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.

# Isoperimetric "sandwiches", free boundary boundary problems and approximation by analytic and harmonic functions

# Obtaining Weyl estimates on manifolds with cusps

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.

# Micro-randomized Trials & mHealth

Biostatistic seminar

*Abstract: Micro-randomized trials are trials in which individuals are randomized 100's or 1000's of times over the course of the study. The goal of these trials is to assess the impact of momentary interventions, e.g. interventions that are intended to impact behavior over small time intervals. A fast growing area of mHealth concerns the use of mobile devices for both collecting real-time data, for processing this data and for providing momentary interventions. We discuss the design and analysis of micro-randomized trials for use in mHealth.*

# Lecture by Louis-Pierre Arguin, 2015 André-Aisenstadt Prize Recipient

**Maximum of strongly correlated random variables
**One of the main goal of probability theory is to find "universal laws". This is well-illustrated by the Law of Large Numbers and the Central Limit Theorem, dating back to the 18th century, which show convergence of the sum of random variables with minimal assumptions on their distributions. Much of current research in probability is concerned with finding universal laws for the maximum of random variables. One universality class of interest (in mathematics and in physics) consists of stochastic processes whose correlations decay logarithmically with the distance. In this talk, we will survey recent results on the subject and their connection to problems in mathematics such as the maxima of the Riemann zeta function on the critical line and of the characteristic polynomial of random matrices.

** Présentation en français avec diapos en anglais. **

Le café sera servi à 15h30 et une réception suivra la conférence au Salon Maurice-L’Abbé (salle 6245).

*Coffee will be served before the conference and a reception will follow at Salon Maurice-L’Abbé (Room 6245).*

# Workshop: Moduli spaces, integrable systems, and topological recursions

**Organizers :**

Dmitry Korotkin (Concordia University)

Jacques Hurtubise (McGill University)

http://www.crm.umontreal.ca/2016/Moduli16/index_e.php

# SUMM 2016 - Seminars in Undergraduate Mathematics in Montreal

**Organizers:**

Renaud Raquépas (McGill University)

Antoine Giard (Université de Montréal)

Jida Hussami (Concordia University)

Joey Litalien (McGill University)

Fabrice Nonez (École Polytechnique de Montréal)

Erick Schulz (McGill University)

http://summ.math.uqam.ca/index.php

# Seminar LACIM -- Arbres non-ambigus : nouveaux résultats et généralisation

Les arbres non-ambigus ont initialement été définis comme des dessins particuliers d'arbres binaires sur le quadrillage. Mais on peut également les voir comme des tableaux boisés de forme rectangulaire. Dans ce dernier cadre, Steingrímsson et Williams ont montré qu'ils sont en bijection avec les permutations dont toutes les excédences (strictes) sont au début. Ehren-borg and Steingrímsson avait prouvé une formule alternante pour énumérer ces permutations particulières. Dans cette exposé je vais donner une nouvelle définition des arbres non-ambigus en terme d'arbres binaires étiquetés croissants le long des arêtes gauches, et indépendamment, le long des arêtes droites. Grâce à cette nouvelle vision, on prouve bijectivement une nouvelle formule énumérative qui a la bonne propriété d'être à termes positifs, et on trouve une formulation compacte de la série génératrice doublement exponentielle des arbres non-ambigus. Je terminerai l'exposé par la présentation d'une généralisation des arbres non-ambigus à toutes dimensions finies.

# An integral formula with geometric applications in Riemannian and Pseudo-Riemannian manifolds

In this talk, we will present a recent joint work with Chao Xia. We first prove a general integral formula for bounded domains in Riemannian manifolds. This formula includes Reilly's integral formula and the recent work of Qiu-Xia as special cases. In the second part of the talk, we will apply this formula to prove 1) Heitz-Karcher type inequalities, 2) Minkowski inequality, 3) two almost Schur type of Theorems. All these geometric inequalities hold for the so-called substatic Riemannian manifolds which consists of a large family Riemannian manifolds including all the space forms. We note that Heitze-Karcher inequality naturally leads to an Alexandrov rigity theorem for substatic warped product spaces. Thus we recovered S. Brendle's recent work by a completely different approach. The results in this talk are focused on Riemannian manifolds, however it has deep roots from Pseudo-Riemannian spaces.

# Seminar Physique Mathématique -- KPZ universality, particle systems and polymer models

KPZ universality describes a scaling behaviour that differs from the central limit theorem by the size of the fluctuations ($n^{1/3}$ instead of $n^{1/2}$) and the limiting distribution. Instead of the Gaussian, the Tracy-Widom distributions from random matrix theory appear in the limit. It is a long standing conjecture that the KPZ universality class contains a large group of models, including particle systems and polymer models. I will discuss two particular examples: a polymer model with gamma weights and the asymmetric exclusion process (ASEP) started from flat and half-flat initial conditions.

# Working Seminar Mathematical Physics -- Cluster algebras and generalized pentagram maps

We discuss how the generalized pentagram maps are related to cluster algebras. The talk is based on the paper: http://arxiv.org/abs/1503.02057