## 2018-19 Montreal Analysis Seminar

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

For suggestions, questions etc. please contact Dmitry Jakobson
(dmitry.jakobson@mcgill.ca), Iosif Polterovich
(iossif@dms.umontreal.ca) or
Galia Dafni (galia.dafni@concordia.ca)

## SUMMER 2019

** Joint seminar with Geometric Analysis**

** Friday, August 9, 13:30-14:30, McGill,
Burnside Hall, Room 1104
**

**Ali Aleyasin** (Stony Brook)

Singular and degenerate Monge-Ampere equations

** Abstract:**
It is well known that several non-linear elliptic partial differential
equations have applications in various fields of geometry and analysis,
including but not limited to the Calabi problem, the Weyl, and Minkowski
problems, and optimal transport. An important class of such non-linear
equations are the real and complex Monge-Amp\`ere equations. Although the
case of strictly elliptic equations with smooth source term has been
rather well-understood, the behaviour of solutions in the vicinity of
possible singularities or degeneracies of the source term is far from being
understood. This corresponds to the vanishing or blowing up of the prescribed
curvature in the Weyl problem. In this talk, I will present an application
of a differential geometric approach to the study of certain singularities
and degeneracies of elliptic complex Monge-Amp\`ere equations. This approach
will allow new estimates for solutions to be derived. I shall also outline
how the idea works in the case of several other important geometric partial
differential equations.

## WINTER 2019

** Friday, June 7, 13:30-14:30, McGill, Burnside Hall, Room 1104
**

**Felix Finster** (Regensburg)

A positive mass theorem for static causal fermion systems

** Abstract:**
After a short general introduction to causal fermion systems and to the
positive mass theorem for asymptotically flat Riemannian manifolds, I will
explain how to describe an asymptotically flat manifold by a static causal
fermion system. The total mass of a static, asymptotically flat causal
fermion system is defined abstractly, and it is shown that this definition
reduces to the ADM mass in a certain limiting case. I will also outline the
proof that for a minimizer of the causal action principle, the total mass
is always non-negative.
I am reporting on joint work with Andreas Platzer.

** Friday, May 31, Concordia, 13:30-14:30, Concordia, Library Building,
Room LB 921-4**

**Kinvi Kangni** (Universite Felix Houphouet-Boigny)

Spherical Grassmannian on a reductive Lie group

** Abstract:**
Let $G$ be a locally compact group, $K$ a compact subgroup of $G$
and $\delta$ an arbitrary class of irreducible unitary representations of $K$.
The $p$-$\delta$-spherical Grassmannian $\mathcal{G}_{p,\delta}$ is an
equivalence class of spherical functions of type $\delta$-positive of
height $p$.

In this talk, we construct some elements of $\mathcal{G}_{p,\delta}$ on
reductive Lie group using a generalized Abel transform.

And if the discret serie is not empty,we give a extension of Paley Wiener
theorem using a compact Cartan subgroup of $G.$

** Joint seminar with Geometric Analysis**

** Friday, May 3, 13:30-14:30, McGill, Burnside Hall, Room 1104**

**Luca Martinazzi** (University of Padua, Italy)

News on the Moser-Trudinger inequality: from sharp estimates to the
Leray-Schauder degree

**Abstract:** The existence of critical points for the Moser-Trudinger
inequality for large energies has been open for a long time. We will first
show how a collaboration with G. Mancini allows to recast the
Moser-Trudinger inequality and the existence of its extremals (originally
due to L. Carleson and A. Chang) under a new light, based on sharp
energy estimate. Building upon a recent subtle work of O. Druet and P-D.
Thizy, in a work in progress with O. Druet, A. Malchiodi and P-D. Thizy,
we use these estimates to compute the Leray-Schauder degree of the
Moser-Trudinger equation (via a suitable use of the Poincare-Hopf theorem),
hence proving that for any bounded non-simply connected domain the
Moser-Trudinger inequality admits critical points of arbitrarily high
energy. In a work in progress with F. De Marchis, O. Druet, A. Malchiodi
and P-D. Thizy, we expect to use a variational argument to treat the case
of a closed surface.

** Friday, April 5, 13:30-14:30, McGill, Burnside Hall, Room 1104**

** Michael Levitin** (University of Reading)

Sharp eigenvalue asymptotics for Steklov problem on curvilinear
polygons

** Abstract:**
I will discuss a work in progress (joint with Leonid Parnovski, Iosif
Polterovich and David Sher) on Steklov (or Dirichlet-to-Neumann map)
eigenvalue asymptotics for curvilinear polygonal domains in R^2. The
results are unexpected, and the asymptotics depends on the arithmetic
properties of the angles of the polygon. There are also connections to
classical probleems of hydrodynamics (the sloping beach problem and the
sloshing problem) and to the Laplacian on quantum graphs.

** Friday, March 29, 13:30-14:30, McGill, Burnside Hall, Room 1104**

** Ilia Binder** (Toronto)

Computability and Complexity in Complex Analysis and Complex Dynamics.

** Abstract:**
In this talk, I will discuss recent advances in the computability of
various objects arising in Complex Dynamics and Theory of Univalent functions.
After a brief introduction to the general Computability and Complexity
Theory, I will talk about the computational properties of polynomial
Julia sets and conformal maps. I will also consider
the computability questions related to the boundary extensions of
conformal maps. Based on joint work with M. Braverman (Princeton),
C. Rojas (Universidad Andres Bello), and M. Yampolsky (University of
Toronto).

** Friday, March 22, 13:30-14:30, McGill, Burnside Hall, Room 1104
**

** Mikhail Karpukhin** (Irvine)

Applications of algebra and topology to isoperimetric
eigenvalue inequalities.

** Absract:**
Spectrum of the Laplace-Beltrami operator is one of the fundamental
invariants of a Riemannian manifold.
Finding the optimal isoperimetric inequalities for its eigenvalues
is a classical problem of spectral geometry going back to J. Hersch,
P. Li and S.-T. Yau. One of the main attractions of this problem is the
variety of methods employed to study it. In the present talk we demonstrate
this feature and, in particular, outline the connections to the theory of
minimal surfaces, algebraic geometry and topology. These include recent
applications of moduli spaces and cobordism theory.
The talk is based on joint works with V. Medvedev, N. Nadirashvili,
A. Penskoi and I. Polterovich.

** Friday, March 15, 13:30-14:30, McGill,
Burnside Hall, Room 1104**

** Siran Li** (CRM, McGill)

Some Problems On Harmonic Maps from B^3 to S^2.

** Abstract:**
Harmonic map equations are an elliptic PDE system
arising from the minimisation of Dirichlet energies between two
manifolds. In this talk we present some recent works concerning the
symmetry and stability of harmonic maps. We construct a new family of
''twisting'' examples of harmonic maps and discuss the existence,
uniqueness and regularity issues. In particular, we characterise the
singularities of minimising general axially symmetric harmonic maps,
and construct non-minimising general axially symmetric harmonic maps
with arbitrary 0- or 1-dimensional singular sets on the symmetry axis.
Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$
to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data for
$p>=2$.
(Joint work with Prof. Robert Hardt.)

** Friday, March 1, 13:30-14:30, McGill, Burnside Hall, Room 1104
**

** Alexandre Girouard** (Laval)

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

** Abstract:**
The Dirichlet-to-Neumann map is a first order pseudodifferential operator
acting on the smooth functions of the boundary of a compact Riemannian
manifold M. Its spectrum is known as the Steklov spectrum of M. The
asymptotic behaviour (as j tends to infinity) of the Steklov eigenvalues
s_j is determined by the Riemannian metric on the boundary of M.
Neverthless, each individual eigenvalue can become arbitrarily big if the
Riemannian metric is perturbed adequately. This can be achieved while
keeping the geometry of the boundary unchanged, but it requires wild
perturbations of the metric in arbitrarily small neighborhoods of the
boundary. In recent work with Bruno Colbois and Asma Hassannezhad, we
impose constraints on the geometry of M on and near its boundary. This
allows the comparison of each Steklov eigenvalue s_j with the
corresponding eigenvalues l_j of the Laplace operator acting on the
boundary. This control is uniform in the index j. The proof of is based
on a generalized Pohozaev identity and on comparison results for the
principal curvatures of hypersurfaces that are parallel to the boundary.

** Joint seminar, Analysis and Number Theory**

** Monday, February 25, McGill, 13:30-14:30, McGill, Burnside Hall,
Room 1104**

** Dmitry Logachev** (UFAM Manaus)

Anderson t-motives - a parallel world to abelian varieties, in finite
characteristic.

** Abstract: **
Formally, Anderson t-motives (generalizations of Drinfeld modules) are
some modules over a ring of non-commutative polynomials in two variables
over a complete algebraically closed field of finite characteristic.
Surprisingly, it turns out that their properties are very similar to the
properties of abelian varieties (more exactly, of abelian varieties with
multiplication by an imaginary quadratic field). For example, we can define
Tate modules of Anderson t-motives, Galois action on them, lattices,
modular curves, L-functions etc. Nevertheless, this analogy is far to be
complete. There is no functional equation for their L-functions; notion
of the algebraic rank is not known yet; 1 - 1 correspondence between
Anderson t-motives and lattices also is known only for Drinfeld modules.
A survey of the theory of Anderson t-motives and statements of some
research problems will be given.

** Friday, February 15, McGill, 13:30-14:30, McGill, Burnside Hall,
Room 1104**

** Javad Mashreghi** (Laval)

Carleson measures for the Dirichlet space

** Abstract: **
We show that a finite measure $\mu$ on the unit disk is a Carleson
measure for the Dirichlet space $\mathcal{D}$ if it satisfies the one-box
condition $\mu(\, S(I) \,) = O(\, \varphi(|I|) \,)$, where $\varphi$ such
that $\varphi(x)/x$ is integrable. We also show that the integral
condition on $\varphi$ is sharp.

** Friday, February 8, McGill, 13:30-14:30, McGill, Burnside Hall,
Room 1104**

** Michael Lipnowski** (McGill)

Geometry of the smallest 1-form Laplacian eigenvalue on
hyperbolic manifolds.

** Abstract: **
I’ll describe a relationship between the smallest 1-form Laplacian
eigenvalue and surface complexity on hyperbolic manifolds. We'll then
speculate on "surface theft", a prospect based on this relationship
for proving good lower bounds on the smallest 1-form Laplacian eigenvalue
for big congruence arithmetic hyperbolic 3-manifolds M.
This is joint work with Mark Stern.

** Thursday, January 24, 14:00-15:00, Concordia, Room LB 921-4**

** Almut Burchard** (Toronto)

A geometric stability result for Riesz-potentials

** Abstract:**
Riesz' rearrangement inequality implies
that integral functionals (such as the Coulomb energy of a
charge distribution) that are defined by a pair interaction potential
(such as the Newton potential) which decreases with distance
are maximized (under appropriate constraints)
only by densities that are radially decreasing about some
point. I will describe recent and ongoing work
with Greg Chambers on the stability of this inequality for the special
case of the Riesz-potentials in n dimensions (given by the kernels
|x-y|^-(n-s)), for densities that
are uniform on a set of given volume. For 1< s < n, we bound
the square of the symmetric difference of a set from a ball
by the difference in energy of the corresponding uniform
distribution from that of the ball.

## FALL 2018

** Friday, November 23, 13:30-14:30, McGill, Burnside
Hall, Room 1104**

** Herve Lombaert** (ETS Montreal/Inria Sophia-Antipolis)

Spectral Correspondence and Learning of Surface Data -
Example on Brain Surfaces

** Abstract:**
How to analyze complex shapes, such as of the highly folded surface of
the brain? In this talk, I will show how spectral representations of
shapes can benefit neuroimaging and, more generally, problems where data
fundamentally lives on surfaces. Key operations, such as segmentation and
registration, typically need a common mapping of surfaces, often obtained
via slow and complex mesh deformations in a Euclidean space. Here, we
exploit spectral coordinates derived from the Laplacian eigenfunctions of
shapes and also address the inherent instability of spectral shape
decompositions. Spectral coordinates have the advantage over Euclidean
coordinates, to be geometry aware and to parameterize surfaces explicitly.
This change of paradigm, from Euclidean to spectral representations,
enables a classifier to be applied directly on surface data,
via spectral coordinates.
The talk will focus, first, on spectral representations of shapes, with
an example on brain surface matching, and second, on the learning of
surface data, with an example on automatic brain surface parcellation.

** Friday, October 26, 14:00-15:00, McGill, Burnside Hall,
Room 1120 (note the new room!)**

** K. Luli** (UC Davis)

Variational Problems on Arbitrary Sets

** Abstract:**
Let E be an arbitrary subset of R^n. Given real valued functions f
defined on E and g defined on R^n, the classical Obstacle Problem asks
for a minimizer of the Dirichlet energy subject to the following two
constraints: (1) F = f on E and (2) F >= g on R^n. In this talk, we
will discuss how to use extension theory to construct (almost) solutions
directly. We will also explain several recent results that will help lay
the foundation for building a complete theory revolving around the belief
that any variational problems that can be solved using PDE theory can also
be dealt with using extension theory.

** Friday, October 19, 13:30-14:30, McGill, Burnside Hall,
Room 1104**

** Hans Christianson** (UNC Chapel Hill)

Quantum Ergodic properties for eigenfunctions on triangles

** Abstract:**
In this talk we will discuss one aspect of the idea of classical-quantum
correspondence in planar domains called quantum ergodicity. For a planar
domain, the classical problem is to imagine the domain is a billiard table,
while the quantum problem is to imagine it is a drum. Waves on a drum head
tend to follow along billiard trajectories, so if the billiard trajectories
are sufficiently chaotic, we expect the waves to spread out. This is
called quantum ergodicity. We will informally discuss some of the subtle
history of this topic, where detailed information about the billiard flow
is essential for the quantum problem. We will then move on to a very
unsubtle result which hints at quantum ergodicity for triangular domains
without using any dynamical systems information at all. This result
follows from a very short proof using nothing more than integrations by parts.

** Friday, October 5, 13:30-14:30, McGill, Burnside Hall, Room 1104**

** J. De Simoi** (Toronto)

On spectral rigidity for generic symmetric convex billiards

** Abstract:**
In this talk I will present a most recent result in the setting of length-
spectral rigidity for convex billiards. In a joint work with A. Figalli and
V. Kaloshin we show that there exists an open and dense set S of smooth convex
axially-symmetric planar domains so that every non-isometric deformation of
some D in S necessarily changes the length of a periodic orbit of the
billiard in D.

** Friday, September 21, 13:30-14:30, McGill, Burnside Hall, Room 1104**

** Javad Mashreghi** (Laval)

Polynomial approximation in super-harmonically weighted Dirichlet spaces

** Abstract:**
Taylor polynomials are natural objects for approximation in function spaces. Indeed, it works in several
function spaces, e.g., Hardy and Bergman spaces. However, it also fails in some cases and a remedy is needed, e.g.,
disc algebra and weighted Dirichlet spaces. We show that in the latter, Taylor polynomials may diverge. However,
by properly adjusting the last coefficient we produce a convergent sequence in local Dirichlet spaces. We also show
that in super-harmonically weighted Dirichlet spaces, Fejer averages provide a convergent sequence.
Joint work with T. Ransford.

** Monday, September 17, 13:30-14:30, McGill, Burnside Hall,
Room 1104**

** Matthew de Courcy-Ireland ** (Princeton)

Monochromatic waves at shrinking scales

** Abstract:**
We study an ensemble of random functions on a compact Riemannian manifold.
These random functions have been proposed by Berry as a model for
high-frequency Laplace eigenfunctions in chaotic settings. We prove that,
with high probability, they are evenly distributed in the mean square sense
on shrinking geodesic balls. The rate of shrinking comes within a
logarithmic factor of the optimal wave scale, and equidistribution occurs
simultaneously over all possible centers for the ball.

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