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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 ==
  
==Fall 2017==
 
  
 
{| 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)
 
|-
 
|-
|September 8
+
|Sep 12    '''Room 911'''
| [https://sites.google.com/a/wisc.edu/theresa-c-anderson/home/ Tess Anderson] (Madison)
+
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
|[[#September 8: Tess Anderson (Madison) |  A Spherical Maximal Function along the Primes ]]
+
|[[#Sep 12: Gunther Uhlmann (Univ. of Washington)|  Harry Potter's Cloak via Transformation Optics ]]
| Tonghai Yang
+
| Li
 
|
 
|
 
|-
 
|-
|September 15
+
|Sep 14    '''Room 911'''
 +
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
 +
|[[#Sep 14: Gunther Uhlmann (Univ. of Washington) |  Journey to the Center of the Earth  ]]
 +
| Li
 
|
 
|
|[[#|   ]]
+
|-
 +
|Sep 21    '''Room 911'''
 +
| [http://stuart.caltech.edu/  Andrew Stuart] (Caltech) LAA lecture
 +
|[[#Sep 21: Andrew Stuart (Caltech) | The Legacy of Rudolph Kalman  ]]
 +
| Jin
 
|
 
|
 +
|-
 +
|Sep 28
 +
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
 +
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
 +
| Thiffeault
 
|
 
|
 +
|-
 +
|Oct 5
 +
| [http://www.personal.psu.edu/eus25/ Eyal Subag] (Penn State)
 +
|[[#Oct 5: Eyal Subag (Penn State)|  Symmetries of the hydrogen atom and algebraic families  ]]
 +
| Gurevich
 
|
 
|
 
|-
 
|-
|September 22, '''9th floor'''
+
|Oct 12
| Jaeyoung Byeon (KAIST)
+
| [https://www.math.wisc.edu/~andreic/ Andrei Caldararu] (Madison)
|[[#September 22: Jaeyoung Byeon (KAIST) | Patterns formation for elliptic systems with large interaction forces ]]
+
|[[#Oct 12: Andrei Caldararu (Madison) | Mirror symmetry and derived categories ]]
| Paul Rabinowitz & Chanwoo Kim
+
| ...
 
|
 
|
 +
|-
 +
|Oct 19
 +
|  [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  ]]
 +
| 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
 
|
 
|
 
|-
 
|-
|October 6, '''9th floor'''
+
|Nov 2 '''Room 911'''
| [http://www3.nd.edu/~jhauenst/ Jonathan Hauenstein] (Notre Dame)
+
| [https://sites.google.com/view/ruixiang-zhang/home?authuser=0# Ruixiang Zhang] (Madison)
|[[#October 6: Jonathan Hauenstein (Notre Dame) |  Real solutions of polynomial equations ]]
+
|[[#Nov 2: Ruixiang Zhang (Madison) |  The Fourier extension operator  ]]
| Nigel Boston
+
 
|  
 
|  
|-
 
|October 13, '''9th floor'''
 
| [http://www.tomokokitagawa.com/ Tomoko L. Kitagawa] (Berkeley)
 
|[[#October 13: Tomoko Kitagawa (Berkeley) |  A Global History of Mathematics from 1650 to 2017 ]]
 
| Laurentiu Maxim
 
 
|
 
|
 
|-
 
|-
|October 20
+
|Nov 7  '''Wednesday'''
| [http://cims.nyu.edu/~pgermain/ Pierre Germain] (Courant, NYU)  
+
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)
|[[#October 13: Pierre Germain (Courant, NYU) |  Stability of the Couette flow in the Euler and Navier-Stokes equations ]]
+
|[[#Nov 7: Luca Spolaor (MIT) |  (Log)-Epiperimetric Inequality and the Regularity of Variational Problems  ]]
| Minh-Binh Tran
+
| Feldman
 
|
 
|
 
|-
 
|-
|October 27
+
|Nov 12  '''Monday'''
|Stefanie Petermichl (Toulouse)
+
| [http://www.math.tamu.edu/~annejls/ Anne Shiu] (Texas A&M)
|[[#October 27: Stefanie Petermichl (Toulouse) Higher order Journé commutators ]]
+
|[[#Nov 9: Anne Shiu (Texas A&M) |  Dynamics of biochemical reaction systems ]]
| Betsy Stovall, Andreas Seeger
+
| Craciun, Stechmann
 
|
 
|
 
|-
 
|-
|November 1 (Wednesday)
+
|Nov 19 '''Monday'''
|[http://pages.iu.edu/~shaoguo/ Shaoming Guo] (Indiana)
+
| [https://sites.google.com/site/ayomdin/ Alexander Yom Din] (Caltech)  
|[[#November 1: Shaoming Guo (Indiana)| Parsell-Vinogradov systems in higher dimensions ]]
+
|[[#Nov 19: Alexander Yom Din (Caltech) | From analysis to algebra to geometry - an example in representation theory of real groups ]]
|Andreas Seeger
+
| Boston, Gurevitch
|
+
|
+
|
+
|
+
 
|
 
|
 
|-
 
|-
|November 17
+
|Nov 20 '''Tuesday, Room 911'''
| [http://math.mit.edu/~ylio/  Yevgeny Liokumovich] (MIT)
+
| [http://http://www.math.uchicago.edu/~drh/ Denis Hirschfeldt] (University of Chicago)
|[[#November 17:Yevgeny Liokumovich (MIT)|  Recent progress in Min-Max Theory  ]]
+
|[[#Nov 20: Denis Hirschfeldt (University of Chicago)|  Computability and Ramsey Theory ]]
|Sean Paul
+
| Andrews
|-
+
|November 21, '''9th floor'''
+
| [https://web.stanford.edu/~mkemeny/homepage.html  Michael Kemeny] (Stanford)
+
|[[#November 21:Michael Kemeny (Stanford)|  The equations defining curves and moduli spaces ]]
+
|Jordan Ellenberg
+
 
|
 
|
 
|-
 
|-
|November 24
+
|Nov 26 '''Monday, Room 911'''
|'''Thanksgiving break'''
+
| [http://math.mit.edu/directory/profile.php?pid=1415 Vadim Gorin] (MIT)
|
+
|[[#Nov 26: Vadim Gorin (MIT)|  Macroscopic fluctuations through Schur generating functions  ]]
 +
| Anderson
 
|
 
|
 
|-
 
|-
|November 27,
+
|Nov 28 '''Wednesday'''
| [http://www.math.harvard.edu/~tcollins/homepage.html  Tristan Collins] (Harvard)
+
| [http://www.math.ias.edu/~gchen/ Gao Chen](IAS)
|[[#November 27:Tristan Collins (Harvard)| The J-equation and stability  ]]
+
|[[#Nov 28: Gao Chen(IAS) | A Torelli type theorem ]]
|Sean Paul
+
| Paul  
|
+
 
|
 
|
 
|-
 
|-
|December 5 (Tuesday)
+
|Nov 30
| [http://web.sas.upenn.edu/rhynd/ Ryan Hynd] (U Penn)
+
| [https://math.indiana.edu/about/faculty/fisher-david.html David Fisher](Indiana U.)
|[[#December 5: Ryan Hynd (U Penn)| Adhesion dynamics and the sticky particle system]]
+
|[[#Nov 30: David Fisher (Indiana U.) | New Techniques for Zimmer's Conjecture ]]
|Sigurd Angenent
+
| Kent
|
+
 
|-
 
|-
|December 8 (Friday)
+
|Dec 3 '''Monday'''
| [https://cims.nyu.edu/~chennan/  Nan Chen] (Courant, NYU)
+
| [http://bena-tshishiku.squarespace.com/ Bena Tshishiku](Harvard)
|[[#December 8: Nan Chen (Courant, NYU)|  A Conditional Gaussian Framework for Uncertainty Quantification, Data Assimilation and Prediction of Complex Turbulent Dynamical Systems  ]]
+
|[[#Dec 3: Bena Tshishiku (Harvard)|  Surface bundles, monodromy, and arithmetic groups ]]
|Leslie Smith
+
| Paul
|
+
 
|
 
|
 
|-
 
|-
|December 11 (Monday)
+
|Dec 5 '''Wednesday, Room 911'''
| [https://people.math.ethz.ch/~mooneyc/ Connor Mooney] (ETH Zurich)
+
| [http://www.mit.edu/~ssen90/ Subhabrata Sen](MIT and Microsoft Research New England)
|[[#December 11: Connor Mooney (ETH Zurich)| Regularity vs. Singularity for Elliptic and Parabolic Systems]]
+
|[[#Dec 5: Subhabrata Sen (MIT and Microsoft Research New England) | Random graphs, Optimization, and Spin glasses ]]
|Sigurd Angenent
+
| Anderson
 
|
 
|
 
|-
 
|-
|December 13 (Wednesday)
+
|Dec 7 '''Room 911'''
| [http://math.mit.edu/~blwilson/ Bobby Wilson] (MIT)
+
| [https://math.berkeley.edu/people/faculty/leonardo-zepeda-n-ez Leonardo Zepeda-Nunez](Berkeley)
|[[#December 13: Bobby Wilson (MIT) | Projections in Banach Spaces and Harmonic Analysis ]]
+
|[[#Dec 7: Leonardo Zepeda-Nunez (Berkeley) | Accelerating ab-initio molecular dynamics via multi-scale neural networks ]]
|Andreas Seeger
+
| Stechmann
 
|
 
|
 
|-
 
|-
|December 15 (Friday)
+
|Dec 10 '''Monday'''
| [http://roy.lederman.name/ Roy Lederman] (Princeton)
+
| [http://math.mit.edu/~maxe/ Max Engelstein](MIT)
|[[#December 15: Roy Lederman (Princeton) | Inverse Problems and Unsupervised Learning with applications to Cryo-Electron Microscopy (cryo-EM) ]]
+
|[[#Dec 10: Max Engelstein (MIT)| The role of Energy in Regularity  ]]
|Leslie Smith
+
| Feldman
 
|
 
|
|-
 
|December 18 (Monday)
 
| [https://web.stanford.edu/~jchw/ Jenny Wilson] (Stanford)
 
|[[#December 18: Jenny Wilson (Stanford)|  Stability in the homology of configuration spaces]]
 
|Jordan Ellenberg
 
|
 
|-
 
|December 19 (Tuesday)
 
| [https://web.stanford.edu/~amwright/  Alex Wright] (Stanford)
 
|[[#December 19: Alex Wright (Stanford)|  Dynamics, geometry, and the moduli space of Riemann surfaces]]
 
|Jordan Ellenberg
 
 
|}
 
|}
  
== Fall Abstracts ==
+
== Abstracts ==
=== September 8: Tess Anderson (Madison) ===
+
Title: A Spherical Maximal Function along the Primes
+
  
Abstract: Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior. The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example.  In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to.  We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory.  This is joint work with Cook, Hughes, and Kumchev.
+
=== 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.
  
=== September 22: Jaeyoung Byeon (KAIST) ===
+
=== Sep 14: Gunther Uhlmann (Univ. of Washington) ===
Title: Patterns formation for elliptic systems with large interaction forces
+
Journey to the Center of the Earth
  
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions.  The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.
+
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.
  
===October 6: Jonathan Hauenstein (Notre Dame) ===
+
The problem can be recast as a geometric problem: Can one determine the
Title: Real solutions of polynomial equations
+
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: Systems of nonlinear polynomial equations arise frequently in applications with the set of real solutions typically corresponding to physically meaningful solutions.  Efficient algorithms for computing real solutions are designed by exploiting structure arising from the application. This talk will highlight some of these algorithms for various applications such as solving steady-state problems of hyperbolic conservation laws, solving semidefinite programs, and computing all steady-state solutions of the Kuramoto model.
+
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.
  
===October 13: Tomoko Kitagawa (Berkeley) ===
+
=== Sep 21: Andrew Stuart (Caltech) ===
Title: A Global History of Mathematics from 1650 to 2017
+
  
Abstract: This is a talk on the global history of mathematics. We will first focus on France by revisiting some of the conversations between Blaise Pascal (1623–1662) and Pierre de Fermat (1607–1665). These two “mathematicians” discussed ways of calculating the possibility of winning a gamble and exchanged their opinions on geometry. However, what about the rest of the world? We will embark on a long oceanic voyage to get to East Asia and uncover the unexpected consequences of blending foreign mathematical knowledge into domestic intelligence, which was occurring concurrently in Beijing and Kyoto. How did mathematicians and scientists contribute to the expansion of knowledge? What lessons do we learn from their experiences?
+
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) ===
  
===October 20: Pierre Germain (Courant, NYU) ===
+
Stirring and Mixing
Title: Stability of the Couette flow in the Euler and Navier-Stokes equations
+
  
Abstract: I will discuss the question of the (asymptotic) stability of the Couette flow in Euler and Navier-Stokes. The Couette flow is the simplest nontrivial stationary flow, and the first one for which this question can be fully answered. The answer involves the mathematical understanding of important physical phenomena such as inviscid damping and enhanced dissipation. I will present recent results in dimension 2 (Bedrossian-Masmoudi) and dimension 3 (Bedrossian-Germain-Masmoudi).
+
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:
  
===October 27: Stefanie Petermichl (Toulouse)===
+
1. How efficiently can stirring "mix"?
Title: Higher order Journé commutators
+
  
Abstract: We consider questions that stem from operator theory via Hankel and
+
2. What is the interaction between diffusion and mixing.
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
+
more basic terms, let us consider a function on the unit circle in its
+
Fourier representation. Let P_+ denote the projection onto
+
non-negative and P_- onto negative frequencies. Let b denote
+
multiplication by the symbol function b. It is a classical theorem by
+
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
+
only if b is in an appropriate space of functions of bounded mean
+
oscillation. The necessity makes use of a classical factorisation
+
theorem of complex function theory on the disk. This type of question
+
can be reformulated in terms of commutators [b,H]=bH-Hb with the
+
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
+
as in the real variable setting, in the multi-parameter setting or
+
other, these classifications can be very difficult.
+
  
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
+
Both these aspects are rich in open problems whose resolution involves tools from various different areas. I present a brief survey of existing
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of
+
results, and talk about a few open problems.
spaces of bounded mean oscillation via L^p boundedness of commutators.
+
We present here an endpoint to this theory, bringing all such
+
characterisation results under one roof.
+
  
The tools used go deep into modern advances in dyadic harmonic
+
=== Oct 5: Eyal Subag (Penn State)===
analysis, while preserving the Ansatz from classical operator theory.
+
  
===November 1: Shaoming Guo (Indiana) ===
+
Symmetries of the hydrogen atom and algebraic families
Title: Parsell-Vinogradov systems in higher dimensions
+
  
Abstract:
+
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.
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
+
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
+
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.
+
  
===November 17:Yevgeny Liokumovich (MIT)===
+
=== Oct 12: Andrei Caldararu (Madison)===
Title: Recent progress in Min-Max Theory
+
  
Abstract:
+
Mirror symmetry and derived categories
Almgren-Pitts Min-Max Theory is a method of constructing minimal hypersurfaces in Riemannian manifolds. In the last few years a number of long-standing open problems in Geometry, Geometric Analysis and 3-manifold Topology have been solved using this method. I will explain the main ideas and challenges in Min-Max Theory with an emphasis on its quantitative aspect: what quantitative information about the geometry and topology of minimal hypersurfaces can be extracted from the theory?
+
  
===November 21:Michael Kemeny (Stanford)===
+
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.
Title: The equations defining curves and moduli spaces
+
  
Abstract:
+
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).
A projective variety is a subset of projective space defined by polynomial equations. One of the oldest problems in algebraic geometry is to give a qualitative description of the equations defining a variety, together with
+
the relations amongst them. When the variety is an algebraic curve (or Riemann surface), several conjectures
+
made since the 80s give a fairly good picture of what we should expect. I will describe a new variational approach to these conjectures,
+
which reduces the problem to studying cycles on Hurwitz space or on the moduli space of curves.
+
  
 +
===  Oct 19:  Jeremy Teitelbaum (U Connecticut)===
 +
Lessons Learned and New Perspectives:
 +
From Dean and Provost to aspiring Data Scientist
  
===November 27:Tristan Collins (Harvard)===
+
After more than 10 years in administration, including 9 as Dean of
Title: The J-equation and stability
+
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.
  
Abstract: Donaldson and Chen introduced the J-functional in '99, and explained its importance in the existence problem for constant scalar curvature metrics on compact Kahler manifolds. An important open problem is to find algebro-geometric conditions under which the J-functional has a critical point.  The critical points of the J-functional are described by a fully-nonlinear PDE called the J-equation.  I will discuss some recent progress on this problem, and indicate the role of algebraic geometry in proving estimates for the J-equation.
+
=== Oct 26: Douglas Ulmer (Arizona)===
  
===December 5: Ryan Hynd (U Penn)===
+
Rational numbers, rational functions, and rational points
Title: Adhesion dynamics and the sticky particle system.
+
  
Abstract:  The sticky particle system expresses the conservation of mass and
+
One of the central concerns of arithmetic geometry is the study of
momentum for a collection of particles that only interact via perfectly inelastic collisions. 
+
solutions of systems of polynomial equations where the solutions are
The equations were first considered in astronomy in a model for the expansion of
+
required to lie in a "small" field such as the rational numbers. I
matter without pressure. These equations also play a central role in the theory of optimal
+
will explain the landscape of expectations and conjectures in this
transport.  Namely, the geodesics in an appropriately metrized space of probability
+
area, focusing on curves and their Jacobians over global fields
measures correspond to solutions of the sticky particle systemWe will survey what is  
+
(number fields and function fields), and then survey the progress made
known about solutions and discuss connections with Hamilton-Jacobi equations.  
+
over the last decade in the function field caseThe talk is intended
 +
to be accessible to a wide audience.
  
===December 8: Nan Chen (Courant, NYU)===
+
=== Nov 2: Ruixiang Zhang (Madison)===
Title: A Conditional Gaussian Framework for Uncertainty Quantification, Data Assimilation and Prediction of Complex Turbulent Dynamical Systems
+
  
Abstract:
+
The Fourier extension operator
A conditional Gaussian framework for uncertainty quantification, data assimilation and prediction of nonlinear turbulent dynamical systems will be introduced in this talk. Despite the conditional Gaussianity, the dynamics remain highly nonlinear and are able to capture strongly non-Gaussian features such as intermittency and extreme events. The conditional Gaussian structure allows efficient and analytically solvable conditional statistics that facilitates the real-time data assimilation and prediction.
+
  
The talk will include three applications of such conditional Gaussian framework. In the first part, a physics-constrained nonlinear stochastic model is developed, and is applied to predicting the Madden-Julian oscillation indices with strongly non-Gaussian intermittent features. The second part regards the state estimation and data assimilation of multiscale and turbulent ocean flows using noisy Lagrangian tracers. Rigorous analysis shows that an exponential increase in the number of tracers is required for reducing the uncertainty by a fixed amount. This indicates a practical information barrier. In the last part of the talk, an efficient statistically accurate algorithm is developed that is able to solve a rich class of high dimensional Fokker-Planck equation with strong non-Gaussian features and beat the curse of dimensions.
+
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.
  
===December 11: Connor Mooney (ETH Zurich)===
+
=== Nov 7: Luca Spolaor (MIT)===
Title: Regularity vs. Singularity for Elliptic and Parabolic Systems
+
  
Abstract:
+
(Log)-Epiperimetric Inequality and the Regularity of Variational Problems
Hilbert's 19th problem asks if minimizers of &ldquo;natural&rdquo; variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDEs. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, as well as outstanding open problems. Some of this is joint work with O. Savin.
+
  
===December 13: Bobby Wilson (MIT)===
+
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.
Title:  Projections in Banach Spaces and Harmonic Analysis
+
Finally I will describe some future applications of this method both in regularity theory and in other settings.
  
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.
+
=== Nov 9: Anne Shiu (Texas A&M)===
  
===December 15: Roy Lederman (Princeton)===
+
Dynamics of biochemical reaction systems
Title: Inverse Problems and Unsupervised Learning with applications to Cryo-Electron Microscopy (cryo-EM)
+
  
Abstract:
+
Reaction networks taken with mass-action kinetics arise in many settings,
Cryo-EM is an imaging technology that is revolutionizing structural biology; the Nobel Prize in Chemistry 2017 was recently awarded to Jacques Dubochet, Joachim Frank and Richard Henderson “for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution".  
+
from epidemiology to population biology to systems of chemical reactions.  
+
This talk focuses on certain biological signaling networks, namely,  
Cryo-electron microscopes produce a large number of very noisy two-dimensional projection images of individual frozen molecules. Unlike related methods, such as computed tomography (CT), the viewing direction of each image is unknown. The unknown directions, together with extreme levels of noise and additional technical factors, make the determination of the structure of molecules challenging.  
+
phosphorylation networks, and their resulting dynamical systems. For many
+
of these systems, the set of steady states admits a rational
While other methods for structure determination, such as x-ray crystallography and nuclear magnetic resonance (NMR), measure ensembles of molecules together, cryo-EM produces measurements of individual molecules. Therefore, cryo-EM could potentially be used to study mixtures of different conformations of molecules. Indeed, current algorithms have been very successful at analyzing homogeneous samples, and can recover some distinct conformations mixed in solutions, but, the determination of multiple conformations, and in particular, continuums of similar conformations (continuous heterogeneity), remains one of the open problems in cryo-EM.
+
parametrization (that is, the set is the image of a map with
+
rational-function coordinates). We describe how such a parametrization
I will discuss a one-dimensional discrete model problem, Heterogeneous Multireference Alignment, which captures many of the group properties and other mathematical properties of the cryo-EM problem. I will then discuss different components which we are introducing in order to address the problem of continuous heterogeneity in cryo-EM: 1. “hyper-molecules,the mathematical formulation of truly continuously heterogeneous molecules, 2. computational and numerical tools for formulating associated priors, and 3. Bayesian algorithms for inverse problems with an unsupervised-learning component for recovering such hyper-molecules in cryo-EM.
+
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.
  
===December 18: Jenny Wilson (Stanford)===
+
=== Nov 19: Alexander Yom Din (Caltech)===
Title: Stability in the homology of configuration spaces
+
 
 +
From analysis to algebra to geometry - an example in representation theory of real groups
  
Abstract:
+
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).
This talk will illustrate some patterns in the homology of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena -- relationships between unstable homology classes in different degrees -- established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius--Kupers--Randal-Williams.
+
  
===December 19: Alex Wright (Stanford)===
+
=== Nov 20: Denis Hirschfeldt (University of Chicago)===
Title: Dynamics, geometry, and the moduli space of Riemann surfaces
+
  
Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.
+
Computability and Ramsey Theory
  
== Spring 2018 ==
+
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.
  
{| cellpadding="8"
+
=== Nov 26: Vadim Gorin (MIT)===
!align="left" | date 
+
 
!align="left" | speaker
+
Macroscopic fluctuations through Schur generating functions
!align="left" | title
+
 
!align="left" | host(s)
+
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.
|-
+
 
| March 16
+
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.
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)
+
 
|[[# TBA|  TBA  ]]
+
=== Nov 28: Gao Chen (IAS) ===
| WIMAW
+
 
|
+
A Torelli type theorem
|-
+
 
|April 4 (Wednesday)
+
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.
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)
+
 
|[[# TBA|  TBA  ]]
+
=== Nov 30: David Fisher (Indiana U.) ===
| Craciun
+
 
|
+
New Techniques for Zimmer's Conjecture
|-
+
 
| April 6
+
Lattices in higher rank simple Lie groups are known to be
| Reserved
+
extremely rigid. Examples of this are Margulis' superrigidity theorem,
|[[# TBA|  TBA  ]]
+
which shows they have very few linear represenations, and Margulis'
| Melanie
+
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
| April 13
+
manifolds. After providing some history and motivation, I will discuss
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)
+
a recent result that makes dramatic progress on the conjecture in all
|[[# TBA|  TBA  ]]
+
cases and proves it in many of them. I will place some emphasis on
| WIMAW
+
surprising connections to other areas of mathematics that arise in the
|
+
proof.
|-
+
 
| April 25 (Wednesday)
+
=== Dec 3: Bena Tshishiku (Harvard) ===
| Hitoshi Ishii (Waseda University) Wasow lecture
+
|[[# TBA|  TBA  ]]
+
| Tran
+
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+
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+
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+
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+
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+
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+
|[[# TBA|  TBA  ]]
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+
|[[# TBA|  TBA  ]]
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+
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+
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|[[# TBA|  TBA  ]]
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|[[# TBA|  TBA  ]]
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|-
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|[[# TBA|  TBA  ]]
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|}
+
  
== Spring Abstracts ==
+
Surface bundles, monodromy, and arithmetic groups
  
=== <DATE>: <PERSON> (INSTITUTION) ===
+
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.
Title: <TITLE>
+
  
Abstract: <ABSTRACT>
+
=== 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]]
+
[[Colloquia/Blank|Blank]]
 +
 
 +
[[Colloquia/Spring2018|Spring 2018]]
 +
 
 +
[[Colloquia/Fall2017|Fall 2017]]
  
 
[[Colloquia/Spring2017|Spring 2017]]
 
[[Colloquia/Spring2017|Spring 2017]]

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.

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