## 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
(jakobson@math.mcgill.ca), Iosif Polterovich
(iossif@dms.umontreal.ca) or
Alina Stancu (alina.stancu@concordia.ca)

## SUMMER 2016

** Thursday, June 2, 13:30-14:30, Concordia, Room LB 921-04
**

**Vlad Yaskin** (University of Alberta, Edmonton)

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.

## 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.

## ANALYSIS-RELATED TALKS ELSEWHERE, FALL 2015

## 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.

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