## 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
Alina Stancu (alina.stancu@concordia.ca)

## WINTER 2019

** Joint seminar with Geometric Analysis**

** Friday, May 3, 13:30-14:30 (to be confirmed), McGill,
Burnside Hall, Room 1104**

** Luca Martinazzi** (Padua)

Title to be announced

** 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.

2017/2018 Seminars

2016/2017 Seminars

2015/2016 Seminars

2014/2015 Seminars

Fall 2013 Seminars

Winter 2014 Seminars

2012/2013 Seminars

2011/2012 Seminars

2010/2011 Seminars

2009/2010 Seminars

2008/2009 Seminars

2007/2008 Seminars

2006/2007 Seminars

2005/2006 Analysis Seminar
2004/2005 Seminars

2004/2005 Seminar in Nonlinear Analysis and Dynamical Systems

2003/2004 Working Seminar in Mathematical Physics

2002/2003 Seminars

2001/2002 Seminars

2000/2001 Seminars

1999/2000 Seminars