# SCIENTIFIC PROGRAMS AND ACTIVITIES

August 30, 2015

## Fields Analysis Working Group 2007-08

A working group seminar and brown bag lunch devoted to nonlinear dynamics and the calculus of variations meeting once a week for three hours at the Fields Institute. The focus will be on working through some key papers from the current literature with graduate students and postdocs, particularly related to optimal transportation and nonlinear waves, and to provide a forum for presenting research in progress. The format will consist of two 70 minute presentations by different speakers, separated by a brown bag lunch.
More information will be linked to http://tosio.math.toronto.edu/pdewiki/index.php/Main_Page as it becomes available. Interested persons are welcome to attend either or both talks (and to propose talks to the organizers, currently <colliand@math.toronto.edu> and <mccann@math.toronto.edu>), Robert McCann and James Colliander .
 Fields Visitor Resources 2006-07 Working Seminar on Nonlinear Evolution Equation and Fields Analysis Working Group Directions

### Wednesday, 2008

16 April
11:10 AM

Almut Burchard, University of Toronto
On Caffarelli's C(1,alpha)-regularity for Monge-Ampere equations
What is alpha?
I will summarize Caffarelli's geometric arguments (which Alessio Figalli explained here Fall), and then estimate the Holder exponent. The resulting lower bound on alpha shrinks dramatically with the spatial dimension. Time permitting, I will discuss a similar estimate of Forzani and Maldonado, and possible improvements.

Relevant papers:
•L. Caffarelli, Some regularity properties of solutions of Monge--Ampère equation, Comm. Pure Appl. Math. 44 (1991) 965--969.
• Liliana Forzani, Diego Maldonado, Properties of the solutions to the Monge-Ampère equation. Nonlinear Analysis. 57 (2004), no. 5-6, 815--829.

16 April
1:10 PM
Paul Lee, University of Toronto
Optimal mass transportation with nonholonomic constraints
I will talk about my joint work with A. Agrachev on optimal mass transportation with nonholonomic constraints. Nonholonomic constraints are restrictions on velocity which do not arise from the configuration. They can be imposed by fixing a plane field or more generally a control system. The cost function is given by an optimal control problem where we minimize certain Lagrangian among curves which satisfies these constraints. We prove, under certain assumption on the constraints, the existence and uniqueness of solution to Monge-Kantorovich problem generalizing previous work of Brenier, McCann and Bernard & Buffoni. I will also talk about an example where the existence and uniqueness theorem does not apply.
2 April
11:10 AM
Jim Colliander, University of Toronto
Talk: TBA
2 April
1:10 PM
Walid Abou Salem, University of Toronto
On the blind collision of fast solitons, II
Continuation of the previous talk on March 26.
26 March
11:10 AM

Robert Jerrard, University of Toronto
Dynamics of topological defects in semilinear hyperbolic PDEs.
Over the past 30 years, numerous theorems have been provedestablishing ways in which the behavior of some solutions of certain semilinear elliptic partial differential equations arising in mathematical physics are related to the minimal surface equation. Many such results are also known for parabolic PDEs. I will discuss a result that give the first rigorous proof (as far as I know) of a result of this sort in the setting of nonlinear wave equations. This is joint work with Alberto Montero.

26 March
1:10 AM
Walid Abou Salem, University of Toronto
On the blind collision of fast solitons
I discuss recent work on the collision of two fast solitons for the nonlinear Schr\"odinger equation in the presence of a spatially slowly varying external potential. For a high initial relative speed $\|v\|$ of the solitons, one can show that, up to times of order $\log\|v\|$ after the collision, the solitons preserve their shape (in $L^2$-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. I also remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.
12 March
11:10 AM
Hao Li (University of Toronto, Economics) Copy of Slides
Equilibria in a sorting problem.
An interval of types is to be sorted into two equal-sized groups. Each type's payoff depends both on its percentile rank in the group it joins, and on the average type of the group. Equilibrium sorting pattern is considered in three settings. In the first setting, each type chooses between the two groups, with higher types having priority over lower types. In the second setting, types bid competitively for ranks in the groups. In the third setting, the two groups each choose how to allocate a fixed amount of resources according to rank in order to maximize the average type, and then types sort into the two groups as in the first setting.

Note: The content of the talk will be drawn from two of my recent papers,
"First in Village or Second in Rome?" and "Competing for Talents." They can be downloaded from my website http://www.chass.utoronto.ca/~haoli/research/index.html

12 March
1:10 AM
Abdeslem Lyaghfouri (Fields Institute)
Hoelder Continuity of Solutions to Quasilinear Elliptic Equations Involving Measures.
I will present a paper by Tero Kilpela"inen on the Ho"lder continuity of the solutions of the p-Laplace equation with right-hand side measure. The relationship between the growth of the measure and the Ho"lder continuity of the solutions will be discussed.

13 February
11:10 AM
Tristan Roy (U.C.L.A.)
Global well-posedness for the defocusing cubic wave equation
6 February
11:10 AM
Shuanglin Shao (U.C.L.A.)
Restriction estimates for Paraboloids in the Cylindrically Symmetric Case
In this talk, we will discuss the linear and bilinear restriction estimates for paraboloids when functions are assumed to be cylindrically symmetric. For dyadically supported functions, further estimates are available and sharp up to endpoints, which prove to be very useful in establishing the global well-posedness result of certain critical nonlinear Schrödinger equations in the radial case. We also derive that the Restriction Conjecture for paraboloids is true in the cylindrically symmetric case.
6 February
13:10 PM
Gideon Simpson (Columbia University)
The Mathematics of Magma Migration: Nonlinearity, Degeneracy, and Dispersion
Geologic processes occur on time scales that introduce non-standard rheologies. In particular, magma migration is modeled as a poro- viscous flow. We will see that such models lead to nonlinear, nonlocal, dispersive wave equations with the potential for degeneracy. This talk will discuss the well-posedness of these equations, the stability of their solitary waves, and associated open problems.
30 January
11:10 AM
Marina Chugunova (University of Toronto)
Spectral Properties of the Non-Self-Adjoint Operator Associated with the Periodic Heat Equation
The periodic heat equation has been derived as a model of the dynamics of a thin viscous fluid on the inside surface of a cylinder rotating around its axis. It is well known that the related Cauchy problem is generally
ill-posed. We study the spectral properties of the non-self-adjoint operator associated with this equation. Some open questions will be stated.
(joint work with D. Pelinovsky)
23 January
13:10 PM
Abdeslem Lyaghfouri (Fields Institute)
Hoelder Continuity of Solutions to Quasilinear Elliptic Equations Involving Measures.
I will present a paper by Tero Kilpeläinen on the Hölder continuity of the solutions of the p-Laplace equation with right-hand side measure. The relationship between the growth of the measure and the Hölder continuity of the solutions will be discussed.
16 January
11:10 AM
Kiumars Kaveh (University of Toronto)
Isoperimetric inequality, its generalizations and applications
The classical isoperimetric inequality states that if P is the perimeter of a closed simple curve in the plane and A is the area of the region enclosed by the curve, then 4\pi A is less than or equal to P^2 and the equality holds if and only if the curve is a circle. It follows that among all the curves with given perimeter, circle encloses the biggest area. The origin of the inequality goes back to the antiquity. We will discuss this inequality and its generalizations to arbitrary dimensions namely Brunn-Minkowski and Alexanderov-Fechel inequlaities. They invlove mixed volumes of convex bodies in R^n. If there is
time, we mention connecion with algebaric geometry.
9 January
11:10 AM
Larry Guth (Stanford University)
Packing widths and isoperimetric inequalities

### 2007

5 December
11:10 AM
Yuxin Ge (Université Paris XII and University of Washington)
On the $\sigma_2$-scalar curvature and its application
In this talk, we establish an analytic foundation for a fully non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature. This equation arises from conformal geometry. As application, we prove that, if a compact 3-dimensional manifold $M$ admits a riemannian metric with positive scalar curvature and $\int \sigma_2\ge 0$, then topologically $M$ is a quotient of sphere.
5 December
1:10 AM
Alessio Figalli (CNRS Nice)
Caffarelli's Holder regularity theory of Monge-Ampere equations. Part II.
This is a continuation of the talk on Nov. 28.
28 November
11:10AM
Benjamin Stephens (University of Toronto)
Parallel transport in Wasserstein Space
We'll look at John Lott's derivations of the basic formulas of curvature and parallel transport in Wasserstein Space for a smooth compact Riemannian manifold.
We'll also describe what parallel transport looks like on the real line and relate it to a nonlinear diffusion problem.
28 November
1:10PM
Alessio Figalli (CNRS Nice)
Caffarelli's Holder regularity theory of Monge-Ampere equations
We will explain Caffarelli's Holder regularity theory of classical Monge-Ampere equations.
21 November
11:10AM
Alessio Figalli, CNRS Nice and SNS Pisa
A mass transportation approach to quantitative isoperimetric inequalities.
In this talk I will show how one can prove a sharp quantitative version of the anisotropic isoperimetric inequality by exploiting mass transportation theory, especially Gromov's proof of the isoperimetric inequality and the Brenier-McCann Theorem. This is a joint work with F. Maggi and A. Pratelli.
21 November
1:10PM
Young-Heon Kim, University of Toronto
An a priori second order derivative estimate for a Monge-Amp\'ere type
equation 2.

This is a continuation of the talk (with the same title) last week.
Nov. 14, 2007
11:10 a.m.
Maria Sosio, University of Toronto
An application of the continuity method.
I'll present an application of the continuity method that allows to reduce the solution of an elliptic differential equation to that one of the Poisson equation. In this proof the Schauder estimates, which I'll take for granted, play an important role for switching from the Laplacian to the elliptic differential operator. The main reference for this talk will be the book: Jurgen Jost, Partial Differential Equations, Springer (2007).
Nov. 14, 2007
1:10 p.m.

Young-Heon Kim, University of Toronto
An a priori second order derivative estimate for a Monge-Ampere type equation.
This is a learning seminar talk, and can be regarded as a continuation of the talk by Maria Sosio at 11h10. We will discuss the a priori second order derivative estimate given in the paper by Ma, Trudinger, and Wang Regularity of Potential Functions of the Optimal Transportation Problem'(Arch. Rational Mech. Anal. 177(2005) (51--183); see Section 4.

Nov. 7
13:00
Isoperimetric inequalities and universal covers
The Dehn function of a finitely generated group G relates the algebraic structure of G to the geometry of the universal cover of a finite 2-complex X where \pi_1(X) = G. In 1999 Alonso, Wang, and Pride introduced q-dimensional analogues of the Dehn function for groups G of type F_q and proved a similar theorem relating these functions to the universal covers of finite, highly acyclic q- complexes, and in 1992 Epstein introduced similar functions with chains in place of sphere and disk maps. It is natural to ask whether these latter functions are the same, and what role the "highly acyclic" condition plays if any. I will give brief definitions of Alonso's and Epstein's functions, as well as generalizations to maps M \to X where M is a compact oriented manifold with boundary, and present partial results relating the functions and further generalizing AWP's results.
October 31
11:10 - 12:00
Almut Burchard, University of Toronto
Eternal solutions to the Ricci flow on R^2.
I will discuss recent work of Daskalopoulos and Sesum on eternal solutions to the Ricci flow on R^2. My goal is to provide some context for recent work of Robert McCann, Aaron Smith and myself on the evolution of a particular family of asymmetric cigar-shaped surfaces under Ricci flow.
October 10
11:30 -- 2:00*
Fields Analysis With Geometry: Working Group Organizational Meeting
The Fields Analysis Working Group has agreed to join forces with last year's Geometry Seminar for the coming year. A brief organizational meeting to discuss plans for the semester, to be followed by a brown bag lunch.
*Note exceptional time!
August 16
13:00 -- 14:00
Brendan Pass (University of Toronto)
Optimal Transportation on Alexandrov Spaces
I will present a paper by J. Bertrand on the existence and uniqueness of optimal maps on Alexandrov spaces with curvature bounded below.
11:10 -- 12:10

Nathan Killoran (University of Toronto)
Supports of Extremal Doubly and Triply Stochastic Measures
Doubly stochastic measures are Borel probability measures on the unit square which push forward via the projection maps to the Lebesgue measure on each axis. The set of doubly stochastic measures is convex, so its extreme points are of particular interest. I will examine the necessary and sufficient conditions for a set to support an extremal doubly stochastic measure, and demonstrate that such a set can be decomposed into a countable collection of Graphs of functions, called a limb-numbering system.' I will also show how this structure generalizes to triply stochastic measures on the unit cube.

July 5

Abdeslem Lyaghfouri (King Fahd University of Petroleum and Minerals)
On the Dam Problem with Two Fluids Governed by a Nonlinear Darcy's Law
In joint work with S. Challal,we consider the problem of two fluids flow through a porous medium governed by a nonlinear law. We establish the existence of a weak solution and prove that it is locally Lipschitz continuous in the zone above the lower fluid. Then we prove the continuity of the upper free boundary. In the rectangular case, we prove the existence of a monotone solution with respect to the vertical variable, and the continuity of the lower free boundary. Finally, we prove that there is a unique monotone solution with respect to x and y, when the dam is rectangular and the flow is obeying to the linear Darcy law.