_{\mathcal{B}}(\alpha)$ is given by $$C_{\mathcal{B}}(\alpha) = \sup_{E \subset \mathbb{R}^n : 0 < |E| < \infty} \frac{1}{|E|}|\{x \in \mathbb{R}^n : M_{\mathcal{B}}\chi_E(x) > \alpha\}|\;.$$ A maximal operator $M_\mathcal{B}$ such that $\lim_{\alpha \rightarrow 1^-}C_{\mathcal{B}}(\alpha) = 1$ is said to satisfy a \emph{Solyanik estimate}. In this talk we will prove that the uncentered Hardy-Littlewood maximal operator satisfies a Solyanik estimate. Moreover, we will indicate applications of Solyanik estimates to smoothness properties of Tauberian constants and to weighted norm inequalities. We will also discuss several fascinating open problems regarding Solyanik estimates. This research is joint with Ioannis Parissis.
Monday, February 22, 13:30-14:30, Burnside 920
The talk has been rescheduled from Friday, February 12
Catherine Beneteau (University South Florida)
Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants
Abstract: In this talk, I will discuss some polynomials that solve a particular optimization problem in Dirichlet-type spaces. Examples of Dirichlet-type spaces include the Hardy space (analytic functions in the disk whose coefficients are square summable), the Bergman space (analytic functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (analytic functions in the disk whose image has finite area, counting multiplicity). The optimal approximants p(z) in question minimize the Dirichlet- type norm of p(z) f(z) - 1, for a given function f(z). I will examine the connections between these optimal approximants, orthogonal polynomials and reproducing kernels, and exploit these connections to describe what is currently known about the zeros. This work is joint with D. Khavinson, C. Liaw, D. Seco, and A. Sola.

The talk has been cancelled
Joint Analysis/Mathematical Physics/Probability seminar
Friday, February 26, 13:30-14:30, Burnside 920
Armen Shirikyan (Cergy-Pontoise)
Global controllability to trajectories for the viscous Burgers equation and applications
Abstract: We study the problem of global controllability by an external force for the viscous Burgers equation on a bounded interval. Assuming that the force is localised in space, we prove that any non-stationary trajectory can be exponentially stabilised. We next discuss various consequences of this result, such as global exact controllability to trajectories, approximate controllability by a localised two-dimensional control, and mixing for the stochastic Burgers equation.
Special Analysis and Group Theory seminar
Thursday, March 10, 15:30-16:30, Burnside 719A
Simone Gutt (Univ Libre de Bruxells)
Completions of group algebras, growth and nuclearity
Abstract: If G is a finitely generated infinite group, we define completions A\sigma(G) of the group algebra C[G] in the space of formal power series in G, using norms which are defined using a growth function\sigma, i.e. an unbounded nowhere decreasing function \sigma: N \to [1,infty) which is submultiplicative (i.e. \sigma(n + m) <= \sigma(n)\sigma(m)) or almost submultiplicative (i.e. for every epsilon > 0, there exists a constant c > 0 such that \sigma(n + m) <= c\sigma(n)^{1+epsilon} \sigma(m)^{1+epsilon}. We show that A\sigma(G) is a Frechet-Hopf *-algebra. We relate nuclearity of such a completion to a growth property of the group. This is joint work with Michel Cahen and Stefan Waldmann.

Monday, March 14, 13:30-14:30, Burnside 920 (to be confirmed)
Daniel Ueltchi (Warwick)
From condensed matter physics to probability theory
Abstract: The basic laws governing atoms and electrons are well understood, but it is impossible to make predictions about the behaviour of large systems of condensed matter. A common approach is to introduce simple models and to use notions of statistical mechanics. I will review certain quantum spin systems such as the Heisenberg model. As it turns out, they can be represented by models of random permutations and of random loops.
Thursday, March 24, 13:30-14:30, Burnside 920
Boris Hanin (MIT)
Nodal Sets of Random Eigenfunctions of the Harmonic Oscillator
Abstract: Random eigenfunctions of energy E for the isotropic harmonic oscillator in R^d have a U(d) symmetry and are in some ways analogous to random spherical harmonics of fixed degree on S^d, whose nodal sets have been the subject of many recent studies. However, there is a fundamentally new aspect to this ensemble, namely the existence of allowed and forbidden regions. In the allowed region, the Hermite functions behave like spherical harmonics, while in the forbidden region, Hermite functions are exponentially decaying and it is unclear to what extent they oscillate and have zeros. The purpose of this talk is to present several results about the expected volume of the zero set of a random Hermite function in both the allowed and forbidden regions as well as an explicit formula for the scaling limit around the caustic of the fixed energy spectral projector for the isotropic harmonic oscillator. This is joint work with Steve Zelditch and Peng Zhou.
Friday, April 1, 13:30-14:30, Burnside 920
Malabika Pramanik (UBC)
A Roth type theorem for large subsets of multidimensional Euclidean spaces
Abstract: We prove that sets of positive upper density contain 3-term progressions of all sufficiently large gaps when the gap size is measured in certain metrics. This is known to be false in the ordinary l^2-metric. We plan to discuss the contrast between the two situations. Joint work with Brian Cook and Akos Magyar.
Friday, April 8, 13:30-14:30, Burnside 920
Jiuyi Zhu (Johns Hopkins University)
Nodal geometry of Steklov eigenfunctions
Abstract: The eigenvalue and eigenfunction problem is fundamental and essential in mathematical analysis. The Steklov problem is an eigenvalue problem with spectrum at the boundary of a compact Riemannian manifold. Recently the study of Steklov eigenfunctions has been attracting much attention. We obtain the sharp doubling inequality for Steklov eigenfunctions on the boundary and interior of manifolds using delicate Carleman estimates. As an application, the optimal vanishing order is derived, which describes quantitative behavior of strong unique continuation property. We can ask Yau's type conjecture for the Hausdorff measure of nodal sets of Steklov eigenfunctions on the boundary and interior of the manifold. I will describe some recent progress about this challenging direction. Part of work is joint with C. Sogge and X. Wang.
Friday, May 6, 13:30-14:30, Room TBA
Justin Solomon (Princeton/MIT)
Computational Spectral Geometry: Tutorial and Modern Applications
Abstract: This talk will be part-tutorial and part-research presentation. I will begin by summarizing some applications of spectral geometry, in particular the geometry of the Laplacian operator, appearing in the discrete geometry processing, computer graphics, and machine learning literatures. Using these applications as motivation, I will construct discretizations of the Laplacian suitable for calculations on triangulated surfaces, volumes bounded by a discretized surface, and point clouds. The talk will conclude with some of my own research in spectral geometry for constructing and analyzing maps between surfaces through the "functional maps" framework.
Monday, May 9, 13:30-14:30, Room TBA
Dmitry Jakobson (McGill)
On small gaps in the lngth spectrum
Abstract: This is joint work with Dmitry Dolgopyat. We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrary small gaps is topologically generic: this is established both for surfaces of constant negative curvature, and for the space of negatively curved metrics. While arbitrary small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric. We present a result in that direction.
Friday, May 13, 13:30-14:30, Room TBA
Eugenia Malinnikova (Trondheim/Purdue)
On ratios of harmonic functions Abstract: We consider pairs of harmonic functions in the unit ball of R^n with the same zero set Z and prove that the ratio is a well-defined real-analytic function that satisfies the maximum principle, the Harnack inequality and a certain gradient estimate. Some examples of such pairs will be discussed. We will show that he constants in these inequalities depend only on the zero set Z, moreover, in dimension two the dependence is only on the length of the zero set. This is a joint work with A. Logunov.

## FALL 2015

Geometric Analysis Seminar
Wednesday, September 9, 13:30-14:30, Burnside 920
M. Moller (Princeton and ICTP, Trieste)
Gluing of Solutions to Nonlinear PDEs of Mean Curvature Type

Montreal Mathematical Sciences Colloquium
Friday, September 25, UQAM - Pavillon Sherbrooke, Salle SH-3420, 16:00-17:00
D. Vassiliev (University College, London)
Analysis of first order systems of PDEs on manifolds without boundary
Abstract: In layman's terms a typical problem in this subject area is formulated as follows. Suppose that our universe has finite size but does not have a boundary. An example of such a situation would be a universe in the shape of a 3-dimensional sphere embedded in 4-dimensional Euclidean space. And imagine now that there is only one particle living in this universe, say, a massless neutrino. Then one can address a number of mathematical questions. How does the neutrino field (solution of the massless Dirac equation) propagate as a function of time? What are the eigenvalues (stationary energy levels) of the particle? Are there nontrivial (i.e. without obvious symmetries) special cases when the eigenvalues can be evaluated explicitly? What is the difference between the neutrino (positive energy) and the antineutrino (negative energy)? What is the nature of spin? Why do neutrinos propagate with the speed of light? Why are neutrinos and photons (solutions of the Maxwell system) so different and, yet, so similar? The speaker will approach the study of first order systems of PDEs from the perspective of a spectral theorist using techniques of microlocal analysis and without involving geometry or physics. However, a fascinating feature of the subject is that this purely analytic approach inevitably leads to differential geometric constructions with a strong theoretical physics flavour.
References:
[1] See items 98-101, 103 and 104 on my publications page
http://www.homepages.ucl.ac.uk/~ucahdva/publicat/publicat.html
[2] Futurama TV series, Mars University episode (1999): Fry: Hey, professor. What are you teaching this semester? Professor Hubert Farnsworth: Same thing I teach every semester. The Mathematics of Quantum Neutrino Fields. I made up the title so that no student would dare take it.
Monday, October 5, 13:30-14:30, Burnside 920
B. Ou (University of Toledo)
An equality for the geodesic curvature of certain curves on a two-dimensional Riemann surface
Abstract: We prove an equality for the geodesic curvature function of certain closed curves in a local domain of a two-dimensional Riemannian surface. We address its connection to the local Gauss-Bonnet theorem. We also show that the equality leads to a four-vertex theorem for simple and closed curves on a two-dimensional Riemannian surface with a constant Gauss curvature.
Friday, October 23 (date changed!), 13:30-14:30, Burnside 920
D. Kinzebulatov (CRM and McGill)
A new approach to the L^p theory of -Delta + b \nabla, and its applications to Feller processes with general drifts
Abstract: pdf

Friday, November 6, 13:30-14:30, Burnside 920
Thierry Daude (Cergy-Pontoise)
Non-uniqueness results for the anisotropic Calderon problem with data measured on disjoint sets.
Abstract: In this talk, we shall give some simple counterexamples to uniqueness for the Calderon problem on Riemannian manifolds with boundary when the Dirichlet and Neumann data are measured on disjoint sets of the boundary. We provide counterexamples in the case of two and three dimensional Riemannian manifolds with boundary having the topology of circular cylinders in dimension two and toric cylinders in dimension three. The construction could be easily extended to higher dimensional Riemannian manifolds. This is joint work with Niky Kamran (McGill University) and Francois Nicoleau (Universite de Nantes).
Friday, November 20, 13:30-14:30, Burnside 920
Nikolay Dimitrov (Berlin)
Discrete uniformization via hyper-ideal circle patterns
Abstract: In this talk I will present a discrete version of the classical uniformization theorem based on the theory of hyper-ideal circle patterns. It applies to surfaces represented as finite branched covers over the Riemann sphere as well as to compact polyhedral surfaces with non-positive cone singularities. The former include all Riemann surfaces realized as algebraic curves, and more generally, any closed Riemann surface with a choice of a meromorphic function on it. The latter include any closed Riemann surface with a choice of a quadratic differential on it. We show that for such surfaces discrete uniformization via hyper-ideal circle patterns always exists and is unique (up to isometry). This kind of discrete uniformization is the result of an interplay between realization theorems for ideal (Rivin) and hyper-ideal (Bao and Bonahon) polyhedra in hyperbolic three-space, and their generalization to hyper-ideal circle patterns on surfaces with cone-singularities (Schlenker). We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.
Friday, November 27, 14:30-15:30, Univ. de Montreal/CRM, Room 5448
Guillaume Roy-Fortin (Northwestern)
$L^q$ norms and nodal sets of Laplace eigenfunctions
Abstract: We will discuss a recent result that exhibits a relation between the average local growth of a Laplace eigenfunction on a compact, smooth Riemannian surface and the global size of its nodal (zero) set. More precisely, we provide a lower and an upper bound for the Hausdorff measure of the nodal set in terms of the average of the growth exponents of an eigenfunction on disks of small radius. Combined with Yau's conjecture and the work of Donnelly-Fefferman, the result implies that the average local growth of eigenfunctions on an analytic manifold with analytic metric is bounded by constants in the semi-classical limit.

## SUMMER 2015

Thursday, June 4, Burnside 920, 13:00-14:00
Junehyuk Jung (KAIST)
Title TBA

Analysis/Geometric Analysis seminar
Wednesday, August 26, 13:30-14:30, Burnside 920
Baojun Bian (Tongji University)
Minimal Viscosity Solution of HJB equation and Applications
Abstract： We consider a singular stochastic control problem arising from continuous-time investment and consumption with capital gains tax, where the associated Hamilton-Jacobi-Bellman (HJB) equation admits many viscosity solutions. We show that the value function corresponds to the minimal viscosity solution of the HJB equation. Moreover, we prove by an explicit construction that the optimal strategy can be approximated by a sequence of sub-optimal strategies with bounded control. This is the first such kind of results to explicitly construct such approximations in the singular control literature. We also prove the comparison principle for approximating HJB equation. This is a joint work with X. Chen and M. Dai.

2005/2006 Analysis Seminar