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Lecture Series

The 2018 Jacqueline B. Lewis Memorial Lecturer is Camillo De Lellis

Institute for Advanced Study, Princeton, NJ

26th Jacqueline B. Lewis Memorial Lectures

Open to the Public – Use Lots 60A or 64 behind Hill Center
For a Parking Permit, call 848-445-6991
Wednesday, October 24   |   Thursday, October 25   |   Friday, October 26
Wed Oct 24 @12:00PM - 01:00PM
The Oriented Plateau Problem
Camillo De Lellis, IAS, Princeton

Abstract: The Plateau problem, named by Henry Lebesgue after the Belgian physicist, consists in finding the surface of least area which spans a given contour.  In order to tackle such question, generations of mathematicians have investigated the very fundamental notions of ``surface", ``boundary" and ``area", proposing a variety of different theories.  In this talk I will give a brief exposition of the so-called theory of currents, introduced by Federer and Fleming in the 60's after the pioneering work of De Giorgi in the case of hypersurfaces.  I will then discuss an open question relating the shapes of the contour and that of the minimizer, posed by Almgren in the early eighties and recently solved in a joint work with Guido de Philippis, Jonas Hirsch and Annalisa Massaccesi.

Hill 705
Thu Oct 25 @ 3:30PM - 04:40PM
Regularity of Area Minimizing Currents: Interior
Camillo De Lellis, IAS, Princeton

Abstract: The theory of integral currents, developed by Federer and Fleming in the 60's, gives a powerful framework to solve the Plateau's problem in every dimension and codimension and general ambient manifolds.  The interior and boundary regularity theory for the codimension one case is rather well understood, thanks to the work of several mathematicians in the 60's, 70's and 80's.  In codimension higher than one the phenomenon of branching causes instead very serious problems.  A celebrated monograph of Almgren provided a far-reaching interior regularity theorem.  However, his original (typewritten) manuscript was more than 1700 pages long.  Four years ago, in a series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program.  This has allowed us to go beyond Almgren's result in several directions.

Hill 705
Hospitality 2:30-3:30pm in Room 703
Fri Oct 26 @ 2:00PM - 03:00PM
Regularity of Area Minimizing Currents: boundary
Camillo De Lellis, IAS, Princeton

Abstract: At the end of his monograph Almgren addresses the question of regularity of solutions at the boundary.  Full regularity was proved by Allard in his Ph.D. thesis when the ambient manifold is the Euclidean space and the boundary surface lies in the boundary of uniformly convex open set.  The general case in codimension 1 ws then settled by Hardt and Simon in the early 80's.  But in codimension higher than 1 and in general ambient manifolds the current state of the art does not even give guarantee the existence of a single boundary regular point.  This prevents the understanding of seemingly innocent questions like the following: does the connectedness of the boundary imply the connectedness of the minimizer?  In a joint work with Guido de Philippis, Jonas Hirsch and Annalisa Massaccesi we give a first general boundary regularity theory which allows us to answer positively to the question above.

Hill 705
Hospitality 1:00-2:00pm in Room 703
Professor Jacqueline B. Lewis was a mathematician and former Dean of Rutgers' University College in New Brunswick who died in 1982 after a career spanning nearly 20 years at Rutgers. She served as an Associate Dean of University College from 1974 to 1978 and as Vice Dean of the school from 1978 to 1981, when she was appointed Dean. She also served as Acting Dean of the Faculty of Professional Studies from June 1981 to September 1982. The lectures were endowed in 1983 by a gift from Dean Lewis' aunt, Lillian Nassau.
Previous Lewis Lectures
Nolan Wallach October 2015 Basic Geometric Invariant Theory I, II
Some Applications of Geometric Invariant Theory
Yum-Tong Siu December 2013 The Complex Neumann Problem and Multiplier Ideal Sheaves
Methods of Partial Differential Equations in Complex Algebraic Geometry
Analytic Methods of Constructing Bundle Sections and their Geometric Applications
Jennifer Tour Chayes November 2010 The Mathematics of Dynamic Random Networks
Andrei Zelevinsky April 2010* Introduction to Cluster Algebras
Cluster Algebras of Finite Type and their Geometric Realizations
Quivers with Potentials and Generalized Reflection Functors
* – Postponed from 2009
Terence Tao March 2008 Szemerédi's Regularity Lemma Revisited
Three Lectures on Discrete Random Matrices:
  Singularity and Determinant of Discrete Random Matrices
  The Least Singular Value of Discrete Random Matrices
  Eigenvalue Distributions of Discrete Random Matrices
Andrei Okounkov April 2007 Moduli of Curves and Combinatorics
Shmuel Weinberger October 2004 Themes in Quantitative Topology:
  Problems and naive examples
  Computation, entropy, and variational problems
  Embeddings, symmetry, and rigidity
Jean-Michel Coron October 2003 Controllability and nonlinearity for some flow control systems
Richard P. Stanley October 2002 Six Recent Developments in Algebraic Combinatorics:
  The Laurent phenomenon; longest increasing subsequences
  The Saturation Conjecture; the n! Conjecture
  Gromov-Witten invariants; graphical degree sequences
Graeme Segal April 2001 The Mathematical Structure of Quantum Field Theory;
Quantum Field Theory and K-Theory;
Quantum Field Theory and Representation Theory
Yves Meyer March 2000 Wavelets and Image Processing: Theory and Application to Denoising
  Hubble Space Telescope Images;
Oscillations, Vibrations, Time-Frequency Analysis and the Virgo Program
  (Detection of Gravitati onal Waves);
The Role of Oscillations in Some Non-Linear Evolution Equations:
  Application to Navier-Stokes Equations
Don Zagier March 1997 Modular Forms and Differential Operators
Alain Connes April 1996 Gravity Coupled with Matter and the Foundation of Noncommutative Geometry
Karen Uhlenbeck September 1994 Moduli Spaces of Solutions to PDE
V. I. Arnold November 1993 On Some Problems in Singularity Theory
Ian Macdonald March 1993 Symmetric Functions and Orthogonal Polynomial
L. Craig Evans March 1992 Compactness for Solutions of Nonlinear Partial Differential Equations
Jurgen Moser March 1991 Stability in Dynamics and Minimal Solutions in the Calculus of Variations
Yasha G. Sinai April 1990 Random Behavior of Eigenvalues of Laplacians
Paul Rabinowitz April 1989 Periodic Solutions of Hamiltonian Systems
H. Blaine Lawson April 1988 Algebraic Cycles
David Kazhdan March 1987 Representations of Reductive P-Adic Groups
Lipman Bers October 1985 Teichmüller Theory for Beginners
G. D. Mostow March 1985 Braids, Hypergeometric Functions, and Lattices
Stephen Smale March 1984 The Topology of Classical Algorithms
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Department of Mathematics

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Rutgers University
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