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__NOTOC__
 
 
 
= 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 2017  ==
 
 
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
Line 15: Line 14:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|'''Monday, January 9, 9th floor'''
+
|Sep 12    '''Room 911'''
| [http://www.stat.berkeley.edu/~racz/ Miklos Racz] (Microsoft)
+
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
|[[#Monday, January 9: Miklos Racz (Microsoft) |  ''Statistical inference in networks and genomics'' ]]
+
|[[#Sep 12: Gunther Uhlmann (Univ. of Washington)|  Harry Potter's Cloak via Transformation Optics ]]
| Valko
+
| Li
 
|
 
|
 
|-
 
|-
|January 13, B239
+
|Sep 14    '''Room 911'''
| [https://math.berkeley.edu/people/faculty/mihaela-ifrim/ Mihaela Ifrim] (Berkeley)
+
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
|[[#Friday, January 13: Mihaela Ifrim (Berkeley) |  ''Two dimensional water waves'' ]]
+
|[[#Sep 14: Gunther Uhlmann (Univ. of Washington) |  Journey to the Center of the Earth ]]
| Angenent
+
| Li
 
|
 
|
 
|-
 
|-
|'''Tuesday, January 17, B139'''
+
|Sep 21    '''Room 911'''
| [https://web.math.princeton.edu/~fabiop/ Fabio Pusateri] (Princeton)
+
| [http://stuart.caltech.edu/ Andrew Stuart] (Caltech) LAA lecture
|[[#Tuesday, January 17: Fabio Pusateri (Princeton) |  ''The Water Waves problem'' ]]
+
|[[#Sep 21: Andrew Stuart (Caltech) |  The Legacy of Rudolph Kalman ]]
| Angenent
+
| Jin
 
|
 
|
 
|-
 
|-
|January 20, B239
+
|Sep 28
| [http://math.mit.edu/~sraskin/ Sam Raskin] (MIT)
+
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
|[[#Friday, January 20: Sam Raskin (MIT) |   Tempered local geometric Langlands  ]]
+
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
| Arinkin
+
| Thiffeault
 
|
 
|
 
|-
 
|-
|'''Monday, January 23, B239'''
+
|Oct 5
| [http://www.math.umd.edu/~tdarvas/ Tamas Darvas] (Maryland)
+
| [http://www.personal.psu.edu/eus25/ Eyal Subag] (Penn State)
|[[#Monday, January 23: Tamas Darvas (Maryland) |  Geometry on the space of Kahler metrics and applications to canonical metrics ]]
+
|[[#Oct 5: Eyal Subag (Penn State)|  Symmetries of the hydrogen atom and algebraic families  ]]
| Viaclovsky
+
| Gurevich
 
|
 
|
 
|-
 
|-
|January 27
+
|Oct 12
|Reserved for possible job talks
+
| [https://www.math.wisc.edu/~andreic/ Andrei Caldararu] (Madison)
|[[# |   ]]
+
|[[#Oct 12: Andrei Caldararu (Madison) | Mirror symmetry and derived categories  ]]
|
+
| ...
 
|
 
|
 
|-
 
|-
|February 3, 9th floor
+
|Oct 19
| Melanie Matchett Wood (UW-, Madison)
+
| [https://teitelbaum.math.uconn.edu/# Jeremy Teitelbaum] (U Connecticut)
|[[#Friday, February 3: Melanie Matchett Wood (UW-Madison) | Random groups from generators and relations ]]
+
|[[#Oct 19:   Jeremy Teitelbaum (U Connecticut)| Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist ]]
|
+
| Boston
|
 
|-
 
|Monday, February 6, B239 (Wasow lecture)
 
| Benoit Perthame (University of Paris VI)
 
|[[#Monday, February 6: Benoit Perthame (University of Paris VI)| Models for neural networks; analysis, simulations and behaviour ]]
 
| Jin
 
|  
 
|-
 
|February 10 (WIMAW lecture), B239
 
| Alina Chertock (NC State Univ.)
 
|[[# |  ]]  
 
| WIMAW
 
 
|
 
|
 
|-
 
|-
|February 17, 9th floor
+
|Oct 26
| [http://web.math.ucsb.edu/~ponce/ Gustavo Ponce] (UCSB)
+
| [http://math.arizona.edu/~ulmer/index.html Douglas Ulmer] (Arizona)
| [[#Friday, February 17: Gustavo Ponce(UCSB) | The Korteweg-de Vries equation vs. the Benjamin-Ono equation ]]  
+
|[[#Oct 26: Douglas Ulmer (Arizona) | Rational numbers, rational functions, and rational points ]]
| Minh-Binh Tran
+
| Yang
 
|
 
|
 
|-
 
|-
|February 24
+
|Nov 2  '''Room 911'''
| [http://acms.nd.edu/people/faculty/jonathan-hauenstein/ Jonathan Hauenstein] (Notre Dame)
+
| [https://sites.google.com/view/ruixiang-zhang/home?authuser=0# Ruixiang Zhang] (Madison)
|[[#February 24: Jonathan Hauenstein (Notre Dame) |  Real solutions of polynomial equations ]]
+
|[[#Nov 2: Ruixiang Zhang (Madison) |  The Fourier extension operator ]]
| Boston
+
|  
 
|
 
|
 
|-
 
|-
|March 3
+
|Nov 7  '''Wednesday'''
| [http://www.math.utah.edu/~bromberg/ Ken Bromberg] (University of Utah)
+
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)
|[[# |   ]]
+
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems  ]]
|Dymarz
+
| Feldman
 
|
 
|
 
|-
 
|-
|Tuesday, March 7, 4PM (Distinguished Lecture)
+
|Nov 12  '''Monday'''
| [http://pages.iu.edu/~temam/ Roger Temam] (Indiana University)  
+
| [http://www.math.tamu.edu/~annejls/ Anne Shiu] (Texas A&M)
|[[#  |    ]]
+
|[[#Nov 9: Anne Shiu (Texas A&M) |  Dynamics of biochemical reaction systems ]]
|Smith
+
| Craciun, Stechmann
 
|
 
|
 
|-
 
|-
|'''Wednesday, March 8, 4PM, B239 '''
+
|Nov 19 '''Monday'''
| [http://pages.iu.edu/~temam/ Roger Temam] (Indiana University)  
+
| [https://sites.google.com/site/ayomdin/ Alexander Yom Din] (Caltech)  
|[[#  |    ]]
+
|[[#Nov 19: Alexander Yom Din (Caltech) | From analysis to algebra to geometry - an example in representation theory of real groups ]]
|Smith
+
| Boston, Gurevitch
 
|
 
|
 
|-
 
|-
|March 10
+
|Nov 20 '''Tuesday, Room 911'''
| '''No Colloquium'''  
+
| [http://http://www.math.uchicago.edu/~drh/ Denis Hirschfeldt] (University of Chicago)
|[[# |   ]]
+
|[[#Nov 20: Denis Hirschfeldt (University of Chicago)| Computability and Ramsey Theory  ]]
|
+
| Andrews
 
|
 
|
 
|-
 
|-
|'''Wednesday, March 15, 4PM '''
+
|Nov 26 '''Monday, Room 911'''
| [http://verso.mat.uam.es/web/ezuazua/zuazua.html Enrique Zuazua] (Universidad Autónoma de Madrid)
+
| [http://math.mit.edu/directory/profile.php?pid=1415 Vadim Gorin] (MIT)
|[[#   TBA|   TBA ]]
+
|[[#Nov 26: Vadim Gorin (MIT)| Macroscopic fluctuations through Schur generating functions ]]
| Jin & Minh-Binh Tran
+
| Anderson
 
|
 
|
 
|-
 
|-
|March 17
+
|Nov 28 '''Wednesday'''
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)  
+
| [http://www.math.ias.edu/~gchen/ Gao Chen](IAS)
| TBA
+
|[[#Nov 28: Gao Chen(IAS) | A Torelli type theorem ]]
| M. Matchett Wood
+
| Paul
 
|
 
|
 
|-
 
|-
|March 24
+
|Nov 30
| '''Spring Break'''
+
| [https://math.indiana.edu/about/faculty/fisher-david.html David Fisher](Indiana U.)
|[[# |   ]]
+
|[[#Nov 30: David Fisher (Indiana U.) | New Techniques for Zimmer's Conjecture ]]
|
+
| Kent
|
 
 
|-
 
|-
|'''Wednesday, March 29  at 3:30PM (Wasow)'''
+
|Dec 3 '''Monday'''
| [https://math.nyu.edu/faculty/serfaty/ Sylvia Serfaty] (NYU)  
+
| [http://bena-tshishiku.squarespace.com/ Bena Tshishiku](Harvard)
|[[# TBA|   TBA]]
+
|[[#Dec 3: Bena Tshishiku (Harvard)| Surface bundles, monodromy, and arithmetic groups ]]
|Tran
+
| Paul
 
|
 
|
 
|-
 
|-
|March 31
+
|Dec 5 '''Wednesday, Room 911'''
| '''No Colloquium'''  
+
| [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
 
|
 
|
 
|-
 
|-
|April 7
+
|Dec 7 '''Room 911'''
| [http://www.math.uiuc.edu/~schenck/ Hal Schenck]
+
| [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 ]]
|Erman
+
| Stechmann
 
|
 
|
 
|-
 
|-
|April 14
+
|Dec 10 '''Monday'''
| Wilfrid Gangbo
+
| [http://math.mit.edu/~maxe/ Max Engelstein](MIT)
|[[# |   ]]
+
|[[#Dec 10: Max Engelstein (MIT)| The role of Energy in Regularity  ]]
|Feldman & Tran
+
| Feldman
 
|
 
|
|-
 
|April 21
 
|  [http://www.math.stonybrook.edu/~mde/ Mark Andrea de Cataldo]  (Stony Brook)
 
|TBA
 
| Maxim
 
|
 
|-
 
|April 28
 
| [http://users.cms.caltech.edu/~hou/ Thomas Yizhao Hou] 
 
|[[# TBA|  TBA  ]]
 
|Li
 
 
|}
 
|}
  
==Fall 2017==
+
== Abstracts ==
  
{| cellpadding="8"
+
=== Sep 12: Gunther Uhlmann (Univ. of Washington) ===
!align="left" | date 
+
Harry Potter's Cloak via Transformation Optics
!align="left" | speaker
+
 
!align="left" | title
+
Can we make objects invisible? This has been a subject of human
!align="left" | host(s)
+
fascination for millennia in Greek mythology, movies, science fiction,
|-
+
etc. including the legend of Perseus versus Medusa and the more recent
|September 8
+
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
|[[# TBA|  TBA  ]]
+
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 15
 
|
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
| '''Wednesday, September 20, LAA lecture
 
| Andrew Stuart (Caltech)
 
|[[# TBA|  TBA  ]]
 
| Jin
 
|
 
|-
 
|September 22
 
|
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|September 29
 
|
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|October 6
 
|
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|October 13
 
|
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|October 20
 
|
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|October 27
 
|
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|November 3
 
|
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|November 10
 
| Reserved for possible job talks
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|November 17
 
| Reserved for possible job talks
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|November 24
 
|'''Thanksgiving break'''
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|December 1
 
| Reserved for possible job talks
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|December 8
 
| Reserved for possible job talks
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
  
|}
+
=== Sep 14: Gunther Uhlmann (Univ. of Washington) ===
 +
Journey to the Center of the Earth
  
== Abstracts ==
+
We will consider the inverse problem of determining the sound
=== September 16: Po-Shen Loh (CMU) ===
+
speed or index of refraction of a medium by measuring the travel times of
Title: Directed paths: from Ramsey to Pseudorandomness
+
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.
  
Abstract: Starting from an innocent Ramsey-theoretic question regarding directed
+
The problem can be recast as a geometric problem: Can one determine the
paths in graphs, we discover a series of rich and surprising connections
+
Riemannian metric of a Riemannian manifold with boundary by measuring
that lead into the theory around a fundamental result in Combinatorics:
+
the distance function between boundary points? This is the boundary
Szemeredi's Regularity Lemma, which roughly states that every graph (no
+
rigidity problem. We will also consider the problem of determining
matter how large) can be well-approximated by a bounded-complexity
+
the metric from the scattering relation, the so-called lens rigidity
pseudorandom object. Using these relationships, we prove that every
+
problem. The linearization of these problems involve the integration
coloring of the edges of the transitive N-vertex tournament using three
+
of a tensor along geodesics, similar to the X-ray transform.
colors contains a directed path of length at least sqrt(N) e^{log^* N}
 
which entirely avoids some color. The unusual function log^* is the
 
inverse function of the tower function (iterated exponentiation).
 
  
=== September 23: Gheorghe Craciun (UW-Madison) ===
+
We will also describe some recent results, join with Plamen Stefanov
Title: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture
+
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.
  
Abstract: The Global Attractor Conjecture says that a large class of polynomial dynamical systems, called toric dynamical systems, have a globally attracting point within each linear invariant space. In particular, these polynomial dynamical systems never exhibit multistability, oscillations or chaotic dynamics.
+
=== Sep 21: Andrew Stuart (Caltech) ===
  
The conjecture was formulated by Fritz Horn in the early 1970s, and is strongly related to Boltzmann's H-theorem.
+
The Legacy of Rudolph Kalman
  
We discuss the history of this problem, including the connection between this conjecture and the Boltzmann equation. Then, we introduce toric differential inclusions, and describe how they can be used to prove this conjecture in full generality.  
+
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.
  
=== September 30: Akos Magyar (University of Georgia) ===  
+
=== Sep 28: Gautam Iyer (CMU) ===
Title: Geometric Ramsey theory
 
  
Abstract: Initiated by Erdos, Graham, Montgomery and others in the 1970's, geometric Ramsey theory studies geometric configurations, determined up to translations, rotations and possibly dilations, which cannot be destroyed by finite partitions of Euclidean spaces. Later it was shown by ergodic and Fourier analytic methods that such results are also possible in the context of sets of positive upper density in Euclidean spaces or the integer lattice. We present a new approach, motivated by developments in arithmetic combinatorics, which provide new results as well new proofs of some classical results in this area.
+
Stirring and Mixing
  
=== October 14: Ling Long (LSU) ===
+
Mixing is something one encounters often in everyday life (e.g. stirring cream into coffee). I will talk about two mathematical
Title: Hypergeometric functions over finite fields
+
aspects of mixing that arise in the context of fluid dynamics:
  
Abstract: Hypergeometric functions are special functions with lot of
+
1. How efficiently can stirring "mix"?
symmetries.  In this talk, we will introduce hypergeometric functions over finite
 
fields, originally due to Greene, Katz and McCarthy, in a way that is
 
parallel to the classical hypergeometric functions, and discuss their
 
properties and applications to character sums and the arithmetic of
 
hypergeometric abelian varieties.
 
This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher, and Fang-Ting Tu.
 
  
=== Tuesday, October 25, 9th floor: Stefan Steinerberger (Yale) ===
+
2. What is the interaction between diffusion and mixing.
Title: Three Miracles in Analysis
 
  
Abstract: I plan to tell three stories: all deal with new points of view on very classical objects and have in common that there is a miracle somewhere. Miracles are nice but difficult to reproduce, so in all three cases the full extent of the underlying theory is not clear and many interesting open problems await. (1) An improvement of the Poincare inequality on the Torus that encodes a lot of classical Number Theory. (2) If the Hardy-Littlewood maximal function is easy to compute, then the function is sin(x). (Here, the miracle is both in the statement and in the proof). (3) Bounding classical integral operators (Hilbert/Laplace/Fourier-transforms) in L^2 -- but this time from below (this problem originally arose in medical imaging). Here, the miracle is also known as 'Slepian's miracle' (this part is joint work with Rima Alaifari, Lillian Pierce and Roy Lederman).
+
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.
  
=== October 28: Linda Reichl (UT Austin) ===
+
=== Oct 5: Eyal Subag (Penn State)===
Title: Microscopic hydrodynamic modes in a binary mixture
 
  
Abstract: Expressions for propagation speeds and decay rates of hydrodynamic modes in a binary mixture can be obtained directly from spectral properties of the Boltzmann equations describing the mixture. The derivation of hydrodynamic behavior from the spectral properties of the kinetic equation provides an alternative to Chapman-Enskog theory, and removes the need for lengthy calculations of transport coefficients in the mixture. It also provides a sensitive test of the completeness of kinetic equations describing the mixture. We apply the method to a hard-sphere binary mixture and show that it gives excellent agreement with light scattering experiments on noble gas mixtures.
+
Symmetries of the hydrogen atom and algebraic families
  
===Monday, October 31: Kathryn Mann (Berkeley) ===
+
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: Groups acting on the circle
 
  
Abstract: Given a group G and a manifold M, can one describe all the actions of G on M?  This is a basic and natural question from geometric topology, but also a very difficult one -- even in the case where M is the circle, and G is a familiar, finitely generated group. 
+
=== Oct 12: Andrei Caldararu (Madison)===
  
In this talk, I’ll introduce you to the theory of groups acting on the circle, building on the perspectives of Ghys, Calegari, Goldman and others.  We'll see some tools, old and new, some open problems, and some connections between this theory and themes in topology (like foliated bundles) and dynamics. 
+
Mirror symmetry and derived categories
  
===November 7: Gaven Martin (New Zealand Institute for Advanced Study) ===
+
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: Siegel's problem on small volume lattices
 
  
Abstract: We outline in very general terms the history and the proof of the identification
+
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).
of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3
 
Coxeter group extended by the involution preserving the symmetry of this
 
diagram. This gives us the smallest regular tessellation of hyperbolic 3-space.
 
This solves (in three dimensions) a problem posed by Siegel in 1945.  Siegel solved this problem in two dimensions by deriving the
 
signature formula identifying the (2,3,7)-triangle group as having minimal
 
co-area.
 
 
There are strong connections with arithmetic hyperbolic geometry in
 
the proof, and the result has applications in the maximal symmetry groups
 
of hyperbolic 3-manifolds in much the same way that Hurwitz's 84g-84 theorem
 
and Siegel's result do.
 
  
===Wednesday, November 16 (9th floor): Kathryn Lindsey (U Chicago) ===
+
=== Oct 19:   Jeremy Teitelbaum (U Connecticut)===
Title: Shapes of Julia Sets
+
Lessons Learned and New Perspectives:
 +
From Dean and Provost to aspiring Data Scientist
  
Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is boundedWilliam Thurston asked "What are the possible shapes of polynomial Julia sets?"  For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name?  It turns out the answer to all of these is "yes!"  I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes.
+
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 genomicsIn 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.
  
===November 18: Andrew Snowden (University of Michigan)===
+
=== Oct 26: Douglas Ulmer (Arizona)===
Title: Recent progress in representation stability
 
  
Abstract:  Representation stability is a relatively new field that studies
+
Rational numbers, rational functions, and rational points
somewhat exotic algebraic structures and exploits their properties to
 
prove results (often asymptotic in nature) about objects of interest.
 
I will describe some of the algebraic structures that appear (and
 
state some important results about them), give a sampling of some
 
notable applications (in group theory, topology, and algebraic
 
geometry), and mention some open problems in the area.
 
  
===Monday, November 21:  Mariya Soskova (University of Wisconsin-Madison)===
+
One of the central concerns of arithmetic geometry is the study of
Title: Definability in degree structures
+
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.
  
Abstract: Some incomputable sets are more incomputable than others. We use
+
=== Nov 2: Ruixiang Zhang (Madison)===
Turing reducibility and enumeration reducibility to measure the
 
relative complexity of incomputable sets. By identifying sets of the
 
same complexity, we can associate to each reducibility a degree
 
structure: the partial order of the Turing degrees and the partial
 
order of the enumeration degrees. The two structures are related in
 
nontrivial ways. The first has an isomorphic copy in the second and
 
this isomorphic copy is an automorphism base. In 1969, Rogers asked a
 
series of questions about the two degree structures with a common
 
theme: definability. In this talk I will introduce the main concepts
 
and describe the work that was motivated by these questions.
 
  
===Friday, December 2:  Hao Shen (Columbia)===
+
The Fourier extension operator
Title:  Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?
 
  
Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.
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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.
  
===Monday, December 5: Botong Wang (UW-Madison)===
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=== Nov 7: Luca Spolaor (MIT)===
Title:  Enumeration of points, lines, planes, etc.
 
  
Abstract: It is a theorem of de Bruijn and Erdos that n points in the plane determine at least n lines, unless all the points lie on a line. This is one of the earliest results in enumerative combinatorial geometry. We will present a higher dimensional generalization of this theorem, which confirms a “top-heavy” conjecture of Dowling and Wilson in 1975. I will give a sketch of the key ideas of the proof, which are the hard Lefschetz theorem and the decomposition theorem in algebraic geometry. I will also talk about a log-concave conjecture on the number of independent sets. These are joint works with June Huh.
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(Log)-Epiperimetric Inequality and the Regularity of Variational Problems
  
=== Friday, December 9: Aaron Brown (U Chicago) ===
+
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.
''Lattice actions and recent progress in the Zimmer program''
+
Finally I will describe some future applications of this method both in regularity theory and in other settings.
  
Abstract: The Zimmer Program is a collection of conjectures and questions regarding actions of lattices in higher-rank simple Lie groups on compact manifolds.  For instance, it is conjectured that all non-trivial volume-preserving actions are built from algebraic examples using standard constructions.  In particular—on manifolds whose dimension is below the dimension of all algebraic examples—Zimmer’s conjecture asserts that every action is finite. 
+
=== Nov 9: Anne Shiu (Texas A&M)===
  
I will present some background, motivation, and selected previous results in the Zimmer program.  I will then explain two of my results within the Zimmer program:
+
Dynamics of biochemical reaction systems
(1) a solution to Zimmer’s conjecture for actions of cocompact lattices in SL(n,R) (joint with D. Fisher and S. Hurtado);
 
(2) a classification (up to topological semiconjugacy) of all actions on tori whose induced action on homology satisfies certain criteria (joint with F. Rodriguez Hertz and Z. Wang).
 
  
=== Monday, December 19: Andrew Zimmer (U Chicago) ===
+
Reaction networks taken with mass-action kinetics arise in many settings,  
''Metric spaces of non-positive curvature and applications in several complex variables''
+
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.
  
Abstract: In this talk I will discuss how to use ideas from the theory of metric spaces of non-positive curvature to understand the behavior of holomorphic maps between bounded domains in complex Euclidean space. Every bounded domain has an metric, called the Kobayashi metric, which is distance non-increasing with respect to holomorphic maps. Moreover, this metric often satisfies well-known non-positive curvature type conditions (for instance, Gromov hyperbolicity or visibility) and one can then use these conditions to understand the behavior of holomorphic maps. Some of what I will talk about is joint work with Gautam Bharali.
+
=== Nov 19: Alexander Yom Din (Caltech)===
 +
 
 +
From analysis to algebra to geometry - an example in representation theory of real groups
  
=== Monday, January 9: Miklos Racz (Microsoft) ===
+
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).
''Statistical inference in networks and genomics''
 
  
Abstract: From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas.
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=== Nov 20: Denis Hirschfeldt (University of Chicago)===
  
I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are root-finding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying high-dimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a high-density, durable, and easy-to-manipulate storage medium of digital data.
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Computability and Ramsey Theory
  
=== Friday, January 13: Mihaela Ifrim (Berkeley) ===
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Computability theory can be seen as the study of the fine
''Two dimensional water waves''
+
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.
  
The classical water-wave problem consists of solving the Euler equations in the presence of a free fluid surface (e.g the water-air interface). This talk will provide an overview of recent developments concerning the motion of a two dimensional incompressible fluid with a free surface. There is a wide range of problems that fall under the heading of water waves, depending on a number of assumptions that can be applied: surface tension, gravity, finite bottom, infinite bottom, rough bottom, etc., and combinations thereof. We will present the physical motivation for studying such problems, followed by the discussion of several interesting mathematical questions related to them. The first step in the analysis is the choice of coordinates, where multiple choices are available. Once the equations are derived we will discuss the main issues arising when analysing local well-posedness, as well as the long time behaviour of solutions with small, or small and localized data. In the last part of the talk we will introduce a new, very robust method which allows one to obtain enhanced lifespan bounds for the solutions. If time permits we will also introduce an alternative method to the scattering theory, which in some cases yields a straightforward route to proving global existence results and obtaining an asymptotic description of solutions. This is joint work with Daniel Tataru, and in part with John Hunter.
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=== Nov 26: Vadim Gorin (MIT)===
  
=== Tuesday, January 17:  Fabio Pusateri (Princeton) ===
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Macroscopic fluctuations through Schur generating functions
''The Water Waves problem''
 
  
We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas.
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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.
  
=== Friday, January 20: Sam Raskin (MIT) === 
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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.
''Tempered local geometric Langlands ''
 
  
The (arithmetic) Langlands program is a cornerstone of modern representation theory and number theory. It has two incarnations: local and global. The former conjectures the existence of certain "local terms," and the latter predicts remarkable interactions between these local terms. By necessity, the global story is predicated on the local.
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=== Nov 28: Gao Chen (IAS) ===
  
Geometric Langlands attempts to find similar patterns in the geometry of curves. However, the scope of the subject has been limited by a meager local theory, which has not been adequately explored.
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A Torelli type theorem
  
The subject of this talk is a part of a larger investigation into local geometric Langlands. We will give an elementary overview of the expectations of this theory, discuss a certain concrete conjecture in the area (on "temperedness"), and provide evidence for this conjecture.
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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.  
  
=== Monday, January 23: Tamas Darvas (Maryland) ===
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=== Nov 30: David Fisher (Indiana U.) ===
''Geometry on the space of Kahler metrics and applications to canonical metrics''
 
  
A basic problem in Kahler geometry, going back to Calabi in the 50's, is to find Kahler
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New Techniques for Zimmer's Conjecture
metrics with the best curvature properties, e.g., Einstein metrics. Such special metrics are
 
minimizers of well known functionals on the space of all Kahler metrics H. However these
 
functionals become convex only if an adequate geometry is chosen on H. One such choice of
 
Riemannian geometry was proposed by Mabuchi in the 80's, and was used to address a number of
 
uniqueness questions in the theory. In this talk I will present more general Finsler geometries on
 
H, that still enjoy many of the  properties that Mabuchi's geometry has, and I will give
 
applications related to existence of special Kahler metrics, including the recent resolution of
 
Tian's related properness conjectures. 
 
  
 +
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.
  
=== Friday, February 3: Melanie Matchett Wood (UW-Madison) ===
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=== Dec 3: Bena Tshishiku (Harvard) ===
''Random groups from generators and relations''
 
  
We consider a model of random groups that starts with a free group on n generators and takes the quotient by n random relations.  We discuss this model in the case of abelian groups (starting with a free abelian group), and its relationship to the Cohen-Lenstra heuristics, which predict the distribution of class groups of number fields.  We will explain a universality theorem, an analog of the central limit theorem for random groups, that says the resulting distribution of random groups is largely insensitive to the distribution from which the relations are chosen.  Finally, we discuss joint work with Yuan Liu on the non-abelian random groups built in this way, including the existence of a limit of the random groups as n goes to infinity. 
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Surface bundles, monodromy, and arithmetic groups
  
=== Monday, February 6: Benoit Perthame (University of Paris VI) ===
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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.
''Models for neural networks; analysis, simulations and behaviour''
 
  
Neurons exchange informations via discharges, propagated
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=== Dec 5: Subhabrata Sen (MIT and Microsoft Research New England) ===
by membrane potential,  which trigger firing of the many connected
 
neurons. How to describe large networks of such neurons? What are the properties of these mean-field equations?
 
How can such a network generate a spontaneous activity?
 
Such questions can be tackled using nonlinear integro-differential
 
equations. These are now classically used in the neuroscience community to describe
 
neuronal networks or neural assemblies. Among them, the best known is certainly
 
Wilson-Cowan's equation which
 
describe spiking rates arising in different brain locations.
 
  
Another classical model is the integrate-and-fire equation that describes
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Random graphs, Optimization, and Spin glasses
neurons through their voltage using a particular type of Fokker-Planck equations. Several mathematical results will be presented concerning existence, blow-up, convergence to steady state,
 
for the excitatory and inhibitory neurons, with or without refractory states. Conditions for the transition to spontaneous activity (periodic solutions) will be discussed.
 
  
One can also describe directly the spike time
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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 analysisGraph 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.
distribution which seems to encode more directly the neuronal information.
 
This leads to a structured population equation that describes
 
at time $t$ the probability to find a neuron with time $s$
 
elapsed since its last dischargeHere, we can 
 
show that small or large connectivity
 
leads to desynchronization. For intermediate regimes, sustained
 
periodic activity occurs.
 
A common mathematical tool is the use of the relative entropy method.
 
  
This talk is based on works with K. Pakdaman and D. Salort, M. Caceres, J. A. Carrillo and D. Smets.
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=== 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.
  
=== February 17:  Gustavo Ponce, UCSB===
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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.
  
''The Korteweg-de Vries equation vs. the Benjamin-Ono equation''
+
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.
  
In this talk we shall study the <math>k</math>-generalized Korteweg-de Vries <math>(k-KdV)</math> equation
+
Joint work with Y. Fan, J. Feliu-Faaba, L. Lin,  W. Jia, and L. Ying
  
<math>\partial_t u + \partial_x^3u +u^k\,\partial_xu=0,\;\;\;\;\;\;\;x,t\in\Bbb R,\, k\in \Bbb Z^+, </math>
+
=== Dec 10: Max Engelstein (MIT) ===
  
and the <math>k</math>-generalized Benjamin-Ono (<math>k</math>-BO) equation
+
The role of Energy in Regularity
  
<math>\partial_t u-\partial_x^2\mathcal {H} u+u^k\,\partial_x u=0, \;\;\;\;\;\;\;x,t\in\Bbb R,\, k\in \Bbb Z^+,</math>
+
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.
  
where <math>\mathcal {H}</math> denotes the Hilbert transform,
+
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. 
  
<math>\mathcal {H} f(x)=\frac{1}{\pi}\, {p.v.}\big(\frac{1}{x}\ast f\big)(x)=(-i\,sgn(\xi) \widehat{f}(\xi))^{\vee}(x).</math>
+
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.  
  
The goal is to review and analyze results concerning solutions of the initial value properties associated to these equations.
+
== Past Colloquia ==
 
These include a comparison of the  local and global well-posedness and unique continuation properties
 
as well as special features of the special solutions of these models.
 
  
=== February 24: Jonathan Hauenstein (Notre Dame) ===
+
[[Colloquia/Blank|Blank]]
''Real solutions of polynomial equations''
 
  
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. 
+
[[Colloquia/Spring2018|Spring 2018]]
  
 +
[[Colloquia/Fall2017|Fall 2017]]
  
== Past Colloquia ==
+
[[Colloquia/Spring2017|Spring 2017]]
  
 
[[Archived Fall 2016 Colloquia|Fall 2016]]
 
[[Archived Fall 2016 Colloquia|Fall 2016]]

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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