Welcome to the Maryland Distinguished Lectures in Algebra and Number Theory! This is an annual lecture series concerning recent developments in Algebra and Number Theory, delivered by the world's foremost experts. It is organized by Thomas Haines.

2025 Davesh Maulik

2024 Zhiwei Yun

Theme: Higher theta series 

Abstract: Theta series play an important role in the classical
theory of modular forms. In the modern language of automorphic
representations, they are constructed from a pair of groups $G$ and
$H$ (one orthogonal and one symplectic, or both unitary groups) and
the remarkable Weil representation of $G\times H$.  Kudla introduced
an analogue of theta series in arithmetic geometry, by forming a
generating series of algebraic cycles on Shimura varieties. The
arithmetic theta series has since become a very active program.

In joint work with Tony Feng and Wei Zhang, we consider analogues of
arithmetic theta series over function fields, and try to go further
than what was done over number fields.  Our work concentrated on
unitary groups. We defined a generating series of algebraic cycles on
the moduli stack of unitary Drinfeld Shtukas (called the higher theta
series). We made the Modularity Conjecture:  the higher theta series
is an automorphic form valued in a certain Chow group. This is a
function field analogue of the special cycles generating series
defined by Kudla and Rapoport, but with an extra degree of freedom
namely the number of legs of the Shtukas.

One concrete formula we proved was a higher derivative version of the
Siegel-Weil formula. It is an equality between degrees of
0-dimensional special cycles on the moduli of unitary Shtukas and
higher derivatives of the Siegel-Eisenstein series of another unitary
group. More recently, we have obtained a proof of a weaker version of
the Modularity Conjecture, confirming that the cycle class of the
higher theta series (valued in the cohomology of the generic fiber) is
automorphic.

The series of talks will feature a colloquium-style introduction to
some representation-theoretic and geometric background (the second talk), the other
two being more technical talks in which I will explain some ingredients in
the proofs of the higher Siegel-Weil formula and the weak Modularity
Conjecture. 

Higher Theta series, Part I

Tuesday, February 27 at 3:30 pm
University of Maryland - Kirwan Hall Rm 3206 

Higher Theta series, Part II (Colloquium)

Wednesday, February 28 at 3:15 pm
University of Maryland - Kirwan Hall Rm 3206 

Higher Theta series, Part III

Thursday, February 29 at 3:30 pm
University of Maryland - Kirwan Hall Rm 3206 

2022 Mark Kisin

Theme: Arithmetic of abelian varieties and their moduli

Essential dimension and prismatic cohomology (Colloquium)

Wednesday, December 7 at 3:15pm
University of Maryland - Kirwan Hall Rm 3206

Abstract: The smallest number of parameters needed to define an algebraic covering space is called its essential dimension. Questions about this invariant go back to Klein, Kronecker and Hilbert and are related to Hilbert's 13th problem. In this talk, I will give a little history, and then explain a new approach which relies on recent developments in $p$-adic Hodge theory. This is joint work with Benson Farb and Jesse Wolfson.

Frobenius conjugacy classes attached to abelian varieties

Thursday, December 8 at 2:00pm
University of Maryland - Kirwan Hall Rm 3206

Abstract: The Mumford-Tate group of an abelian variety $A$ over the complex numbers is an algebraic group $G$, defined in terms of the complex geometry of $A$, more specifically its Hodge structure. If $A$ is defined over a number field $K$, then a remarkable result of Deligne asserts that the ℓ -adic cohomology of $A$ gives rise to a $G$ -valued Galois representation ρ_ℓ:Gal($\bar K/K$)→$G(\mathbb Q_\ell)$. We will show that for a place of good reduction v∤ℓ of A, the conjugacy class of Frobenius ρ_ℓ(Frob_v) does not depend on ℓ. This is joint work with Rong Zhou.

Heights in the isogeny class of an abelian variety

Friday, December 9 at 2:00pm
University of Maryland - Kirwan Hall Rm 3206

Abstract: Let $A$ be an abelian variety over $\bar{\mathbb Q}.$ In this talk I will consider the following conjecture of Mocz. Conjecture: Let $c > 0.$ In the isogeny class of $A,$ there are only finitely many isomorphism classes of abelian varieties of height $ c.$ I will sketch a proof of the conjecture when the Mumford-Tate conjecture - which is known in many cases - holds for $A.$ This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field. This is joint work with Lucia Mocz.

2021 Wei Zhang

2019 Ngô Bảo Châu

Dr. Kadir Aziz, who had been a Professor Emeritus at the Department of Mathematics and Statistics at UMBC since 1989, passed away on March 25, 2016 in Chevy Chase, Maryland. He was 92 years old.

Kadir was born in Afghanistan in 1923. He grew up and received his early education in Paris where his father was the Afghan ambassador, and later in Washington, DC, where he obtained a bachelor's degree from Wilson Teachers College (now merged with the University of District of Columbia) in 1952, and a Master's degree from George Washington University in 1954.

Thereafter he entered the doctoral program in mathematics at the University of Maryland. His doctoral dissertation in 1958, titled "On Higher Order Boundary Value Problems for Hyperbolic Partial Differential Equations in Two and Three Variables" was written under the guidance of Dr. Joaquin Diaz.

Upon receiving the PhD degree, he obtained a faculty position at Georgetown University, where he quickly rose through the ranks, and in 1963, only five years after his PhD, was appointed a full Professor of Mathematics. In 1966 he assumed the duties of the Department Chairman there.

A year later, in 1967, Kadir moved to the nascent UMBC campus as one of the original senior faculty members of its College of Sciences. At the same time, he was appointed Adjunct Research Professor at the Institute for Physical Science and Technology at UMCP. Kadir was a major force in setting up the foundations of what has now become a thriving Department of Mathematics and Statistics at UMBC.

Kadir's research focused on the numerical analysis of partial differential equations. He was one of pioneers of what became known as the Finite Element Method (FEM). This method rapidly became one of the most powerful and indispensable tools for treating numerical problems of engineering, physics, and other sciences.

It is remarkable that the very first international conference on the mathematical theory of the FEM was held at UMBC in 1972, when UMBC was only six years old. The proceedings of that conference, a book edited by Kadir, and containing a groundbreaking monograph written by him and Ivo Babuska, went on to become a standard reference on the subject and an inspiration for the future development of the field.

During his years at Georgetown and UMBC, Kadir's research was supported by grants from the National Science Foundation, Office of Naval Research, Air Force Office of Scientific Research, Department of Energy, and the Naval Surface Weapons Center. Kadir supervised the dissertations of 14 doctoral students at Georgetown, UMCP, and UMBC.

In 1999, Kadir donated funds to establish what is known as the Aziz Lecture Series -- initially organized at UMBC, and later at UMCP. The purpose of the series is to provide a forum for expository lectures by experts in the field on the numerical solutions of differential equations. One or two Aziz Lectures have been delivered each year since the establishment of this continuing series.

Kadir was well-known for his joie de vivre -- he loved good wine, good food and good conversation. He will be missed for his heartiness, generosity, and sense of humor.

Lectures on Differential Equations and their Numerical Analysis

The Aziz Lectures were established by a generous gift from Prof. A. Kadir Aziz. The purpose of the lectures is to bring distinguished mathematicians to the University of Maryland, College Park, to give survey lectures on differential equations, their numerical analysis, and related areas.

Kadir Aziz, who died on March 25, 2016 at the age of 92, received his Ph.D. from the University of Maryland, College Park in 1957. He was on the faculty of Georgetown University from 1956 to 1967, and was on the faculty at the University of Maryland, Baltimore County (UMBC) since 1967. He was Professor Emeritus of Mathematics and Statistics at UMBC. Throughout his career Kadir Aziz was an active and highly respected member of the Numerical Analysis community at Maryland.

CV of Kadir Aziz

The Aziz lecture is given at 3:15pm in the Math Colloquium Room (MTH 3206).

Usually the speaker gives a related talk in the Applied Math Colloquium on the previous day at 3:30pm.

Aziz Lectures 2024

Both the Applied Math Colloquium (September 3 at 3:30pm) and the Aziz Lecture (September 4 at 3:15pm) will be broadcast via Zoom

September 4, 2024

Multilevel approximation of Gaussian random fields
Christoph Schwab
ETH, Zurich
Switzerland

Centered Gaussian random fields (GRFs) indexed by compacta as e.g. compact orientable manifolds M are determined by their covariance operators. We consider the numerical analysis of sample-wise, compressive multi-level wavelet-Galerkin approximations of centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with pseudodifferential covariance operator being elliptic, self-adjoint and positive from the Hörmander class. For pathwise approximations with p parameters, tapered covariance or precision matrices have O(p) nonzero entries, can be optimally diagonally preconditioned, and allow O(p) path simulation, covariance estimation and kriging of GRFs.

Joint work with Helmut Harbrecht (Uni Basel), Kristin Kirchner (TU Delft), and Lukas Herrmann (RICAM, Linz).

Aziz Lectures 2023

April 12, 2023 - 3:15pm

Some Mathematical Aspects of Deep Learning and Stochastic Gradient Descent
Lexing Ying
Stanford University

This talk concerns several mathematical aspects of deep learning and stochastic gradient descent. The first aspect is why deep neural networks trained with stochastic gradient descent often generalize. We will make a connection between the generalization and the stochastic stability of the stochastic gradient descent dynamics. The second aspect is to understand the training process of stochastic gradient descent. Here, we use several simple mathematical examples to explain several key empirical observations, including the edge of stability, exploration of flat minimum, and learning rate decay. Based on joint work with Chao Ma.

Aziz Lectures 2019

May 1, 2019

Operator Preconditioning
Ralf Hiptmair
ETH Zurich
Switzerland

This text is printed on a background of locally refined finite element meshes. For the fast iterative solution of finite element models on such meshes preconditioning is indispensable. This is where operator preconditioning enters stage: it offers a general all-purpose recipe for constructing preconditioners for discrete linear operators that have arisen from a Galerkin approach, in particular, from finite element methods and boundary element methods. The key idea is to employ matching Galerkin discretizations of operators of complementary mapping properties. If these can be found, the resulting preconditioners will be robust with respect to the choice of the bases for trial and test spaces. As a consequence, in a finite element setting, they will still perform well even for high-resolution models.

Aziz Lectures 2018

September 12, 2018

Smooth random functions and smooth random ODEs
Lloyd N. Trefethen
Oxford University
United Kingdom

What is a random function? What is noise? The standard answers are nonsmooth, defined pointwise via the Wiener process and Brownian motion. In the Chebfun project, we have found it more natural to work with smooth random functions defined by finite Fourier series with random coefficients. The length of the series is determined by a wavelength parameter lambda. Integrals give smooth random walks, which approach Brownian paths as lambda shrinks to 0, and smooth random ODEs, which approach stochastic DEs of the Stratonovich variety. Numerical explorations become very easy in this framework. There are plenty of conceptual challenges in this subject, starting with the fact that white noise has infinite amplitude and infinite energy, a paradox that goes back in two different ways to Einstein in 1905.

May 4, 2018

Stochastic Nonlinear Schrödinger Equations
Arnaud Debussche
École Normale Supérieure de Rennes 

The nonlinear Schrödinger equation is a prototype model to describe propagation of waves in dispersive media. It arises in several models and noise appears naturally. It may represent the noise due to amplifiers or random dispersion in the fiber. In this talk I will present some aspects of wellposedness and influence on blow-up phenomena for the stochastic nonlinear Schrödinger equation.

February 7, 2018

Mathematical theory and computational approaches for modern materials science
Claude Le Bris
Ecole des Ponts and Inria

The talk, intended for a general audience, will survey some challenging mathematical and numerical problems in contemporary materials science. Questions such as the passage from the microscale to the macroscale, the insertion of uncertainties, defects and heterogeneities in the models, will be examined. We will discuss the interesting issues raised for mathematical analysis (theory of partial differential equations, stochastic processes, homogenization theory) and for numerical analysis (finite element methods, discrete to continuum, Monte Carlo methods, etc).

Aziz Lectures 2016

December 9, 2016

Modeling traffic flow on a network of roads
Prof. Alberto Bressan
Department of Mathematics
Penn State University

The talk will present various PDE models of traffic flow on a network of roads. These comprise a set of conservation laws, determining the density of traffic on each road, together with suitable boundary conditions, describing the dynamics at intersections. While conservation laws determine the evolution of traffic from given initial data, actual traffic patterns are best studied from the point of view of optimal decision problems, where each driver chooses the departure time and the route taken to reach destination. Given a cost functional depending on the departure and arrival times, a relevant mathematical problem is to determine (i) global optima, minimizing the sum of all costs to all drivers, and (ii) Nash equilibria, where no driver can lower his own cost by changing departure time or route to destination. Several results and open problems will be discussed.

May 4, 2016

Quantum Dots and Dislocations: Dynamics of Materials
Prof. Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University

The formation and assembly patterns of quantum dots have a significant impact on the optoelectronic properties of semiconductors. We will address short time existence for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained three-dimensional films. Further, the nucleation of misfit dislocations will be analyzed.

This is joint work with Nicola Fusco, Giovanni Leoni and Massimiliano Morini.

Aziz Lectures 2015

November 18, 2015

Tensor Sparsity - a Regularity Notion for High Dimensional PDEs
Prof. Wolfgang Dahmen
Institute für Geometrie und Praktische Mathematik
RWTH Aachen University (Germany)

The numerical solution of PDEs in a spatially high-dimensional regime (such as the electronic Schrödinger or Fokker-Planck equations) is severely hampered by the “curse of dimensionality”: the computational cost required for achieving a desired target accuracy increases exponentially with respect to the spatial dimension.

We explore a possible remedy by exploiting a typically hidden sparsity of the solution to such problems with respect to a problem dependent basis or dictionary. Here sparsity means that relatively few terms from such a dictionary suffice to realize a given target accuracy. Specifically, sparsity with respect to dictionaries comprised of separable functions – rank-one tensors – would significantly mitigate the curse of dimensionality. The main result establishes such tensor-sparsity for elliptic problems over product domains when the data are tensor-sparse, which can be viewed as a structural regularity theorem.

April 15, 2015

Waves in random media: the story of the phase
Prof. Lenya Ryzhik
Department of Mathematics Stanford University

The macroscopic description of wave propagation in random media typically focuses on the scattering of the wave intensity, while the phase is discarded as a uselessly random object. At the same time, most of the beauty in wave scattering come from the phase correlations. I will describe some of the miracles, as well as some limit theorems for the wave phase.

May 6, 2015

Mathematical challenges in kinetic models of dilute polymers: analysis, approximation and computation
Prof Endre Süli
Mathematical Institute University of Oxford
United Kingdom

We survey recent analytical and computational results for coupled macroscopic-microscopic bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric fluids with noninteracting polymer chains, involving the unsteady Navier–Stokes system in a bounded domain and a high-dimensional Fokker–Planck equation. The Fokker–Planck equation emerges from a system of (Ito) stochastic differential equations, which models the evolution of a vectorial stochastic process comprised by the centre-of-mass position vector and the orientation (or configuration) vectors of the polymer chain. We discuss the existence of large-data global-in-time weak solutions to the coupled Navier–Stokes–Fokker–Planck system. The Fokker–Planck equation involved in the model is a high-dimensional partial differential equation, whose numerical approximation is a formidable computational challenge, complicated by the fact that for practically relevant spring potentials, such as finitely extensible nonlinear elastic potentials, the drift coefficient in the Fokker–Planck equation is unbounded.

Aziz Lectures 2014

November 12, 2014

The interplay between geometric modeling and simulation of partial differential equations
Prof. Annalisa Buffa
Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes"
Pavia, Italy

Computer-based simulation of partial differential equations involves approximation of the unknown fields and a description of geometrical entities such as the computational domain and the properties of the media. There are a variety of situations: in some cases this description is very complex, in some other the governing equations are very difficult to discretize. Starting with an historical perspective, I will describe the recent efforts to improve the interplay between the mathematical approaches characterizing these two aspects of the problem.

Aziz Lectures 2013

November 8, 2013

April 2-3, 2013

Aziz Lectures 2012

February 22, 2012

Aziz Lectures 2011

December 2, 2011

February 11, 2011

Aziz Lectures 2010

November 12, 2010

March 5, 2010

Aziz Lectures 2009

October 12, 2009

May 1, 2009

Aziz Lectures 2008

November 14, 2008

March 28, 2008

Aziz Lectures 2007

Nov. 16, 2007

May 4, 2007

Aziz Lectures 2006

December 1, 2006

April 21, 2006

February 3, 2006

Aziz Lectures 2005

December 9, 2005

Aziz Lectures 2004

November 19, 2004

May 7, 2004

Aziz Lectures 2003

November 14, 2003

March 7, 2003

Aziz Lectures 2002

Dec. 6, 2002

May 3, 2002

Aziz Lectures 2001

Nov. 16, 2001

April 23, 2001

Aziz Lectures 2000

March 15, 2000

Aziz Lectures 1999

December 10, 1999

September 24, 1999

February 19, 1999

Avron Douglis (1918-1995) received an AB degree in economics from the University of Chicago in 1938. After working as an economist for three years and serving in World War II he began graduate studies in mathematics at New York University. He received his doctorate in 1949 under the direction of Richard Courant. He held a one-year post-doctoral appointment at the California Institute of Technology, and then returned to New York University as an assistant and then associate professor. In 1956 he accepted an appointment as associate professor at the University of Maryland, where he remained for the rest of his career, except for visiting appointments at the Universities of Minnesota, Oxford, and Newcastle upon Tyne. He was promoted to full professor in 1958 and became an emeritus in 1988.

Avron Douglis's research, noted for its depth, precision, and richness, covered the entire range of the theory of partial differential equations: linear and nonlinear; elliptic, parabolic, and hyperbolic. The famous papers he had written with S. Agmon and L. Nirenberg are among the most frequently cited in all of mathematics.

The Avron Douglis Library is housed in the department.

The Avron Douglis Lectures were established by the family and friends of Avron Douglis to honor his memory. Each academic year it brings to Maryland a distinguished expert to speak on a subject related to partial differential equations.

November 20, 2024 - Free Boundary Regularity in Obstacle Problems

Alessio Figalli
ETH Zürich

The obstacle problem arises naturally when studying an elastic membrane constrained by contact with an object (the obstacle). This classical problem has captivated mathematicians for over 60 years. In this lecture, we will first review the fundamental theory of obstacle problems and then discuss recent developments in the regularity theory of free boundaries.

About the Speaker

Alessio Figalli is a Chaired Professor of Mathematics and FIM Director at ETH Zurich,  working primarily on calculus of variations and partial differential equations. He was awarded the Peccot-Vimont Prize and the Peccot Lectures in 2012, the EMS Prize in 2012, the Stampacchia Medal in 2015, the Feltrinelli Prize in 2017, and the Fields Medal in 2018. He was an invited speaker at the International Congress of Mathematicians 2014. He received the Doctorate Honoris Causa from the Université Côte d'Azur in 2018, the Polytechnic University of Catalonia  in 2019, and University of Sussex in 2022. He is a member of numerous Academies of Sciences including the European Academy, Royal Spanish Academy, Academy of Bologna, Academia Europaea and Accademia dei Lincei.

April 17, 2024 - Long Time Dynamics of Fluids and Plasmas

Peter Constantin
Princeton University

Fluids interacting with charged particles belong to many realms of scientific endeavor, ranging from electrochemistry and electrophysiology, to astrophysics and nuclear fusion engineering. I will discuss first the case of electrostatic interactions, relevant to electrochemistry and elctrophysiology. The models consist of Navier-Stokes equations for the fluid, coupled to equations describing the evolution of charge densities, with negligible magnetic effects. The resulting systems, Nernst-Planck-Navier-Stokes and electroconvection systems, have a rich dynamical behavior under non-equilibrium boundary conditions, but exhibit entropic relaxation under equilibrium boundary conditions. Mathematically, this relaxation is due to the presence of nontrivial dissipative structures. I will present some of the ideas of proofs of regularity and stability, some of the results about the existence of non-equilibrium states, and the problem of electroneutrality. I will then discuss strong magnetic fields and relaxation to magneto-hydrostatic (MHS) equilibria via viscous and non-dissipative mechanisms. This is a mathematical and potentially numerical approach to reach MHS configurations which are of interest to stellarator fusion design. I will present some of the challenges of this area. Finally, if time permits, I will also briefly discuss problems of relativistic plasma kinetics.

About the Speaker

Peter Constantin is the John von Neumann Professor of Mathematics and Applied and Computational Mathematics at Princeton University. Peter Constantin received his B.A and M.A. summa cum laude from the University of Bucharest, Faculty of Mathematics and Mechanics. He obtained his Ph.D.  from The Hebrew University of Jerusalem under the direction of Shmuel Agmon. Constantin's work is focused on the analysis of PDE and nonlocal models arising in statistical and nonlinear physics. Constantin worked on scattering for Schrodinger operators, on finite dimensional aspects of the dynamics of Navier-Stokes equations, on blow up for models of Euler equations. He introduced active scalars, and, with Jean-Claude Saut, local smoothing for general dispersive PDE. Constantin worked on singularity formation in fluid interfaces, on turbulence shell models, on upper bounds for turbulent transport, on the inviscid limit, on stochastic representation of Navier-Stokes equations, on the Onsager conjecture. He worked on critical nonlocal dissipative equations, on complex fluids, and on ionic diffusion in fluids. Constantin has advised thirteen graduate students in mathematics, and served in the committee of seven graduate students in physics. He mentored twenty-five postdoctoral associates. Constantin served as Chair of the Mathematics Department of the University of Chicago and is the Director of the Program in Applied and Computational Mathematics at Princeton University. Constantin is a Fellow of the Institute of Physics, a SIAM Fellow, and Inaugural Fellow of the American Mathematical Society, a Fellow of the American Academy of Arts and Sciences and a member of the National Academy of Sciences.

February 14, 2023 - Internal Waves in 2D Aquaria and Homeomorphisms of the Circle

Maciej Zworski
UC Berkeley

The connections between the formation of internal waves in fluids, spectral theory, and homeomorphisms of the circle were investigated by oceanographers in the 90s and resulted in novel experimental observations (Leo Maas et al, 1997). The specific homeomorphism is given by a ``chess billiard" and has been considered by many authors (Fritz John 1941, Vladimir Arnold 1957, Jim Ralston 1973... ). The relation between the nonlinear dynamics of this homeomorphism and linearized internal waves provides a striking example of classical/quantum correspondence (in a classical and surprising setting of fluids!). I will illustrate the results with numerical and experimental examples and explain how classical concepts such as rotation numbers of homeomorphisms (introduced by Henri Poincare) are related to solutions of the Poincare evolution problem (so named by Elie Cartan). The talk is based on joint work with Semyon Dyatlov and Jian Wang. I will also mention recent progress by Zhenhao Li on the case of irrational rotation numbers.

March 4, 2020 - Flows of Vector Fields: Classical and Modern

Camillo De Lellis
IAS, Princeton

Consider a (possibly time-dependent) vector field v on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindelöf) Theorem states that, if the vector field v is Lipschitz in space, for every initial datum x there is a unique trajectory γ starting at x at time 0 and solving the ODE.(t) = v(t, γ(t). The theorem looses its validity as soon as v is slightly less regular. However, if we bundle all trajectories into a global map allowing x to vary, a celebrated theory put forward by DiPerna and Lions in the 80's show that there is a unique such flow under very reasonable conditions and for much less regular vector fields. A long-standing open question is whether this theory is the byproduct of a stronger classical result which ensures the uniqueness of trajectories for almost every initial datum. I will give a complete answer to the latter question and draw connections with partial differential equations, harmonic analysis, probability theory and Gromov’s h-principle.

December 8, 2017 - The Threshold Theorems for the Hyperbolic Yang-Mills Equations

Daniel Tataru
UC Berkeley

This talk will aim to provide an overview of a recent series of papers, joint with Sung-Jin Oh, devoted to the energy critical 4+1 dimensional hyperbolic Yang-Mills equation. These papers provide a comprehensive analysis of the large data problem, ultimately providing a proof of the Threshold Conjecture for Yang-Mills, and more. We will cover an array of ideas, ranging from gauge theory to hard core pde estimates to geometry and blow-up analysis.

November 2, 2016 - Effective Models for Ginzburg-Landau Vortices

Sylvia Serfaty
NYU

Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.

March 31, 2016 - The h-principle in Fluid Mechanics

László Székelyhidi Jr.
Institute of Mathematics, University of Leipzig

It is known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence. In joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we interpret the wild behaviour of weak solutions of Euler as an instance of Gromov’s h-principle. In this lecture I will explain this connection and outline recent progress concerning Onsager’s conjecture.

April 9, 2015 - From molecular dynamics to kinetic theory and fluid mechanics

Laure Saint-Raymond
Harvard and MIT on leave from Ecole Normale Superieure

In his sixth problem, Hilbert asked for an axiomatization of gas dynamics, and he suggested to use the Boltzmann equation as an intermediate description between the (microscopic) atomic dynamics and (macroscopic) fluid models. The main difficulty to achieve this program is to prove the asymptotic decorrelation between the local microscopic interactions, referred to as propagation of chaos, on a time scale much larger than the mean free time. This is indeed the key property to observe some relaxation towards local thermodynamic equilibrium. This control of the collision process can be obtained in fluctuation regimes [1, 2]. In [2], we have established a long time convergence result to the linearized Boltzmann equation, and eventually derived the acoustic and incompressible Stokes equations in dimension 2. The proof relies crucially on symmetry arguments, combined with a suitable pruning procedure to discard super exponential collision trees.
Keywords: system of particles, low density, Boltzmann-Grad limit, kinetic equation, fluid models
[1] T. Bodineau, I. Gallagher, L. Saint-Raymond. The Brownian motion as the limit of a deterministic system of hard-spheres, to appear in Invent. Math. (2015).
[2] T. Bodineau, I. Gallagher, L. Saint-Raymond. From hard spheres to the linearized Boltzmann equation: an L2 analysis of the Boltzmann-Grad limit, in preparation.

April 17, 2014 - Regularity, blow up, and small scale creation in fluids

Alexander Kiselev
University of Wisconsin at Madison

The Euler equation of fluid mechanics describes a flow of inviscid and incompressible fluid, and has been first written in 1755. The equation is both nonlinear and nonlocal, and its solutions often create small scales easily and tend to be unstable. I will review some of the background, and then discuss a recent sharp result on small scale creation in solutions of the 2D Euler equation. I will also indicate links to the long open question of finite time blow up for solutions of the 3D Euler equation.

April 19, 2013 - Topology-Preserving Diffusion of Divergence-Free Vector Fields

Yann Brenier
École Polytechnique

The usual heat equation is not suitable to preserve the topology of divergence-free vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, one can find examples of topology-preserving diffusion equations for divergence-free vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of ”dissipative solutions”, which shares common features with both P.-L. Lions’ dissipative solutions to the Euler equations and the concept of ”curves of maximal slopes”, à la De Giorgi, recently used by Gigli and collaborators to study the scalar heat equation in very general metric spaces. We show that the initial value problem admits global "dissipative" solutions (at least for two space dimensions) and that they are unique whenever they are smooth.

February 8, 2012 - On the rigidity of black holes

Sergiu Klainerman
Princeton University

The rigidity conjecture states that all regular, stationary solutions of the Einstein field equations in vacuum are isometric to the Kerr solution. The simple motivation behind this conjecture is that one expects, due to gravitational radiation, that general, dynamic, solutions of the Einstein field equation settle down, asymptotically, into a stationary regime. A well known result of Carter, Robinson and Hawking has settled the conjecture in the class of real analytic spacetimes. The assumption of real analyticity is however very problematic; there is simply no physical or mathematical justification for it. During the last five years I have developed, in collaboration with A. Ionescu and S. Alaxakis, a strategy to dispense of it. In my lecture I will these results and concentrate on some recent results obtained in collaboration with A. Ionescu.

February 25, 2011 - Mathematical Strategies for Real Time Filtering of Turbulent Dynamical Systems

Andrew Majda
Courant Institute of Mathematical Sciences -- New York University

An important emerging scientific issue in many practical problems ranging from climate and weather prediction to biological science involves the real time filtering and prediction through partial observations of noisy turbulent signals for complex dynamical systems with many degrees of freedom as well as the statistical accuracy of various strategies to cope with the .curse of dimensions.. The speaker and his collaborators, Harlim (North Carolina State University), Gershgorin (CIMS Post doc), and Grote (University of Basel) have developed a systematic applied mathematics perspective on all of these issues. One part of these ideas blends classical stability analysis for PDE's and their finite difference approximations, suitable versions of Kalman filtering, and stochastic models from turbulence theory to deal with the large model errors in realistic systems. Many new mathematical phenomena occur. Another aspect involves the development of test suites of statistically exactly solvable models and new NEKF algorithms for filtering and prediction for slow-fast system, moist convection, and turbulent tracers. Here a stringent suite of test models for filtering and stochastic parameter estimation is developed based on NEKF algorithms in order to systematically correct both multiplicative and additive bias in an imperfect model. As briefly described in the talk, there are both significantly increased filtering and predictive skill through the NEKF stochastic parameter estimation algorithms provided that these are guided by mathematical theory. The recent paper by Majda et al (Discrete and Cont. Dyn. Systems, 2010, Vol. 2, 441-486) as well as a forthcoming introductory graduate text by Majda and Harlim (Cambridge U. Press) provide an overview of this research.

April 24, 2009 - The global behavior of solutions to critical nonlinear dispersive and wave equations

Carlos E. Kenig
University of Chicago

In this lecture we will describe a method (which I call the concentration-compactness/rigidity theorem method) which Frank Merle and I have developed to study global well-posedness and scattering for critical non-linear dispersive and wave equations. Such problems are natural extensions of non-linear elliptic problems which were studied earlier, for instance in the context of the Yamabe problem and of harmonic maps. We will illustrate the method with some concrete examples and also mention other applications of these ideas.

April 25, 2008 - Surface Waves and Images

Joseph B. Keller
Stanford University

March 30, 2007 - Steady Water Waves: Theory and Computation

Walter Strauss
Brown University

September 30, 2005 - A New Perspective on Motion by Curvature

Robert V. Kohn
Courant Institute of Mathematical Sciences, New York University

April 15, 2005 - Conservation Laws and Some Consequences

Cathleen Synge Morawetz
Courant Institute of Mathematical Sciences, New York University

March 5, 2004 - Hyperbolic Conservation Laws with Dissipation

Constantine Dafermos
Brown University, Division of Applied Mathematics

October 8, 2002 - Topology and Sobolev Spaces

Haim Brezis
Universite de Paris VI, Insitiut Universitaire de France, and Rutgers University

April 12, 2002 - Navier-Stokes and Other Super-critical Equations

Vladmir Sverak
University of Minnesota

April 20, 2001 - Shock Wave Theory

Tai-Ping Liu
Academia Sinica, Taiwan & Stanford University

March 31, 2000 - Effective Hamiltonians

Lawrence C. Evans
University of California, Berkeley

April 23, 1999 - Some remarks on homogenization

Luis Caffarelli
University of Texas, Austin

April 17, 1998 - An Example of Diffusion-Induced Blowup of a Parabolic System

Hans Weinberger
University of Minnesota

April 4, 1997 - The Zero Dispersion Limit

Peter Lax
Courant Institute

May 9, 1996 - Degree Theory Beyond Continuous Maps

Louis Nirenberg
Courant Institute