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 ==Spring 2019==   ==Spring 2019== 

 
 == Fall 2018 ==
 

 

 

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 !align="left"  host(s)   !align="left"  host(s) 
     
 Sep 12 '''Room 911'''   Jan 25 
  [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series    [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW 
 [[#Sep 12: Gunther Uhlmann (Univ. of Washington) Harry Potter's Cloak via Transformation Optics ]]   [[#Beata Randrianantoanina (Miami University Ohio)  Some nonlinear problems in the geometry of Banach spaces and their applications ]] 
  Li    Tullia Dymarz 
     
     
 Sep 14 '''Room 911'''   Jan 30 '''Wednesday''' 
  [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series    [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University) 
 [[#Sep 14: Gunther Uhlmann (Univ. of Washington)  Journey to the Center of the Earth ]]   [[#Lillian Pierce (Duke University)  Short character sums ]] 
  Li    Boston and Street 
     
     
 Sep 21 '''Room 911'''   Jan 31 '''Thursday''' 
  [http://stuart.caltech.edu/ Andrew Stuart] (Caltech) LAA lecture    [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M) 
 [[#Sep 21: Andrew Stuart (Caltech)  The Legacy of Rudolph Kalman ]]   [[#Dean Baskin (Texas A&M)  Radiation fields for wave equations ]] 
  Jin    Street 
     
     
 Sep 28   Feb 1 
  [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)    [https://services.math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke University) 
 [[#Sep 28: Gautam Iyer (CMU) Stirring and Mixing ]]   [[# TBA TBA ]] 
  Thiffeault    Qin 
     
     
 Oct 5   Feb 5 '''Tuesday''' 
  [http://www.personal.psu.edu/eus25/ Eyal Subag] (Penn State)    [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University) 
 [[#Oct 5: Eyal Subag (Penn State) Symmetries of the hydrogen atom and algebraic families ]]   [[# TBA TBA ]] 
  Gurevich    Denisov 
     
     
 Oct 12   Feb 8 
  [https://www.math.wisc.edu/~andreic/ Andrei Caldararu] (Madison)    [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern) 
 [[#Oct 12: Andrei Caldararu (Madison)  Mirror symmetry and derived categories ]]   [[#Aaron Naber (Northwestern)  A structure theory for spaces with lower Ricci curvature bounds ]] 
  ...    Street 
     
     
 Oct 19   Feb 15 
  [https://teitelbaum.math.uconn.edu/# Jeremy Teitelbaum] (U Connecticut)    
 [[#Oct 19: Jeremy Teitelbaum (U Connecticut) Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist ]]
  [[# TBA TBA ]] 
  Boston
 
 
 
 
 
 Oct 26
 
  [http://math.arizona.edu/~ulmer/index.html Douglas Ulmer] (Arizona)
 
 [[#Oct 26: Douglas Ulmer (Arizona)  Rational numbers, rational functions, and rational points ]]
 
  Yang
 
 
 
 
 
 Nov 2 '''Room 911'''
 
  [https://sites.google.com/view/ruixiangzhang/home?authuser=0# Ruixiang Zhang] (Madison)
 
 [[#Nov 2: Ruixiang Zhang (Madison)  The Fourier extension operator ]]  
     
     
     
 Nov 7 '''Wednesday'''   Feb 22 
  [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)    [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State) 
 [[#Nov 7: Luca Spolaor (MIT)  (Log)Epiperimetric Inequality and the Regularity of Variational Problems ]]   [[# TBA TBA ]] 
  Feldman    Erman and Corey 
     
     
 Nov 12 '''Monday'''   March 4 
  [http://www.math.tamu.edu/~annejls/ Anne Shiu] (Texas A&M)    [http://wwwusers.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture 
 [[#Nov 9: Anne Shiu (Texas A&M)  Dynamics of biochemical reaction systems ]]   [[# TBA TBA ]] 
  Craciun, Stechmann    Kim 
     
     
 Nov 19 '''Monday'''   March 8 
  [https://sites.google.com/site/ayomdin/ Alexander Yom Din] (Caltech)    [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State) 
 [[#Nov 19: Alexander Yom Din (Caltech)  From analysis to algebra to geometry  an example in representation theory of real groups ]]   [[# TBA TBA ]] 
  Boston, Gurevitch    Erman 
     
     
 Nov 20 '''Tuesday, Room 911'''   March 15 
  [http://http://www.math.uchicago.edu/~drh/ Denis Hirschfeldt] (University of Chicago)    Maksym Radziwill (Caltech) 
 [[#Nov 20: Denis Hirschfeldt (University of Chicago) Computability and Ramsey Theory ]]   [[# TBA TBA ]] 
  Andrews    Marshall 
     
     
 Nov 26 '''Monday, Room 911'''   March 29 
  [http://math.mit.edu/directory/profile.php?pid=1415 Vadim Gorin] (MIT)    Jennifer Park (OSU) 
 [[#Nov 26: Vadim Gorin (MIT) Macroscopic fluctuations through Schur generating functions ]]   [[# TBA TBA ]] 
  Anderson    Marshall 
     
     
 Nov 28 '''Wednesday'''   April 5 
  [http://www.math.ias.edu/~gchen/ Gao Chen](IAS)    JuLee Kim (MIT) 
 [[#Nov 28: Gao Chen(IAS)  A Torelli type theorem ]]   [[# TBA TBA ]] 
  Paul    Gurevich 
     
     
 Nov 30   April 12 
  [https://math.indiana.edu/about/faculty/fisherdavid.html David Fisher](Indiana U.)    Evitar Procaccia (TAMU) 
 [[#Nov 30: David Fisher (Indiana U.)  New Techniques for Zimmer's Conjecture ]]   [[# TBA TBA ]] 
  Kent
   Gurevich 
 
 
 Dec 3 '''Monday'''
 
  [http://benatshishiku.squarespace.com/ Bena Tshishiku](Harvard)
 
 [[#Dec 3: Bena Tshishiku (Harvard) Surface bundles, monodromy, and arithmetic groups ]]
 
  Paul  
     
     
 Dec 5 '''Wednesday, Room 911'''   April 19 
  [http://www.mit.edu/~ssen90/ Subhabrata Sen](MIT and Microsoft Research New England)    [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University) 
 [[#Dec 5: Subhabrata Sen (MIT and Microsoft Research New England)  Random graphs, Optimization, and Spin glasses ]]   [[# TBA TBA ]] 
  Anderson    JeanLuc 
     
     
 Dec 7 '''Room 911'''   April 26 
  [https://math.berkeley.edu/people/faculty/leonardozepedanez Leonardo ZepedaNunez](Berkeley)    [https://www.brown.edu/academics/appliedmathematics/faculty/kavitaramanan/home Kavita Ramanan] (Brown University) 
 [[#Dec 7: Leonardo ZepedaNunez (Berkeley)  Accelerating abinitio molecular dynamics via multiscale neural networks ]]   [[# TBA TBA ]] 
  Stechmann    WIMAW 
     
     
 Dec 10 '''Monday'''   May 3 
  [http://math.mit.edu/~maxe/ Max Engelstein](MIT)    Tomasz Przebinda (Oklahoma) 
 [[#Dec 10: Max Engelstein (MIT) The role of Energy in Regularity ]]   [[# TBA TBA ]] 
  Feldman    Gurevich 
     
 }   } 
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 == Abstracts ==   == Abstracts == 

 

 === Sep 12: Gunther Uhlmann (Univ. of Washington) ===   ===Beata Randrianantoanina (Miami University Ohio)=== 
 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 nontechnical fashion
 
 one of them, the socalled "traansformation optics"
 
 in a nontechnical fashion n the socalled 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 socalled lens rigidity
 
 problem. The linearization of these problems involve the integration
 
 of a tensor along geodesics, similar to the Xray 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 spacecraft 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 HarishChandra 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 (socalled GromovWitten 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 CalabiYau three folds (joint work with Lev Borisov); and a recent computation of a categorical GromovWitten 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 (JAXGM) 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 highlevel administrator, talk a bit about
 
 the research environment at an independent research institute like JAXGM, 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 blowup, using as a model case the socalled Obstacle problem. Then I will state the (Log)epiperimetric inequality and explain how it is used to prove uniqueness of the blowup 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 freeboundary 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 massaction 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
 
 rationalfunction 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 noncompact 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 HarishChandra for the former and by BeilinsonBernstein 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 CasselmanJacquet 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 computabilitytheoretic strength. As has become increasingly clear, they can also be seen in the computabilitytheoretic analysis of objects whose definitions come from notions that arise naturally in combinatorics. The heuristic here is that
 
 computabilitytheoretically natural notions tend to be combinatorially
 
 natural, and viceversa. I will discuss some results and open questions in
 
 the computabilitytheoretic analysis of combinatorial principles, in
 
 particular Ramseytheoretic 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)===
  Title: Some nonlinear problems in the geometry of Banach spaces and their applications. 

 

 Macroscopic fluctuations through Schur generating functions
  Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics. 

 

 I will talk about a special class of largedimensional stochastic systems with strong correlations. The main examples will be random tilings, noncolliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.
  ===Lillian Pierce (Duke University)=== 

 

 It is believed that macroscopic fluctuations in such systems are universally described by logcorrelated 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.
  Title: Short character sums 

 

 === Nov 28: Gao Chen (IAS) ===
  Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a socalled character sum. For example, both understanding the Riemann zeta function or Dirichlet Lfunctions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations. 

 

 A Torelli type theorem   ===Dean Baskin (Texas A&M)=== 

 

 The length of a circle determines the shape of it. In this talk, we will discuss nontrivial generalizations of this fact for Riemann surfaces, hyperkähler 4manifolds, CalabiYau threefolds and G_2, Spin(7) manifolds.
  Title: Radiation fields for wave equations 

 

 === Nov 30: David Fisher (Indiana U.) ===
  Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space. 

 

 New Techniques for Zimmer's Conjecture
  ===Aaron Naber (Northwestern)=== 

 

 Lattices in higher rank simple Lie groups are known to be
  Title: A structure theory for spaces with lower Ricci curvature bounds. 
 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) ===
  Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li. 

 

 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 4manifolds 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 MaxCut and Minbisection are canonical examples in this class. On the other hand, physicists study these questions using the nonrigorous ``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 meanfield spin glasses. The new techniques are broadly applicable, and lead to novel algorithmic and statistical consequences.
 

 
 === Dec 7: Leonardo ZepedaNunez (Berkeley) ===
 

 
 Accelerating abinitio molecular dynamics via multiscale neural networks
 

 
 Deep learning has rapidly become a large field with an evergrowing 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 multiscale architecture inspired in Hmatrices (and H2matrices) to efficiently approximate, within 34 digits, several challenging nonlinear maps arising from the discretization of PDEs, whose evaluation would otherwise require computationally intensive iterative methods.
 

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

 
 Joint work with Y. Fan, J. FeliuFaaba, 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 EulerLagrange 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 EulerLagrange 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/BlankBlank]]   [[Colloquia/BlankBlank]] 
  
  [[Colloquia/Fall2018Fall 2018]] 

 

 [[Colloquia/Spring2018Spring 2018]]   [[Colloquia/Spring2018Spring 2018]] 