Current Seminars

Graduate Student Seminar:
Location: MSRI: Baker Board Room Speakers: Melvin Leok (University of California, San Diego)Updated on Sep 18, 2018 12:26 PM PDT 
Hamiltonian Seminar: Stability for PDEs, the Maslov Index, and Spatial Dynamics
Location: MSRI: Simons Auditorium Speakers: Margaret Beck (Boston University)Understanding the stability of solutions to PDEs is important, because it is typically only stable solutions which are observable. For many PDEs in one spatial dimension, stability is wellunderstood, largely due to a formulation of the problem in terms of socalled spatial dynamics, where one views the single spatial variable as a timelike evolution variable. This allows for many powerful techniques from the theory of dynamical systems to be applied. In higher spatial dimensions, this perspective is not clearly applicable. In this talk, I will discuss recent work that suggests both that the Maslov index could be a important tool for understanding stability when the system has a symplectic structure, particularly in the multidimensional setting, and also suggests a possible analogue of spatial dynamics in the multidimensional setting.
Updated on Sep 14, 2018 08:46 AM PDT
Upcoming Seminars

Combinatorics Seminar: NearEquality of Ribbon Schur Functions
Location: UC Berkeley Math (Evans Hall 939) Speakers: Tom Foster (University of California, Berkeley)We consider the problem of when the difference of two ribbon Schur functions is a single Schur function. We prove that this nearequality phenomenon occurs in fourteen infinite families and we conjecture that these are the only possible cases. Towards this converse, we prove that under certain additional assumptions the only instances of nearequality are among our fourteen families. In particular, we prove that our first ten families are a complete classification of all cases where the difference of two ribbon Schur functions is a single Schur function whose corresponding partition has at most two parts at least 2. We also provide a framework for interpreting the remaining four families and we explore some ideas towards resolving our conjecture in general. We also determine some necessary conditions for the difference of two ribbon Schur functions to be Schurpositive.
Updated on Sep 13, 2018 01:18 PM PDT 
Hamiltonian Colloquium: From Hamiltonian systems with infinitely many periodic orbits to pseudorotations via symplectic topology
Location: MSRI: Simons Auditorium Speakers: Basak Gurel (University of Central Florida)Ever since the ConleyZehnder proof of the Arnold conjecture for tori, the study of periodic orbits has arguably been the most important interface between Hamiltonian dynamical systems and symplectic topology. A general feature of Hamiltonian systems is that they tend to have numerous periodic orbits. In fact, for a broad class of closed symplectic manifolds, every Hamiltonian diffeomorphism has infinitely many simple periodic orbits.
There are, however, notable exceptions. Namely, an important class of symplectic manifolds including the twosphere admits Hamiltonian diffeomorphisms with finitely many periodic orbits — the socalled pseudorotations — which are of particular interest in dynamical systems. Furthermore, recent works by Bramham (in dimension two) and by Ginzburg and myself (in dimensions greater than two) show that one can obtain a lot of information about the dynamics of pseudorotations, going far beyond periodic orbits, via symplectic techniques.
In this talk I will discuss various aspects of the existence question for periodic orbits of Hamiltonian systems, focusing on recent higher dimensional results about pseudorotations.Updated on Sep 20, 2018 02:36 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:26 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Lunch with Hamilton: Topological dynamics in threedimensional volumepreserving maps
Location: MSRI: Baker Board Room Speakers: Kevin Mitchell (University of California, Merced)Symbolic dynamics, and the associated topological entropy, are well
developed tools for analyzing twodimensional areapreserving
dynamics, such as arise in 2D symplectic maps and the chaotic mixing
of 2D fluids. For example, topological entropy has been useful in
quantifying the mixing of fluids stirred by periodically braiding
rods. However, at present no analogous symbolic techniques exist for
extracting topological dynamics from symplectic maps in higher
dimensions. Here, we address chaotic, volumepreserving maps in
threedimensions, which is a stepping stone to 4D symplectic maps and
a system of intrinsic interest for mixing of 3D fluids. We address
this challenge using the topology of intersecting codimensionone
stable and unstable manifolds. This leads to a symbolic dynamics of
2D surfaces based on homotopy theory. This symbolic dynamics can be
understood as resulting from stirring by loops that undergo a kind of
3D braiding. The resulting theory provides a rigorous lower bound on
the growth rates of both twodimensional surfaces and onedimensional
curves. We illustrate our theory with a mathematical model of a
chaotic ring vortex. Finally, we will present results that hint at
the presence of a subtle duality in the topological dynamics.Updated on Sep 20, 2018 11:46 AM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:26 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Celestial Mechanics: Scattering Ct'd
Location: MSRI: Baker Board Room Speakers: Richard Montgomery (University of California, Santa Cruz)I will describe the classical Rutherford
scattering, following Knauf's treatment
in which the scattering cross section is the
pushforward of the Lebesque measure
to the sphere of outgoing directions.
Created on Sep 21, 2018 08:36 AM PDT 
UC Berkeley Colloquium: You can hear the shape of a billiard table
Location: UC Berkeley Math (Evans Hall 60) Speakers: Moon Duchin (Tufts University)A great deal of fundamental mathematics has been directed at the question of "hearing the shape of a drum," or reading geometric features of a plane domain or manifold off from its Laplace spectrum. I'll address a parallel question in symbolic dynamics: if you have a Euclidean polygon and only know the sequences of sides struck in succession by billiard trajectories—that is, the bounce spectrum—does it determine the polygon? Spoiler: The answer is basically yes. This is joint work with Erlandsson, Leininger, and Sadanand.
Updated on Sep 06, 2018 11:14 AM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Hamiltonian Seminar: Lagrangian spectral invariants, graph selector and AubryMather theory
Location: MSRI: Simons Auditorium Speakers: YongGeun Oh (Institute for basic science)In this talk, I will first introduce BernadOliviera dos Santos's symplectic description of Mane critical value, Aubry set and Mane set, and Arnaud's graphicality theorem of invariant Lagrangian submanifold under the flow of Tonelli Hamiltonians. Then I will explain construction of the graph selector of exact Lagrangian submanifold and its extension to the class of Lipschitzexact Lagrangian submanifolds. Finally I will explain generalization of above mentioned results in AubryMather theory to this class of Lipschitzexact Lagrangian submanifolds. The main ingredient of the constructions is the Floer homology theory in symplectic topology. This talk is based on the joint work with Amorim and Oliviera dos Santos.
Updated on Sep 19, 2018 10:18 AM PDT 
Combinatorics Seminar
Location: UC Berkeley Math (Evans Hall 939) Speakers: Thomas McConville (Massachusetts Institute of Technology)Created on Sep 13, 2018 11:21 AM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:27 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:27 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
An introduction to Delay Differential Equations and the Infinite Limit Cycle Bifurcation
Location: UC Berkeley Engineering (Etcheverry Hall 3110) Speakers: Richard Rand (University of California Berkeley)The differential equation x(t)'' + x(t) + x(t)^3 = 0 is conservative and admits no limit cycles. If the linear term x(t)
is replaced by a delayed term x(tT), where T is the delay, the resulting delay differential equation exhibits an
infinite number of limit cycles. The amplitudes of the limit cycles go to infinity in the limit as T approaches zero.
This newly discovered bifurcation will be illustrated after a general introduction to delay differential equations.
This work is based on a 2017 paper with graduate students M. Davidow and B. Shayak.Created on Sep 13, 2018 04:03 PM PDT 
Combinatorics Seminar: The Taylor coefficients of the Jacobi theta constant Î¸3
Location: UC Berkeley Math (Evans Hall 939) Speakers: Dan Romik (University of California, Davis)We study the Taylor expansion around the point x=1 of a classical modular form, the Jacobi theta constant θ3. This leads naturally to a new sequence (d(n))∞n=0=1,1,−1,51,849,−26199,… of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of θ3. We prove several results about the numbers d(n) and conjecture that they satisfy the congruence d(n)≡(−1)n−1 (mod 5) and other similar congruence relations.
Updated on Sep 13, 2018 01:25 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:28 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:29 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Combinatorics Seminar: Walks, groups and Difference Equations
Location: UC Berkeley Math (Evans Hall 939) Speakers: Michael Singer (University College)Many questions in combinatorics, probability and statistical mechanics can be reduced to counting lattice paths (walks) in regions of the plane. A standard approach to counting problems is to consider properties of the associated generating function. These functions have long been well understood for walks in the full plane and in a half plane. Recently much attention has focused on walks in the first quadrant of the plane and has now resulted in a complete characterization of those walks whose generating functions are algebraic, holonomic (solutions of linear differential equations) or at least differentially algebraic (solutions of algebraic differential equations).
I will give an introduction to this topic, discuss previous work of BousquetMelou, Kauers, Mishna, and others and then present recent work by Dreyfus, Hardouin, Roques and myself applying the theory of QRT maps and Galois theory of difference equations to determine which generating functions satisfy differential equations and which do not.Updated on Sep 13, 2018 01:36 PM PDT 
Hamiltonian Colloquium:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 01:40 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:35 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Lunch with Hamilton:
Location: MSRI: Baker Board RoomCreated on Aug 24, 2018 02:30 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:27 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Hamiltonian Seminar:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 03:29 PM PDT 
Hamiltonian Colloquium:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 01:40 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:29 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Lunch with Hamilton:
Location: MSRI: Baker Board RoomCreated on Aug 24, 2018 02:30 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:29 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Hamiltonian Seminar:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 03:29 PM PDT 
Combinatorics Seminar
Location: UC Berkeley Math (Evans Hall 939) Speakers: Mariel Supina (University of California, Berkeley)Created on Sep 13, 2018 11:21 AM PDT 
Hamiltonian Colloquium:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 01:40 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:30 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Lunch with Hamilton:
Location: MSRI: Baker Board RoomCreated on Aug 24, 2018 02:30 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:30 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Hamiltonian Seminar:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 03:29 PM PDT 
Combinatorics Seminar
Location: UC Berkeley Math (Evans Hall 939)Created on Sep 13, 2018 11:21 AM PDT 
Hamiltonian Colloquium:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 01:40 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:30 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Lunch with Hamilton:
Location: MSRI: Baker Board RoomCreated on Aug 24, 2018 02:30 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:31 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Hamiltonian Seminar:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 03:29 PM PDT 
Hamiltonian Colloquium:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 01:40 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:31 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Lunch with Hamilton:
Location: MSRI: Baker Board RoomCreated on Aug 24, 2018 02:30 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:31 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Hamiltonian Colloquium:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 01:40 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:32 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Lunch with Hamilton:
Location: MSRI: Baker Board RoomCreated on Aug 24, 2018 02:30 PM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Hamiltonian Seminar:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 03:29 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:32 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:33 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Hamiltonian Colloquium:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 01:40 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:33 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Lunch with Hamilton:
Location: MSRI: Baker Board RoomCreated on Aug 24, 2018 02:30 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:34 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Hamiltonian Seminar:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 03:29 PM PDT 
Hamiltonian Colloquium:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 01:40 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:34 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Lunch with Hamilton:
Location: MSRI: Baker Board RoomCreated on Aug 24, 2018 02:30 PM PDT 
Chancellor Course: Topics in Analysis
Location: UC Berkeley: Evans Hall, Room 748 Speakers: Wilfrid Gangbo (University of California, Los Angeles)This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.
Updated on Aug 17, 2018 03:34 PM PDT 
Weak KAM Theory, Homogenization and Symplectic Topology
Location: UC Berkeley Math Speakers: Fraydoun Rezakhanlou (University of California, Berkeley)In this course we will explore the connection between HamiltonJacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to KolmogorovArnoldMoser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. AubryMather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated HamiltonJacobi PDE to construct AubryMather invariant measures. Recently there have been several interesting works to understand the connection between AubryMather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions. In this course, we also explore the connection between AubryMather Theory and the homogenization phenomena for HamiltonJacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.
Created on Aug 24, 2018 03:42 PM PDT 
Graduate Student Seminar
Location: MSRI: Baker Board RoomCreated on Sep 07, 2018 01:47 PM PDT 
Hamiltonian Seminar:
Location: MSRI: Simons AuditoriumCreated on Aug 24, 2018 03:29 PM PDT
Past Seminars

Seminar Weak KAM Theory, Homogenization and Symplectic Topology
Created on Aug 24, 2018 03:42 PM PDT 
Seminar Celestial Mechanics: Scattering in the Nbody problem
Updated on Sep 18, 2018 11:43 AM PDT 
Seminar Chancellor Course: Topics in Analysis
Updated on Aug 17, 2018 03:25 PM PDT 
Seminar Lunch with Hamilton: Lie group and homogeneous variational integrators and their applications to geometric optimal control theory
Updated on Sep 13, 2018 03:50 PM PDT 
Seminar Weak KAM Theory, Homogenization and Symplectic Topology
Created on Aug 24, 2018 03:42 PM PDT 
Seminar Chancellor Course: Topics in Analysis
Updated on Aug 17, 2018 03:25 PM PDT 
Seminar Hamiltonian Colloquium: Almostinvariant tori
Updated on Sep 13, 2018 09:09 AM PDT 
Seminar Combinatorics Seminar: Divisors on matroids and their volumes
Updated on Sep 14, 2018 01:10 PM PDT 
Seminar Hamiltonian Seminar: Introducing symplectic billiards
Updated on Sep 06, 2018 09:53 AM PDT 
Seminar Graduate Student Seminar
Created on Sep 07, 2018 10:39 AM PDT 
Seminar Mathematics Department Colloquium: Cantor invariant subsets of conservative 2dimensional dynamics
Created on Sep 10, 2018 09:22 AM PDT 
Seminar Weak KAM Theory, Homogenization and Symplectic Topology
Created on Aug 24, 2018 03:42 PM PDT 
Seminar Special Seminar: Simultaneous Binary Collisions and the Mysterious 8/3
Created on Sep 07, 2018 01:57 PM PDT 
Seminar Chancellor Course: Topics in Analysis
Updated on Aug 17, 2018 03:25 PM PDT 
Seminar Lunch with Hamilton: A new necessary and sufficient condition for the stability of linear Hamiltonian systems with periodic coefficients
Updated on Sep 06, 2018 08:44 AM PDT 
Seminar Special Seminar: On polynomially integrable billiards on surfaces of constant curvature
Created on Sep 06, 2018 09:56 AM PDT 
Seminar Weak KAM Theory, Homogenization and Symplectic Topology
Created on Aug 24, 2018 03:42 PM PDT 
Seminar Chancellor Course: Topics in Analysis
Updated on Aug 17, 2018 03:24 PM PDT 
Seminar Hamiltonian Colloquium: Capturing photo electron motion with guiding centers: a Hamiltonian approach
Updated on Aug 29, 2018 10:35 AM PDT 
Seminar Hamiltonian Seminar: Ground states are generically a periodic orbit
Updated on Aug 30, 2018 08:59 AM PDT 
Seminar Postdoc Lunch with Senior Researcher
Updated on Sep 06, 2018 04:55 PM PDT 
Seminar Weak KAM Theory, Homogenization and Symplectic Topology
Created on Aug 24, 2018 03:42 PM PDT 
Seminar Chancellor Course: Topics in Analysis
Updated on Aug 17, 2018 03:24 PM PDT 
Seminar Lunch with Hamilton: Parabolic resonances and other nonseparable structures
Updated on Aug 29, 2018 02:08 PM PDT 
Seminar Weak KAM Theory, Homogenization and Symplectic Topology
Created on Aug 24, 2018 03:42 PM PDT 
Seminar Chancellor Course: Topics in Analysis
Updated on Aug 17, 2018 03:24 PM PDT 
Seminar Hamiltonian Seminar: A local systolic inequality in contact and symplectic geometry
Updated on Aug 23, 2018 01:40 PM PDT 
Seminar Five Minute Talks
Updated on Aug 24, 2018 09:08 AM PDT 
Seminar Weak KAM Theory, Homogenization and Symplectic Topology
Created on Aug 23, 2018 04:01 PM PDT 
Seminar Chancellor Course: Topics in Analysis
Updated on Aug 17, 2018 03:23 PM PDT