## 2020-21 CRM/MONTREAL/QUEBEC ANALYSIS ZOOM SEMINARS

Seminars are usually held on Mondays or Fridays. In person seminars in Montreal are held at Concordia, McGill or Universite de Montreal; in person seminars in Quebec City are held at Laval.
To attend a zoom session, and for suggestions, questions etc. please contact Galia Dafni (galia.dafni@concordia.ca), Alexandre Girouard (alexandre.girouard@mat.ulaval.ca), Dmitry Jakobson (dmitry.jakobson@mcgill.ca), Damir Kinzebulatov (damir.kinzebulatov@mat.ulaval.ca) or Iosif Polterovich (iossif@dms.umontreal.ca)

Montreal Analysis seminar is currently held online on zoom, organized jointly with Laval University in Quebec City. Please, contact one of the organizers for the seminar zoom links.

The talks are recorded and posted on the CRM Youtube channel, on Mathematical Analysis Lab playlist

## WINTER 2021

Friday, January 15, 14:00-15:00 Eastern time, zoom seminar
Anush Tserunyan (McGill)
Ergodic theorems along trees
Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $\{x, T(x), T^2(x), \hdots, T^n(x)\}$ in front of the point $x$. We prove a “backward” ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over subtrees of the graph of T that are rooted at $x$ and lie behind $x$ (in the direction of $T^{-1}$). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank, where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This strengthens Bufetov’s theorem from 2000, which was the most general result in this vein. This is joint work with Jenna Zomback.
Friday, February 12, 11:00 Eastern time, zoom seminar
Amir Vig (CRM, McGill)
Spectral invariants and Birkhoff Billiards
Abstract: Consider a smooth, bounded, strictly convex domain in the plane. There are two games one can play. The classical one is billiards, in which a billiard ball orbits around the domain and reflects elastically at the boundary. The “quantum” analogue involves the study of wave propagation in the domain and understanding the frequencies at which such waves oscillate. In this talk, we discuss recent progress on the inverse spectral problem of determining a billiard table from its Laplace spectrum. In particular, we introduce a new class of spectral invariants for a generic class of billiard tables obtained from an explicit Hadamard-Riesz type parametrix for the wave propagator, microlocally near geodesic loops of small rotation number. These same techniques also allow us to prove (infinitesimal) Robin spectral rigidity of the ellipse, when both the boundary and boundary conditions are allowed to deform simultaneously. Finally, we mention ongoing work together with Vadim Kaloshin to cancel singularities in the wave trace for special types of domains.
Friday/Saturday, February 26-27
Zoom conference on the occasion of Alexander Shnirelman's birthday
Conference web page
Friday, March 5, 11:00 Eastern time, zoom seminar
Benjamin Jaye (Georgia Tech)
Multi-scale analysis of Jordan curves
Abstract: In this talk we will describe how one can detect regularity in Jordan curves through analysis of associated geometric square functions. We will particularly focus on the resolution of a conjecture of L. Carleson. Joint work with Xavier Tolsa and Michele Villa (arXiv:1909.08581).
Friday, March 19, 14:00 Eastern time, zoom seminar
Thomas Ransford (Laval)
Failure of approximation of odd functions by odd polynomials
Abstract: We construct a Hilbert holomorphic function space $H$ on the unit disk such that the polynomials are dense in $H$, but the odd polynomials are not dense in the odd functions in $H$. As a consequence, there exists a function $f$ in $H$ that lies outside the closed linear span of its Taylor partial sums $s_n(f)$, so it cannot be approximated by any triangular summability method applied to the $s_n(f)$. We also show that there exists a function $f$ in $H$ that lies outside the closed linear span of its radial dilates $f_r, r < 1$. (Joint work with Javad Mashreghi and Pierre-Olivier Parise).
Friday, March 26, 14:00 Eastern time, zoom seminar
Dmitry Faifman (Tel Aviv)
A Funk perspective on convex geometry
Abstract: The Funk metric in the interior of a convex set is a lesser-known cousin of the Hilbert metric. The latter generalizes the Beltrami-Klein model of hyperbolic geometry, and both have straight segments as geodesics, thus constituting solutions of Hilbert's 4th problem alongside normed spaces. Unlike the Hilbert metric, the Funk metric is not projectively invariant. I will explain how, nevertheless, the Funk metric gives rise to many projective invariants, which moreover enjoy a duality extending results of Holmes-Thompson and Alvarez Paiva on spheres of normed spaces and Gutkin-Tabachnikov on Minkowski billiards. I will also discuss how the maximal volume problem in Funk geometry yields an extension of the Blaschke-Santalo inequality.
Friday, April 2, 11:30 Eastern time, zoom seminar
Hakim Boumaza (Paris)
Integrated density of states of the periodic Airy-Schrödinger operator
Abstract: In this talk I present, in the semiclassical regime, an explicit formula for the integrated density of states of the periodic Airy-Schrodinger operator on the real line. The potential of this Schrödinger operator is periodic, continuous and piecewise linear. For this purpose, the spectrum of the Schrödinger operator whose potential is the restriction of the periodic Airy-Schrödinger potential to a finite number of periods is studied. We prove that all the eigenvalues of the operator corresponding to the restricted potential are in the spectral bands of the periodic Airy-Schrodinger operator and none of them are in its spectral gaps. In the semiclassical regime, we count the number of these eigenvalues in each of the spectral bands. Note that in these results there are explicit constants which characterize the semiclassical regime. This is joint work with Olivier Lafitte (USPN - CRM Montreal).
Friday, April 9, 14:00 Eastern time, zoom seminar
Konstantin Khanin (Toronto)
On Stationary Solutions to the Stochastic Heat Equation
Abstract: I plan to discuss the problem of uniqueness of global solutions to the random Hamilton-Jacobi equation. I will formulate several conjectures and present results supporting them. Then I will discuss a new uniqueness result for the Stochastic Heat equation in the regime of weak disorder.
Friday, April 16, 10:00 Eastern time, zoom seminar
Nicolai Krylov (Minnesota)
A review of some new results in the theory of linear elliptic equations with drift in $L_{d}$
Abstract: We present an overview of recent results related to the Aleksandrov type estimates with power of summability of the free term $d_0 < d$, the Harnack inequality for $u\in W^{2}_{d_{0},loc}$, Holder continuity of $L$-harmonic and $L$-caloric functions. Under the assumption that the main coefficients are almost in VMO (and $b\in L_{d}$) we also present the results about solvability of the elliptic equations in $W^{2}_{d_{0}}$ in domains and in the whole space. A few relates issues are discussed as well.
Friday, April 23, 14:00 Eastern time, zoom seminar
Vitali Vougalter (Toronto)
Solvability of some integro-differential equations with anomalous diffusion and transport
Abstract: The work deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in the presence of the transport term. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used. We discuss how the introduction of the transport term impacts the regularity of solutions.
Friday, April 30, 14:00 Eastern time, zoom seminar
Krzysztof Bogdan (Wroclaw)
Optimal Hardy identities and inequalites for the fractional Laplacian on $L^p$
Abstract: We will present a route from symmetric Markovian semigroups to Hardy inequalities, to nonexplosion and contractivity results for Feynman-Kac semigroups on $L^p$. We will focus on the fractional Laplacian on $\mathbb{R}^d$, in which case the constants, estimates of the Feynman-Kac semigroups and tresholds for contractivity and explosion are sharp. Namely we will discuss selected results from arXiv:2103.06550, joint with Bartl omiej Dyda, Tomasz Grzywny, Tomasz Jakubowski, Panki Kim, Julia Lenczewska, Katarzyna Pietruska-Pa\l uba or Dominika Pilarczyk.
Wednesday, May 5, 13:30 Eastern time, zoom seminar (Joint with the Geometric Analysis Seminar)
Frederic Naud (Paris)
The spectral gap of random hyperbolic surfaces
Abstract: We will start with a survey on (some very recent) results on the low spectrum of random compact hyperbolic surfaces, for various models including discrete and continuous Teichmuller spaces. We will then give some ideas of the proofs by emphasizing the analogy with radom graphs. We will also dicuss non compact situations where similar results on resonances can be obtained. Joint works with Michael Magee and Doron Puder.
Friday, May 7, 14:00 Eastern time, zoom seminar
Nages Shanmugalingam (U. Cincinnati)
Prime ends for domains in metric measure spaces and their use in potential theory and QC theory.
Abstract: Prime ends were first developed by Caratheodory in order to understand the boundary behavior of conformal mappings from the disk. As such, the construction of Caratheodory and Ahlfors worked for simply connected planar domains, but had to be modified for more general domains. In this talk we will focus on a construction in the setting of domains in metric spaces, and describe their use in potential theory (Dirichlet problem for the p-energy minimizers) and in studying boundary behavior of QC maps.
Friday, May 14, 11:00 Eastern time, zoom seminar
Michael Lipnowski (McGill)
Towards optimal spectral gaps in large genus.
Abstract: I'll discuss recent joint work with Alex Wright (arXiv: 2103.07496) showing that typical large genus hyperbolic surfaces have first Laplacian eigenvalue at least 3/16−epsilon.
Wednesday, May 19, 13:30 Eastern time, zoom seminar (Joint with the Geometric Analysis Seminar)
Laurent Moonens (Paris-Saclay)
Solving the divergence equation with measure data in non-regular domains
Abstract: In this talk, we shall present a recent joint work with E. Russ (Grenoble) concerning the equation $\mathrm{div}\,v=\mu$ in a (rather general) open domain $\Omega$, where $\mu$ is a (signed) Radon measure in $\Omega$ satisfying $\mu(\Omega)=0$. We show in particular that, under mild assumptions on the geometry of $\Omega$ (and some assumptions on $\mu$), one can provide a constructive way to build solutions $v$ in a weighted $L^\infty$ space enjoying weak Neumann-type boundary conditions.
Friday, May 28, 14:00 Eastern time, zoom seminar
Li Chen (Louisiana State)
BV functions and some functional inequalities on nested fractals
Abstract: In this talk, we discuss functions of bounded variations (BV) on fractals which satisfy sub-Gaussian heat kernel bounds. We also prove isoperimetric inequalities and Poincare inequalities on some nested fractals. Our proofs use heat kernel methods and the key ingredient is a weak Bakry-Emery curvature type condition. The talk are based on joint works with Patricia Alonso-Ruiz, Fabrice Baudoin, Luke Rogers, Nageswari Shanmugalingam and Alexander Teplyaev.

## FALL 2020

Friday, September 11, 9:00 Eastern Time, zoom seminar
Jose Maria Martell (ICMAT)
Uniform rectifiability and elliptic operators satisfying a Carleson measure condition
Abstract: In this talk I will study the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. Our setting is that of domains having an Ahlfors regular boundary and satisfying the so-called interior Corkscrew and Harnack chain conditions (these are respectively scale-invariant/quantitative versions of openness and path-connectivity) and we show that for the class of Kenig-Pipher uniformly elliptic operators (operators whose coefficients have controlled oscillation in terms of a Carleson measure condition) the solvability of the $L^p$-Dirichlet problem with some finite $p$ is equivalent to the quantitative openness of the exterior domains or to the uniform rectifiablity of the boundary. Joint work with S. Hofmann, S. Mayboroda, T. Toro, and Z. Zhao.
Friday, October 9, 13:00 Eastern time, zoom seminar
Arick Shao (Queen Mary university)
On Controllability of Waves and Geometric Carleman Estimates
Abstract: We consider the question of exact (boundary) controllability of wave equations: whether one can steer their solutions from any initial state to any final state using appropriate boundary data. In particular, we discuss new and fully general results for linear wave equations on time-dependent domains with moving boundaries. We also discuss the novel geometric Carleman estimates that are the main tools for proving these controllability results.
Friday, October 16, 14:30 Eastern Time, zoom seminar
Guangyu Xi (University of Maryland)
On the Smoluchowski-Kramers approximation of infinite dimensional systems with state-dependent damping
Abstract: We study a class of stochastic nonlinear damped wave equations endowed with Dirichlet boundary conditions. We show that the Smoluchowski-Kramers approximation describes the convergence of solutions of the stochastic wave equations to the solution of a quasilinear stochastic parabolic equation endowed with Dirichlet boundary conditions. This is a joint work with Sandra Cerrai from UMD.
Friday, October 23, 14:30 Eastern time, zoom seminar
Schiffer operator corresponding to Riemann surfaces with finitely many borders
Abstract: In this talk, a brief history of the Schiffer operator on the complex plane, on the Riemann sphere, and (some type of) Riemann surfaces will be reviewed. Some of my thesis' results concerning the development of this operator on Riemann surfaces (open, with finitely many borders, each homeomorphic to S^1) will be presented. We will see when the Schiffer operator is a bounded isomorphism.
Friday, November 13, 14:30 Eastern time, zoom seminar
Hien Nguyen (Iowa State)
Strangling of the fundamental gap in hyperbolic space
Abstract: For the Laplace operator with Dirichlet boundary conditions on convex domains in $H^n, n ≥ 2$, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for domains of any diameter. This property distinguishes hyperbolic spaces from Euclidean and spherical ones, where the quantity is bounded below by $3 \pi^2$.
Friday, November 20, 12noon Eastern time, zoom seminar
Daniel Peralta-Salas (ICMAT)
Optimal domains for the first curl eigenvalue
Abstract: The classical Faber-Krahn inequality for the first eigenvalue of the Dirichlet Laplacian shows that the ball is the unique optimal domain. In this talk I will explore the analogous problem for the curl operator: for a fixed volume, what is the optimal domain for the first positive (or negative) eigenvalue of curl? In spite of being one of the most important vector-valued operators, this question is rather unexplored and remains wide open. In this talk I will show that, even taking into account that the first eigenvalue is uniformly lower bounded in terms of the volume, there are no axisymmetric smooth optimal domains for the curl that satisfy a mild technical assumption. In particular, this rules out the existence of optimal axisymmetric domains with a convex section. This is based on joint work with Alberto Enciso.
Friday, November 27, 14:30 Eastern time, zoom seminar
Alex Brudnyi (Calgary)
On the stable rank of algebras of bounded holomorphic functions
Abstract: The concept of the stable rank introduced by Bass plays an important role in some stabilization problems of algebraic K-theory analogous to that of dimension in topology. Despite a simple definition, the stable rank is often quite difficult to calculate even for relatively uncomplicated rings. We present examples of algebras for which the stable rank is computed. Next, we consider a similar problem for algebras of bounded holomorphic functions on Riemann surfaces. The central result in this area is a theorem of Treil asserting that the Bass stable rank of the algebra of bounded holomorphic functions on the open unit disk is 1. We discuss some generalizations of Treil's result.
Friday, December 11, 9:30-10:30 Eastern time, zoom seminar
Ryan Gibara (Laval)
Boundedness and continuity for rearrangements on spaces defined by mean oscillation
Abstract: In joint work with Almut Burchard and Galia Dafni, we study the boundedness and continuity of rearrangement operators on the space BMO of functions of bounded mean oscillation. Improved bounds are obtained for the BMO-seminorm of the decreasing rearrangement, and the symmetric decreasing rearrangement is shown to be bounded on BMO. Both of these rearrangements are shown to be discontinuous as maps on BMO, but sufficient normalisation conditions are established to guarantee continuity on the subspace VMO of functions of vanishing mean oscillation.
Friday, December 11, 11:00-12:00 Eastern time, zoom seminar
Renaud Raquepas (McGill)
Entropy production in nondegenerate diffusions: large times and small noises
Abstract: Entropy production (EP) is a key quantity from thermodynamics which quantifies the irreversibility of the time evolution of physical systems. I will start the presentation with a general introduction to the different approaches to defining EP. Then, I will focus on the context of nondegenerate diffusions and I will describe the large-deviation properties of EP as time goes to infinity. I will also explain how the behaviour of the corresponding rate function boils down to the study of the smallest eigenvalue for a family of differential operators with a small parameter.
2005/2006 Analysis Seminar