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= Mathematics Colloquium =
 
= Mathematics Colloquium =
  
 
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.
 
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.
  
==[[Tentative Colloquia|Tentative schedule for next semester]] ==
+
The calendar for spring 2019 can be found [[Colloquia/Spring2019|here]].
 +
 
 +
== Fall 2018 ==
 +
 
  
== Spring 2016  ==
 
 
 
 
{| cellpadding="8"
 
{| cellpadding="8"
!align="left" | date
+
!align="left" | date  
 
!align="left" | speaker
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | title
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
| '''January 22'''  
+
|Sep 12    '''Room 911'''
|<!--[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)-->
+
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
| <!-- [[Colloquia#September 11: Speaker (University) | title]] -->
+
|[[#Sep 12: Gunther Uhlmann (Univ. of Washington)| Harry Potter's Cloak via Transformation Optics  ]]
| <!--Host-->
+
| Li
 +
|
 
|-
 
|-
| '''January 28 (Th 4pm VV901)'''  
+
|Sep 14    '''Room 911'''
| [https://web.math.princeton.edu/~ssivek/ Steven Sivek] (Princeton)
+
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
|     [[Colloquia#September 11: Speaker (University) | The augmentation category of a Legendrian knot]]
+
|[[#Sep 14: Gunther Uhlmann (Univ. of Washington) | Journey to the Center of the Earth  ]]
| Ellenberg
+
| Li
 +
|
 
|-
 
|-
| '''January 29'''  
+
|Sep 21    '''Room 911'''
|[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)
+
| [http://stuart.caltech.edu/ Andrew Stuart] (Caltech) LAA lecture
| [[Colloquia#September 11: Ana Caraiani (Princeton) | Locally symmetric spaces, torsion classes, and the geometry of period domains]]
+
|[[#Sep 21: Andrew Stuart (Caltech) | The Legacy of Rudolph Kalman  ]]
| Ellenberg
+
| Jin
 +
|
 
|-
 
|-
| '''February 5'''
+
|Sep 28
|[http://math.uchicago.edu/~souganidis/ Takis Souganidis] (University of Chicago)
+
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
| [[Colloquia#September 11: Takis Souganidis (University of Chicago) | Scalar Conservation Laws with Rough Dependence]]
+
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
| Lin
+
| Thiffeault
 +
|
 
|-
 
|-
| '''February 12'''
+
|Oct 5
|[http://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)  
+
| [http://www.personal.psu.edu/eus25/ Eyal Subag] (Penn State)
| [[Colloquia#February 12: Gautam Iyer (CMU)| Homogenization and Anomalous Diffusion]]
+
|[[#Oct 5: Eyal Subag (Penn State)| Symmetries of the hydrogen atom and algebraic families  ]]
| Jean-Luc
+
| Gurevich
 +
|
 
|-
 
|-
| '''February 19'''
+
|Oct 12
| [https://people.math.osu.edu/lafont.1/ Jean-François Lafont] (Ohio State)  
+
| [https://www.math.wisc.edu/~andreic/ Andrei Caldararu] (Madison)
| [[Colloquia#February 19: Jean-François Lafont (Ohio State) | Rigidity and flexibility of almost-isometries]]
+
|[[#Oct 12: Andrei Caldararu (Madison) | Mirror symmetry and derived categories  ]]
| Dymarz
+
| ...
 +
|
 
|-
 
|-
| '''February 26'''
+
|Oct 19
|Hiroyoshi Mitake (Hiroshima university)    
+
| [https://teitelbaum.math.uconn.edu/# Jeremy Teitelbaum] (U Connecticut)
| [[Colloquia#February 26: Hiroyoshi Mitake (Hiroshima university) | On asymptotic speed of the crystal growth]]
+
|[[#Oct 19:   Jeremy Teitelbaum (U Connecticut)| Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist  ]]
| Tran
+
| Boston
 +
|
 
|-
 
|-
| '''March 4'''
+
|Oct 26
| [http://www.columbia.edu/~gb2030/ Guillaume Bal] (Columbia University)
+
| [http://math.arizona.edu/~ulmer/index.html Douglas Ulmer] (Arizona)
| [[Colloquia#September 11: Guillaume Bal (Columbia University) | Inverse and Control Transport Problems]]
+
|[[#Oct 26: Douglas Ulmer (Arizona) | Rational numbers, rational functions, and rational points ]]
| Li, Jin
+
| Yang
 +
|
 
|-
 
|-
| '''March 11'''  
+
|Nov 2  '''Room 911'''
| [http://math.umn.edu/~luskin Mitchell Luskin] (University of Minnesota)
+
| [https://sites.google.com/view/ruixiang-zhang/home?authuser=0# Ruixiang Zhang] (Madison)
| [[Colloquia#March 11: Mitchell Luskin (UMN) | Mathematical Modeling of Incommensurate 2D Materials]]
+
|[[#Nov 2: Ruixiang Zhang (Madison) | The Fourier extension operator  ]]
| Li
+
|  
 +
|
 
|-
 
|-
| '''March 18'''  
+
|Nov 7  '''Wednesday'''
| [http://www.math.lsa.umich.edu/~spatzier/ Ralf Spatzier] (University of Michigan)  
+
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)
CANCELED
+
|[[#Nov 7: Luca Spolaor (MIT) (Log)-Epiperimetric Inequality and the Regularity of Variational Problems  ]]
| Dymarz
+
| Feldman
 +
|
 
|-
 
|-
| '''March 25'''  
+
|Nov 12  '''Monday'''
| Spring Break<!-- [webpage Speaker Name] (University) -->   
+
| [http://www.math.tamu.edu/~annejls/ Anne Shiu] (Texas A&M)
| <!-- [[Colloquia#September 11: Speaker (University) | title]] -->
+
|[[#Nov 9: Anne Shiu (Texas A&M) | Dynamics of biochemical reaction systems  ]]
| <!-- host -->
+
| Craciun, Stechmann
 +
|
 
|-
 
|-
| '''April 1'''  
+
|Nov 19 '''Monday'''
|  
+
| [https://sites.google.com/site/ayomdin/ Alexander Yom Din] (Caltech)
|   CANCELED
+
|[[#Nov 19: Alexander Yom Din (Caltech) | From analysis to algebra to geometry - an example in representation theory of real groups  ]]
 +
| Boston, Gurevitch
 
|
 
|
 
|-
 
|-
| '''April 8'''  
+
|Nov 20 '''Tuesday, Room 911'''
| [https://web.math.princeton.edu/~aionescu/ Alexandru Ionescu] (Princeton)  
+
| [http://http://www.math.uchicago.edu/~drh/ Denis Hirschfeldt] (University of Chicago)
| [[Colloquia#April 8: Alexandru Ionescu (Princeton) | On long-term existence of solutions of water wave models]]  
+
|[[#Nov 20: Denis Hirschfeldt (University of Chicago)| Computability and Ramsey Theory  ]]
| Wainger/Seeger
+
| Andrews
 +
|
 
|-
 
|-
| '''April 15'''  
+
|Nov 26 '''Monday, Room 911'''
| [https://www.kcl.ac.uk/nms/depts/mathematics/people/atoz/wigmani.aspx Igor Wigman] (King's College - London)  
+
| [http://math.mit.edu/directory/profile.php?pid=1415 Vadim Gorin] (MIT)
| [[Colloquia#September 11: Speaker (University) |Nodal Domains of Eigenfunctions]]
+
|[[#Nov 26: Vadim Gorin (MIT)| Macroscopic fluctuations through Schur generating functions  ]]
| Gurevich/Marshall
+
| Anderson
 +
|
 
|-
 
|-
| '''April 22'''  
+
|Nov 28 '''Wednesday'''
| [http://www.cims.nyu.edu/~bourgade/ Paul Bourgade] (NYU)
+
| [http://www.math.ias.edu/~gchen/ Gao Chen](IAS)
| [[Colloquia#April 22: Paul Bourgade (NYU) | Random matrices beyond mean-field]]
+
|[[#Nov 28: Gao Chen(IAS) | A Torelli type theorem ]]
| Seppalainen/Valko
+
| Paul
 +
|
 
|-
 
|-
| '''April 29'''
+
|Nov 30
| [http://www.physics.upenn.edu/~kamien/kamiengroup/ Randall Kamien] (U Penn)  
+
| [https://math.indiana.edu/about/faculty/fisher-david.html David Fisher](Indiana U.)
| [[Colloquia#April 29: Randall Kamien (U Penn) | Liquid crystals and their (algebraic) topology]]  
+
|[[#Nov 30: David Fisher (Indiana U.) | New Techniques for Zimmer's Conjecture ]]
| Spagnolie
+
| Kent
 
|-
 
|-
| '''May 4'''  
+
|Dec 3 '''Monday'''
| [https://www.math.ias.edu/people/faculty/sarnak Peter Sarnak] (Princeton and IAS)  
+
| [http://bena-tshishiku.squarespace.com/ Bena Tshishiku](Harvard)
| [[Colloquia#May 4: Peter Sarnak (Princeton and IAS) | Strong approximation for Markoff surfaces]] (Note special day; still 4 pm in B239)
+
|[[#Dec 3: Bena Tshishiku (Harvard)| Surface bundles, monodromy, and arithmetic groups ]]
| Marshall
+
| Paul
 +
|
 
|-
 
|-
| '''May 6'''  
+
|Dec 5 '''Wednesday, Room 911'''
|  
+
| [http://www.mit.edu/~ssen90/ Subhabrata Sen](MIT and Microsoft Research New England)
 +
|[[#Dec 5: Subhabrata Sen (MIT and Microsoft Research New England) | Random graphs, Optimization, and Spin glasses ]]
 +
| Anderson
 +
|
 +
|-
 +
|Dec 7 '''Room 911'''
 +
| [https://math.berkeley.edu/people/faculty/leonardo-zepeda-n-ez Leonardo Zepeda-Nunez](Berkeley)
 +
|[[#Dec 7: Leonardo Zepeda-Nunez (Berkeley) | Accelerating ab-initio molecular dynamics via multi-scale neural networks ]]
 +
| Stechmann
 +
|
 +
|-
 +
|Dec 10 '''Monday'''
 +
| [http://math.mit.edu/~maxe/ Max Engelstein](MIT)
 +
|[[#Dec 10: Max Engelstein (MIT)|  The role of Energy in Regularity  ]]
 +
| Feldman
 +
|
 
|}
 
|}
  
 
== Abstracts ==
 
== Abstracts ==
  
 +
=== Sep 12: Gunther Uhlmann (Univ. of Washington) ===
 +
Harry Potter's Cloak via Transformation Optics
  
=== January 28: Steven Sivek (Princeton) ===
+
Can we make objects invisible? This has been a subject of human
Title: The augmentation category of a Legendrian knot
+
fascination for millennia in Greek mythology, movies, science fiction,
 +
etc. including the legend of Perseus versus Medusa and the more recent
 +
Star Trek and Harry Potter. In the last fifteen years or so there have been
 +
several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion
 +
one of them, the so-called "traansformation optics"
 +
in a non-technical fashion n the so-called that has received the most attention in the
 +
scientific literature.
  
Abstract: A well-known principle in symplectic geometry says that information about the smooth structure on a manifold should be captured by the symplectic geometry of its cotangent bundle. One prominent example of this is Nadler and Zaslow's microlocalization correspondence, an equivalence between a category of constructible sheaves on a manifold and a symplectic invariant of its cotangent bundle called the Fukaya category.
+
=== Sep 14: Gunther Uhlmann (Univ. of Washington) ===
 +
Journey to the Center of the Earth
  
The goal of this talk is to describe a model for a relative version of this story in the simplest case, corresponding to Legendrian knots in the standard contact 3-space. This construction, called the augmentation category, is a powerful invariant which is defined in terms of holomorphic curves but can also be described combinatorially.  I will describe some interesting properties of this category and relate it to a category of sheaves on the plane. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Maslow.
+
We will consider the inverse problem of determining the sound
 +
speed or index of refraction of a medium by measuring the travel times of
 +
waves going through the medium. This problem arises in global seismology
 +
in an attempt to determine the inner structure of the Earth by measuring
 +
travel times of earthquakes. It has also several applications in optics
 +
and medical imaging among others.
  
=== January 29: Ana Caraiani (Princeton) ===
+
The problem can be recast as a geometric problem: Can one determine the
Title:  Locally symmetric spaces, torsion classes, and the geometry of period domains
+
Riemannian metric of a Riemannian manifold with boundary by measuring
 +
the distance function between boundary points? This is the boundary
 +
rigidity problem. We will also consider the problem of determining
 +
the metric from the scattering relation, the so-called lens rigidity
 +
problem. The linearization of these problems involve the integration
 +
of a tensor along geodesics, similar to the X-ray transform.
  
Abstract:  The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.
+
We will also describe some recent results, join with Plamen Stefanov
 +
and Andras Vasy, on the partial data case, where you are making
 +
measurements on a subset of the boundary. No previous knowledge of
 +
Riemannian geometry will be assumed.
  
I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.
+
=== Sep 21: Andrew Stuart (Caltech) ===
  
 +
The Legacy of Rudolph Kalman
  
=== February 5: Takis Souganidis (University of Chicago) ===
+
In 1960 Rudolph Kalman published what is arguably the first paper to develop a systematic, principled approach to the use of data to improve the predictive capability of mathematical models. As our ability to gather data grows at an enormous rate, the importance of this work continues to grow too. The lecture will describe this paper, and developments that have stemmed from it, revolutionizing fields such space-craft control, weather prediction, oceanography and oil recovery, and with potential for use in new fields such as medical imaging and artificial intelligence. Some mathematical details will be also provided, but limited to simple concepts such as optimization, and iteration; the talk is designed to be broadly accessible to anyone with an interest in quantitative science.
Title:  Scalar Conservation Laws with Rough Dependence
+
  
I will present a recently developed theory for scalar conservation laws with nonlinear multiplicative rough signal dependence. I will describe the difficulties, introduce the notion of pathwise entropy/kinetic solution and its well-posedness. I will also talk about the long time behavior of the solutions as well as some regularization by noise type results.
+
=== Sep 28: Gautam Iyer (CMU) ===
  
=== February 12:  Gautam Iyer (CMU) ===
+
Stirring and Mixing
  
Homogenization and Anomalous Diffusion
+
Mixing is something one encounters often in everyday life (e.g. stirring cream into coffee). I will talk about two mathematical
 +
aspects of mixing that arise in the context of fluid dynamics:
  
Homogenization is a well known technique used to approximate the macroscopic behaviour of a material with microscopic impurities.
+
1. How efficiently can stirring "mix"?
While this originally arose in the study of composite materials, it has applications to various other fields, and I will focus on a few results
+
motivated by fluid dynamics. One well known result in this direction is by GI Taylor estimating the dispersion rate of a solute in a pipe. The
+
length scales involved in typical pipelines, however, are too short for this result to apply. I will conclude with a few recent "intermediate time" results describing the effective behaviour in scaling regimes outside those of standard homogenization results.
+
  
=== February 19: Jean-François Lafont (Ohio State) ===
+
2. What is the interaction between diffusion and mixing.
  
Rigidity and flexibility of almost-isometries
+
Both these aspects are rich in open problems whose resolution involves tools from various different areas. I present a brief survey of existing
 +
results, and talk about a few open problems.
  
An almost isometry (AI) is a quasi-isometry (QI) with multiplicative
+
=== Oct 5: Eyal Subag (Penn State)===
constant =1. Given two metrics on a closed manifold, Milnor-Swarc implies
+
that the lifted metrics on the universal cover are QI to each other. When are
+
they AI to each other? In the rigidity direction, we give various examples
+
where the only time such lifts are AI is when they are isometric (joint with
+
Kar and Schmidt). In the flexible direction, we show that for higher genus
+
surfaces, any two metrics have lifts which, after possibly scaling one of the
+
lifted metrics, are AI to each other (joint with Schmidt and van Limbeek). In
+
the latter examples, one can further show that the AI is usually not equivariant
+
with respect to the group actions.
+
  
=== February 26: Hiroyoshi Mitake (Hiroshima University) ===
+
Symmetries of the hydrogen atom and algebraic families
In the talk, I will propose a model equation to study the crystal growth as a prototype, which is described by a level-set mean curvature flow equation with driving and source terms. We establish the well-posedness of solutions, and study the asymptotic speed. Interestingly, a new type of nonlinear phenomena in terms of asymptotic speed of solutions appears because of the double nonlinear effects coming from the surface evolution and the source term, which is sensitive to the shapes of source terms. This is a joint work with Y. Giga (U. Tokyo), and H. V. Tran (U. Wisconsin-Madison).
+
  
=== March 11: Mitchell Luskin (UMN) ===
+
The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry.  In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden,  give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules.  I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system.  No prior knowledge on quantum mechanics or representation theory will be assumed.
Title: Mathematical Modeling of Incommensurate 2D Materials
+
  
Abstract: Incommensurate materials are found in crystals, liquid crystals, and quasi-crystals. Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the elastic, electronic, and optical properties of these materials. One of the main issues encountered in the mathematical modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. This leads to complex commensurate-incommensurate transitions and pattern formation.
+
=== Oct 12: Andrei Caldararu (Madison)===
  
Even basic concepts like the Cauchy-Born strain energy density, the electronic density of states, and the Kubo-Greenwood formulas for transport properties have not been given a rigorous analysis in the incommensurate setting. New approximate approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.
+
Mirror symmetry and derived categories
  
===March 18: Ralf Spatzier (UMichigan)===
+
Mirror symmetry is a remarkable phenomenon, first discovered in physics. It relates two seemingly disparate areas of mathematics, symplectic and algebraic geometry. Its initial formulation was rather narrow, as a technique for computing enumerative invariants (so-called Gromov-Witten invariants) of symplectic varieties by solving certain differential equations describing the variation of Hodge structure of “mirror" varieties. Over the past 25 years this narrow view has expanded considerably, largely due to insights of M. Kontsevich who introduced techniques from derived categories into the subject. Nowadays mirror symmetry encompasses wide areas of mathematics, touching on subjects like birational geometry, number theory, homological algebra, etc.
  
CANCELED: Rigidity in Geometry and Dynamics
+
In my talk I shall survey some of the recent developments in mirror symmetry, and I will explain how my work fits in the general picture. In particular I will describe an example of derived equivalent but not birational Calabi-Yau three folds (joint work with Lev Borisov); and a recent computation of a categorical Gromov-Witten invariant of positive genus (work with my former student Junwu Tu).
  
I will survey some rigidity phenomena in dynamics and also geometry, with emphasis on the notion of higher rank.
+
===  Oct 19:  Jeremy Teitelbaum (U Connecticut)===
This first emerged in Margulis’ celebrated work on superrrigidity but has also been important in more recent work on symmetry in dynamical systems.
+
Lessons Learned and New Perspectives:
How special is it for maps commute with each other?  Smale asked this problem fifty years ago, and answers are finally emerging.  Much depends on the differentiability
+
From Dean and Provost to aspiring Data Scientist
of the maps: it gets harder the more differentiable the map is. Sometimes we can even classify such maps.  I’ll discuss this and
+
related phenomena.
+
  
 +
After more than 10 years in administration, including 9 as Dean of
 +
Arts and Sciences and 1 as interim Provost at UConn, I have returned
 +
to my faculty position.  I am spending a year as a visiting scientist
 +
at the Jackson Laboratory for Genomic Medicine (JAX-GM) in Farmington,
 +
Connecticut, trying to get a grip on some of the mathematical problems
 +
of interest to researchers in cancer genomics.  In this talk, I will offer some personal
 +
observations about being a mathematician and a high-level administrator, talk a bit about
 +
the research environment at an independent research institute like JAX-GM, outline
 +
a few problems that I've begun to learn about, and conclude with a
 +
discussion of how these experiences have shaped my view of graduate training in mathematics.
  
 +
=== Oct 26: Douglas Ulmer (Arizona)===
  
=== April 8: Alexandru Ionescu (Princeton) ===
+
Rational numbers, rational functions, and rational points
  
Title:  On long-term existence of solutions of water wave models
+
One of the central concerns of arithmetic geometry is the study of
 +
solutions of systems of polynomial equations where the solutions are
 +
required to lie in a "small" field such as the rational numbers.  I
 +
will explain the landscape of expectations and conjectures in this
 +
area, focusing on curves and their Jacobians over global fields
 +
(number fields and function fields), and then survey the progress made
 +
over the last decade in the function field case.  The talk is intended
 +
to be accessible to a wide audience.
  
I will talk about some recent work on long-term/global regularity of solutions of water wave models in 2 and 3 dimensions. The
+
=== Nov 2: Ruixiang Zhang (Madison)===
models we consider describe the evolution of an inviscid perfect fluid in
+
a free boundary domain, under the influence of gravity and/or surface
+
tension. This is joint work with Fabio Pusateri and, in part, with Yu Deng and
+
Benoit Pausader.
+
  
 +
The Fourier extension operator
  
===April 22: Paul Bourgade (NYU)===
+
I will present an integral operator that originated in the study of the Euclidean Fourier transform and is closely related to many problems in PDE, spectral theory, analytic number theory, and combinatorics. I will then introduce some recent developments in harmonic analysis concerning this operator. I will mainly focus on various new ways to "induct on scales" that played an important role in the recent solution in all dimensions to Carleson's a.e. convergence problem on free Schrödinger solutions.
  
Title: Random matrices beyond mean-field
+
=== Nov 7: Luca Spolaor (MIT)===
  
 +
(Log)-Epiperimetric Inequality and the Regularity of Variational Problems
  
Random matrix statistics were proposed by Eugene Wigner as a new class of universal statistical laws for highly correlated systems.
+
In this talk I will present a new method for studying the regularity of minimizers to variational problems. I will start by introducing the notion of blow-up, using as a model case the so-called Obstacle problem. Then I will state the (Log)-epiperimetric inequality and explain how it is used to prove uniqueness of the blow-up and regularity results for the solution near its singular set. I will then show the flexibility of this method by describing how it can be applied to other free-boundary problems and to (almost)-area minimizing currents.
We will first review established instances of this conjecture for mean-field matrix models. We will then propose an approach towards the spectral analysis of non mean-field models, which are closer to Wigner's original vision.  
+
Finally I will describe some future applications of this method both in regularity theory and in other settings.
A key role is played by a new patching of quantum unique ergodicity estimates.
+
  
 +
=== Nov 9: Anne Shiu (Texas A&M)===
  
=== April 29: Randall Kamien (U Penn) ===
+
Dynamics of biochemical reaction systems
  
Title: Liquid Crystals and their (Algebraic) Topology
+
Reaction networks taken with mass-action kinetics arise in many settings,
 +
from epidemiology to population biology to systems of chemical reactions.
 +
This talk focuses on certain biological signaling networks, namely,
 +
phosphorylation networks, and their resulting dynamical systems. For many
 +
of these systems, the set of steady states admits a rational
 +
parametrization (that is, the set is the image of a map with
 +
rational-function coordinates). We describe how such a parametrization
 +
allows us to investigate the dynamics, including the emergence of
 +
bistability in a network underlying ERK regulation, and the capacity for
 +
oscillations in a mixed processive/distributive phosphorylation network.
  
Liquid Crystals, the materials in your iPhone, are complex materials with varying degrees of internal order.  I will discuss and demonstrate how algebraic topology can be used to identify and characterize long-lived configurations.  I will also describe how conic sections naturally arise in these structures as intersections of simple polynomials.
+
=== Nov 19: Alexander Yom Din (Caltech)===
 +
 
 +
From analysis to algebra to geometry - an example in representation theory of real groups
  
=== May 4: Peter Sarnak (Princeton and IAS) ===
+
Representation theory of non-compact real groups, such as SL(2,R), is a fundamental discipline with uses in harmonic analysis, number theory, physics, and more. This theory is analytical in nature, but in the course of the 20th century it was algebraized and geometrized (the key contributions are by Harish-Chandra for the former and by Beilinson-Bernstein for the latter). Roughly and generally speaking, algebraization strips layers from the objects of study until we are left with a bare skeleton, amenable to symbolic manipulation. Geometrization, again very roughly, reveals how algebraic objects have secret lives over spaces - thus more amenable to human intuition. In this talk, I will try to motivate and present one example - the calculation of the Casselman-Jacquet module of a principal series representation (I will explain the terms in the talk).
  
Title: Strong approximation for Markoff surfaces
+
=== Nov 20: Denis Hirschfeldt (University of Chicago)===
  
We discuss the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated in part by the finite orbits of these actions on the algebraic points. The latter can be determined effectively and in special cases is connected to the problem of determining all algebraic Painleve VI's. Applications to forms of strong approximation for integer points and to sieving on such affine surfaces, as well as to Markoff numbers will be given.
+
Computability and Ramsey Theory
  
Joint work with J.Bourgain and A.Gamburd.
+
Computability theory can be seen as the study of the fine
===
+
structure of definability. Much of its power relies on the deep
 +
connections between definability and computation. These connections can be seen in fundamental results such as Post's Theorem, which establishes a connection between the complexity of formulas needed to define a given set of natural numbers and its computability-theoretic strength. As has become increasingly clear, they can also be seen in the computability-theoretic analysis of objects whose definitions come from notions that arise naturally in combinatorics. The heuristic here is that
 +
computability-theoretically natural notions tend to be combinatorially
 +
natural, and vice-versa. I will discuss some results and open questions in
 +
the computability-theoretic analysis of combinatorial principles, in
 +
particular Ramsey-theoretic ones such as versions of Ramsey's Theorem for colorings of countably infinite sets, and versions of Hindman's Theorem, which states that for every coloring of the natural numbers with finitely many colors, there is an infinite set of numbers such that all nonempty sums of distinct elements of this set have the same color.
 +
 
 +
=== Nov 26: Vadim Gorin (MIT)===
 +
 
 +
Macroscopic fluctuations through Schur generating functions
 +
 
 +
I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.
 +
 
 +
It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.
 +
 
 +
=== Nov 28: Gao Chen (IAS) ===
 +
 
 +
A Torelli type theorem
 +
 
 +
The length of a circle determines the shape of it. In this talk, we will discuss non-trivial generalizations of this fact for Riemann surfaces, hyperkähler 4-manifolds, Calabi-Yau threefolds and G_2, Spin(7) manifolds.
 +
 
 +
=== Nov 30: David Fisher (Indiana U.) ===
 +
 
 +
New Techniques for Zimmer's Conjecture
 +
 
 +
Lattices in higher rank simple Lie groups are known to be
 +
extremely rigid. Examples of this are Margulis' superrigidity theorem,
 +
which shows they have very few linear represenations, and Margulis'
 +
arithmeticity theorem, which shows they are all constructed via number
 +
theory. Motivated by these and other results, in 1983 Zimmer made a
 +
number of conjectures about actions of these groups on compact
 +
manifolds. After providing some history and motivation, I will discuss
 +
a recent result that makes dramatic progress on the conjecture in all
 +
cases and proves it in many of them. I will place some emphasis on
 +
surprising connections to other areas of mathematics that arise in the
 +
proof.
 +
 
 +
=== Dec 3: Bena Tshishiku (Harvard) ===
 +
 
 +
Surface bundles, monodromy, and arithmetic groups
 +
 
 +
Fiber bundles with fiber a surface arise in many areas including hyperbolic geometry, symplectic geometry, and algebraic geometry. Up to isomorphism, a surface bundle is completely determined by its monodromy representation, which is a homomorphism to a mapping class group. This allows one to use algebra to study the topology of surface bundles. Unfortunately, the monodromy representation is typically difficult to ``compute" (e.g. determine its image). In this talk, I will discuss some recent work toward computing monodromy groups for holomorphic surface bundles, including certain examples of Atiyah and Kodaira. This can be applied to the problem of counting the number of ways that certain 4-manifolds fiber over a surface. This is joint work with Nick Salter.
 +
 
 +
=== Dec 5: Subhabrata Sen (MIT and Microsoft Research New England) ===
 +
 
 +
Random graphs, Optimization, and Spin glasses
 +
 
 +
Combinatorial optimization problems are ubiquitous in diverse mathematical applications. The desire to understand their ``typical" behavior motivates a study of these problems on random instances. In spite of a long and rich history, many natural questions in this domain are still intractable to rigorous mathematical analysis.  Graph cut problems such as Max-Cut and Min-bisection are canonical examples in this class. On the other hand, physicists study these questions using the non-rigorous ``replica" and ``cavity" methods, and predict complex, intriguing features. In this talk, I will describe some recent progress in our understanding of their typical properties on random graphs, obtained via connections to the theory of mean-field spin glasses. The new techniques are broadly applicable, and lead to novel algorithmic and statistical consequences.
 +
 
 +
=== Dec 7: Leonardo Zepeda-Nunez (Berkeley) ===
 +
 
 +
Accelerating ab-initio molecular dynamics via multi-scale neural networks
 +
 
 +
Deep learning has rapidly become a large field with an ever-growing range of applications; however, its intersection with scientific computing remains in its infancy, mainly due to the high accuracy that scientific computing problems require, which depends greatly on the architecture of the neural network.
 +
 
 +
In this talk we present a novel deep neural network with a multi-scale architecture inspired in H-matrices (and H2-matrices) to efficiently approximate, within 3-4 digits, several challenging non-linear maps arising from the discretization of PDEs, whose evaluation would otherwise require computationally intensive iterative methods.
 +
 
 +
In particular, we focus on the notoriously difficult Kohn-Sham map arising from Density Functional Theory (DFT). We show that the proposed multiscale-neural network can efficiently learn this map, thus bypassing an expensive self-consistent field iteration. In addition, we show the application of this methodology to ab-initio molecular dynamics, for which we provide examples for 1D problems and small, albeit realistic, 3D systems.
 +
 
 +
Joint work with Y. Fan, J. Feliu-Faaba, L. Lin,  W. Jia, and L. Ying
 +
 
 +
=== Dec 10: Max Engelstein (MIT) ===
 +
 
 +
The role of Energy in Regularity
 +
 
 +
The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes.  A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.
 +
 
 +
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field. 
 +
 
 +
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.
  
 
== Past Colloquia ==
 
== Past Colloquia ==
 +
 +
[[Colloquia/Blank|Blank]]
 +
 +
[[Colloquia/Spring2018|Spring 2018]]
 +
 +
[[Colloquia/Fall2017|Fall 2017]]
 +
 +
[[Colloquia/Spring2017|Spring 2017]]
 +
 +
[[Archived Fall 2016 Colloquia|Fall 2016]]
 +
 +
[[Colloquia/Spring2016|Spring 2016]]
  
 
[[Colloquia/Fall2015|Fall 2015]]
 
[[Colloquia/Fall2015|Fall 2015]]

Latest revision as of 16:04, 4 December 2018

Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

The calendar for spring 2019 can be found here.

Fall 2018

date speaker title host(s)
Sep 12 Room 911 Gunther Uhlmann (Univ. of Washington) Distinguished Lecture series Harry Potter's Cloak via Transformation Optics Li
Sep 14 Room 911 Gunther Uhlmann (Univ. of Washington) Distinguished Lecture series Journey to the Center of the Earth Li
Sep 21 Room 911 Andrew Stuart (Caltech) LAA lecture The Legacy of Rudolph Kalman Jin
Sep 28 Gautam Iyer (CMU) Stirring and Mixing Thiffeault
Oct 5 Eyal Subag (Penn State) Symmetries of the hydrogen atom and algebraic families Gurevich
Oct 12 Andrei Caldararu (Madison) Mirror symmetry and derived categories ...
Oct 19 Jeremy Teitelbaum (U Connecticut) Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist Boston
Oct 26 Douglas Ulmer (Arizona) Rational numbers, rational functions, and rational points Yang
Nov 2 Room 911 Ruixiang Zhang (Madison) The Fourier extension operator
Nov 7 Wednesday Luca Spolaor (MIT) (Log)-Epiperimetric Inequality and the Regularity of Variational Problems Feldman
Nov 12 Monday Anne Shiu (Texas A&M) Dynamics of biochemical reaction systems Craciun, Stechmann
Nov 19 Monday Alexander Yom Din (Caltech) From analysis to algebra to geometry - an example in representation theory of real groups Boston, Gurevitch
Nov 20 Tuesday, Room 911 Denis Hirschfeldt (University of Chicago) Computability and Ramsey Theory Andrews
Nov 26 Monday, Room 911 Vadim Gorin (MIT) Macroscopic fluctuations through Schur generating functions Anderson
Nov 28 Wednesday Gao Chen(IAS) A Torelli type theorem Paul
Nov 30 David Fisher(Indiana U.) New Techniques for Zimmer's Conjecture Kent
Dec 3 Monday Bena Tshishiku(Harvard) Surface bundles, monodromy, and arithmetic groups Paul
Dec 5 Wednesday, Room 911 Subhabrata Sen(MIT and Microsoft Research New England) Random graphs, Optimization, and Spin glasses Anderson
Dec 7 Room 911 Leonardo Zepeda-Nunez(Berkeley) Accelerating ab-initio molecular dynamics via multi-scale neural networks Stechmann
Dec 10 Monday Max Engelstein(MIT) The role of Energy in Regularity Feldman

Abstracts

Sep 12: Gunther Uhlmann (Univ. of Washington)

Harry Potter's Cloak via Transformation Optics

Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc. including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last fifteen years or so there have been several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion one of them, the so-called "traansformation optics" in a non-technical fashion n the so-called that has received the most attention in the scientific literature.

Sep 14: Gunther Uhlmann (Univ. of Washington)

Journey to the Center of the Earth

We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.

Sep 21: Andrew Stuart (Caltech)

The Legacy of Rudolph Kalman

In 1960 Rudolph Kalman published what is arguably the first paper to develop a systematic, principled approach to the use of data to improve the predictive capability of mathematical models. As our ability to gather data grows at an enormous rate, the importance of this work continues to grow too. The lecture will describe this paper, and developments that have stemmed from it, revolutionizing fields such space-craft control, weather prediction, oceanography and oil recovery, and with potential for use in new fields such as medical imaging and artificial intelligence. Some mathematical details will be also provided, but limited to simple concepts such as optimization, and iteration; the talk is designed to be broadly accessible to anyone with an interest in quantitative science.

Sep 28: Gautam Iyer (CMU)

Stirring and Mixing

Mixing is something one encounters often in everyday life (e.g. stirring cream into coffee). I will talk about two mathematical aspects of mixing that arise in the context of fluid dynamics:

1. How efficiently can stirring "mix"?

2. What is the interaction between diffusion and mixing.

Both these aspects are rich in open problems whose resolution involves tools from various different areas. I present a brief survey of existing results, and talk about a few open problems.

Oct 5: Eyal Subag (Penn State)

Symmetries of the hydrogen atom and algebraic families

The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system. No prior knowledge on quantum mechanics or representation theory will be assumed.

Oct 12: Andrei Caldararu (Madison)

Mirror symmetry and derived categories

Mirror symmetry is a remarkable phenomenon, first discovered in physics. It relates two seemingly disparate areas of mathematics, symplectic and algebraic geometry. Its initial formulation was rather narrow, as a technique for computing enumerative invariants (so-called Gromov-Witten invariants) of symplectic varieties by solving certain differential equations describing the variation of Hodge structure of “mirror" varieties. Over the past 25 years this narrow view has expanded considerably, largely due to insights of M. Kontsevich who introduced techniques from derived categories into the subject. Nowadays mirror symmetry encompasses wide areas of mathematics, touching on subjects like birational geometry, number theory, homological algebra, etc.

In my talk I shall survey some of the recent developments in mirror symmetry, and I will explain how my work fits in the general picture. In particular I will describe an example of derived equivalent but not birational Calabi-Yau three folds (joint work with Lev Borisov); and a recent computation of a categorical Gromov-Witten invariant of positive genus (work with my former student Junwu Tu).

Oct 19: Jeremy Teitelbaum (U Connecticut)

Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist

After more than 10 years in administration, including 9 as Dean of Arts and Sciences and 1 as interim Provost at UConn, I have returned to my faculty position. I am spending a year as a visiting scientist at the Jackson Laboratory for Genomic Medicine (JAX-GM) in Farmington, Connecticut, trying to get a grip on some of the mathematical problems of interest to researchers in cancer genomics. In this talk, I will offer some personal observations about being a mathematician and a high-level administrator, talk a bit about the research environment at an independent research institute like JAX-GM, outline a few problems that I've begun to learn about, and conclude with a discussion of how these experiences have shaped my view of graduate training in mathematics.

Oct 26: Douglas Ulmer (Arizona)

Rational numbers, rational functions, and rational points

One of the central concerns of arithmetic geometry is the study of solutions of systems of polynomial equations where the solutions are required to lie in a "small" field such as the rational numbers. I will explain the landscape of expectations and conjectures in this area, focusing on curves and their Jacobians over global fields (number fields and function fields), and then survey the progress made over the last decade in the function field case. The talk is intended to be accessible to a wide audience.

Nov 2: Ruixiang Zhang (Madison)

The Fourier extension operator

I will present an integral operator that originated in the study of the Euclidean Fourier transform and is closely related to many problems in PDE, spectral theory, analytic number theory, and combinatorics. I will then introduce some recent developments in harmonic analysis concerning this operator. I will mainly focus on various new ways to "induct on scales" that played an important role in the recent solution in all dimensions to Carleson's a.e. convergence problem on free Schrödinger solutions.

Nov 7: Luca Spolaor (MIT)

(Log)-Epiperimetric Inequality and the Regularity of Variational Problems

In this talk I will present a new method for studying the regularity of minimizers to variational problems. I will start by introducing the notion of blow-up, using as a model case the so-called Obstacle problem. Then I will state the (Log)-epiperimetric inequality and explain how it is used to prove uniqueness of the blow-up and regularity results for the solution near its singular set. I will then show the flexibility of this method by describing how it can be applied to other free-boundary problems and to (almost)-area minimizing currents. Finally I will describe some future applications of this method both in regularity theory and in other settings.

Nov 9: Anne Shiu (Texas A&M)

Dynamics of biochemical reaction systems

Reaction networks taken with mass-action kinetics arise in many settings, from epidemiology to population biology to systems of chemical reactions. This talk focuses on certain biological signaling networks, namely, phosphorylation networks, and their resulting dynamical systems. For many of these systems, the set of steady states admits a rational parametrization (that is, the set is the image of a map with rational-function coordinates). We describe how such a parametrization allows us to investigate the dynamics, including the emergence of bistability in a network underlying ERK regulation, and the capacity for oscillations in a mixed processive/distributive phosphorylation network.

Nov 19: Alexander Yom Din (Caltech)

From analysis to algebra to geometry - an example in representation theory of real groups

Representation theory of non-compact real groups, such as SL(2,R), is a fundamental discipline with uses in harmonic analysis, number theory, physics, and more. This theory is analytical in nature, but in the course of the 20th century it was algebraized and geometrized (the key contributions are by Harish-Chandra for the former and by Beilinson-Bernstein for the latter). Roughly and generally speaking, algebraization strips layers from the objects of study until we are left with a bare skeleton, amenable to symbolic manipulation. Geometrization, again very roughly, reveals how algebraic objects have secret lives over spaces - thus more amenable to human intuition. In this talk, I will try to motivate and present one example - the calculation of the Casselman-Jacquet module of a principal series representation (I will explain the terms in the talk).

Nov 20: Denis Hirschfeldt (University of Chicago)

Computability and Ramsey Theory

Computability theory can be seen as the study of the fine structure of definability. Much of its power relies on the deep connections between definability and computation. These connections can be seen in fundamental results such as Post's Theorem, which establishes a connection between the complexity of formulas needed to define a given set of natural numbers and its computability-theoretic strength. As has become increasingly clear, they can also be seen in the computability-theoretic analysis of objects whose definitions come from notions that arise naturally in combinatorics. The heuristic here is that computability-theoretically natural notions tend to be combinatorially natural, and vice-versa. I will discuss some results and open questions in the computability-theoretic analysis of combinatorial principles, in particular Ramsey-theoretic ones such as versions of Ramsey's Theorem for colorings of countably infinite sets, and versions of Hindman's Theorem, which states that for every coloring of the natural numbers with finitely many colors, there is an infinite set of numbers such that all nonempty sums of distinct elements of this set have the same color.

Nov 26: Vadim Gorin (MIT)

Macroscopic fluctuations through Schur generating functions

I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.

It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.

Nov 28: Gao Chen (IAS)

A Torelli type theorem

The length of a circle determines the shape of it. In this talk, we will discuss non-trivial generalizations of this fact for Riemann surfaces, hyperkähler 4-manifolds, Calabi-Yau threefolds and G_2, Spin(7) manifolds.

Nov 30: David Fisher (Indiana U.)

New Techniques for Zimmer's Conjecture

Lattices in higher rank simple Lie groups are known to be extremely rigid. Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory. Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds. After providing some history and motivation, I will discuss a recent result that makes dramatic progress on the conjecture in all cases and proves it in many of them. I will place some emphasis on surprising connections to other areas of mathematics that arise in the proof.

Dec 3: Bena Tshishiku (Harvard)

Surface bundles, monodromy, and arithmetic groups

Fiber bundles with fiber a surface arise in many areas including hyperbolic geometry, symplectic geometry, and algebraic geometry. Up to isomorphism, a surface bundle is completely determined by its monodromy representation, which is a homomorphism to a mapping class group. This allows one to use algebra to study the topology of surface bundles. Unfortunately, the monodromy representation is typically difficult to ``compute" (e.g. determine its image). In this talk, I will discuss some recent work toward computing monodromy groups for holomorphic surface bundles, including certain examples of Atiyah and Kodaira. This can be applied to the problem of counting the number of ways that certain 4-manifolds fiber over a surface. This is joint work with Nick Salter.

Dec 5: Subhabrata Sen (MIT and Microsoft Research New England)

Random graphs, Optimization, and Spin glasses

Combinatorial optimization problems are ubiquitous in diverse mathematical applications. The desire to understand their ``typical" behavior motivates a study of these problems on random instances. In spite of a long and rich history, many natural questions in this domain are still intractable to rigorous mathematical analysis. Graph cut problems such as Max-Cut and Min-bisection are canonical examples in this class. On the other hand, physicists study these questions using the non-rigorous ``replica" and ``cavity" methods, and predict complex, intriguing features. In this talk, I will describe some recent progress in our understanding of their typical properties on random graphs, obtained via connections to the theory of mean-field spin glasses. The new techniques are broadly applicable, and lead to novel algorithmic and statistical consequences.

Dec 7: Leonardo Zepeda-Nunez (Berkeley)

Accelerating ab-initio molecular dynamics via multi-scale neural networks

Deep learning has rapidly become a large field with an ever-growing range of applications; however, its intersection with scientific computing remains in its infancy, mainly due to the high accuracy that scientific computing problems require, which depends greatly on the architecture of the neural network.

In this talk we present a novel deep neural network with a multi-scale architecture inspired in H-matrices (and H2-matrices) to efficiently approximate, within 3-4 digits, several challenging non-linear maps arising from the discretization of PDEs, whose evaluation would otherwise require computationally intensive iterative methods.

In particular, we focus on the notoriously difficult Kohn-Sham map arising from Density Functional Theory (DFT). We show that the proposed multiscale-neural network can efficiently learn this map, thus bypassing an expensive self-consistent field iteration. In addition, we show the application of this methodology to ab-initio molecular dynamics, for which we provide examples for 1D problems and small, albeit realistic, 3D systems.

Joint work with Y. Fan, J. Feliu-Faaba, L. Lin, W. Jia, and L. Ying

Dec 10: Max Engelstein (MIT)

The role of Energy in Regularity

The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.

However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.

We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.

Past Colloquia

Blank

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012