Difference between revisions of "Colloquia"

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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'''.
  
== Spring 2018 ==
+
The calendar for spring 2019 can be found [[Colloquia/Spring2019|here]].
 +
 
 +
== Fall 2018 ==
 +
 
  
 
{| cellpadding="8"
 
{| cellpadding="8"
Line 11: Line 14:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|January 29 (Monday)
+
|Sep 12    '''Room 911'''
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
+
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
|[[#January 29 Li Chao (Columbia)|  Elliptic curves and Goldfeld's conjecture ]]
+
|[[#Sep 12: Gunther Uhlmann (Univ. of Washington)|  Harry Potter's Cloak via Transformation Optics ]]
| Jordan Ellenberg
+
| Li
 
|
 
|
 
|-
 
|-
|February 2 (Room: 911)
+
|Sep 14    '''Room 911'''
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)
+
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
|[[#February 2 Thomas Fai (Harvard)|  The Lubricated Immersed Boundary Method ]]
+
|[[#Sep 14: Gunther Uhlmann (Univ. of Washington) |  Journey to the Center of the Earth  ]]
| Spagnolie, Smith
+
| Li
 
|
 
|
 
|-
 
|-
|February 5 (Monday, Room: 911)
+
|Sep 21    '''Room 911'''
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
+
| [http://stuart.caltech.edu/ Andrew Stuart] (Caltech) LAA lecture
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]
+
|[[#Sep 21: Andrew Stuart (Caltech) | The Legacy of Rudolph Kalman  ]]
| Ellenberg, Gurevitch
+
| Jin
 
|
 
|
 
|-
 
|-
|February 6 (Tuesday 2 pm, Room 911)
+
|Sep 28
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
+
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]
+
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
| Ellenberg, Gurevitch
+
| Thiffeault
 
|
 
|
 
|-
 
|-
|February 9
+
|Oct 5
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)
+
| [http://www.personal.psu.edu/eus25/ Eyal Subag] (Penn State)
|[[#February 9 Wes Pegden (CMU)|  The fractal nature of the Abelian Sandpile ]]
+
|[[#Oct 5: Eyal Subag (Penn State)|  Symmetries of the hydrogen atom and algebraic families  ]]
| Roch
+
| Gurevich
 
|
 
|
 
|-
 
|-
|March 2
+
|Oct 12
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)
+
| [https://www.math.wisc.edu/~andreic/ Andrei Caldararu] (Madison)
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]
+
|[[#Oct 12: Andrei Caldararu (Madison) | Mirror symmetry and derived categories  ]]
| Caldararu
+
| ...
 
|
 
|
 
|-
 
|-
| March 16
+
|Oct 19
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)
+
| [https://teitelbaum.math.uconn.edu/# Jeremy Teitelbaum] (U Connecticut)
|[[# TBATBA ]]
+
|[[#Oct 19:  Jeremy Teitelbaum (U Connecticut)Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist ]]
| WIMAW
+
| Boston
 
|
 
|
 
|-
 
|-
|April 4 (Wednesday)
+
|Oct 26
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)
+
| [http://math.arizona.edu/~ulmer/index.html Douglas Ulmer] (Arizona)
|[[# TBA| TBA  ]]
+
|[[#Oct 26: Douglas Ulmer (Arizona) | Rational numbers, rational functions, and rational points ]]
| Craciun
+
| Yang
 
|
 
|
 
|-
 
|-
| April 6
+
|Nov 2  '''Room 911'''
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)
+
| [https://sites.google.com/view/ruixiang-zhang/home?authuser=0# Ruixiang Zhang] (Madison)
|[[# Edray GoinsTBA ]]
+
|[[#Nov 2: Ruixiang Zhang (Madison) The Fourier extension operator ]]
| Melanie
+
|  
 
|
 
|
 
|-
 
|-
| April 13
+
|Nov 7  '''Wednesday'''
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)
+
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)
|[[# TBATBA ]]
+
|[[#Nov 7: Luca Spolaor (MIT) (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]
| WIMAW
+
| Feldman
 
|
 
|
 
|-
 
|-
| April 20
+
|Nov 12  '''Monday'''
| Xiuxiong Chen(Stony Brook University)
+
| [http://www.math.tamu.edu/~annejls/ Anne Shiu] (Texas A&M)
|[[# Xiuxiong ChenTBA ]]
+
|[[#Nov 9: Anne Shiu (Texas A&M) Dynamics of biochemical reaction systems ]]
| Bing Wang
+
| Craciun, Stechmann
 
|
 
|
 
|-
 
|-
| April 25 (Wednesday)
+
|Nov 19 '''Monday'''
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture
+
| [https://sites.google.com/site/ayomdin/ Alexander Yom Din] (Caltech)  
|[[# TBA| TBA ]]
+
|[[#Nov 19: Alexander Yom Din (Caltech) | From analysis to algebra to geometry - an example in representation theory of real groups ]]
| Tran
+
| Boston, Gurevitch
 
|
 
|
 
|-
 
|-
|date
+
|Nov 20 '''Tuesday, Room 911'''
| person (institution)
+
| [http://http://www.math.uchicago.edu/~drh/ Denis Hirschfeldt] (University of Chicago)
|[[# TBATBA ]]
+
|[[#Nov 20: Denis Hirschfeldt (University of Chicago)Computability and Ramsey Theory ]]
| hosting faculty
+
| Andrews
 
|
 
|
 
|-
 
|-
|date
+
|Nov 26 '''Monday, Room 911'''
| person (institution)
+
| [http://math.mit.edu/directory/profile.php?pid=1415 Vadim Gorin] (MIT)
|[[# TBATBA ]]
+
|[[#Nov 26: Vadim Gorin (MIT)Macroscopic fluctuations through Schur generating functions ]]
| hosting faculty
+
| Anderson
 
|
 
|
 
|-
 
|-
|date
+
|Nov 28 '''Wednesday'''
| person (institution)
+
| [http://www.math.ias.edu/~gchen/ Gao Chen](IAS)
|[[# TBA| TBA  ]]
+
|[[#Nov 28: Gao Chen(IAS) | A Torelli type theorem ]]
| hosting faculty
+
| Paul
 
|
 
|
 
|-
 
|-
|date
+
|Nov 30
| person (institution)
+
| [https://math.indiana.edu/about/faculty/fisher-david.html David Fisher](Indiana U.)
|[[# TBA| TBA  ]]
+
|[[#Nov 30: David Fisher (Indiana U.) | New Techniques for Zimmer's Conjecture ]]
| hosting faculty
+
| Kent
|
+
 
|-
 
|-
|date
+
|Dec 3 '''Monday'''
| person (institution)
+
| [http://bena-tshishiku.squarespace.com/ Bena Tshishiku](Harvard)
|[[# TBATBA  ]]
+
|[[#Dec 3: Bena Tshishiku (Harvard)Surface bundles, monodromy, and arithmetic groups ]]
| hosting faculty
+
| Paul
 
|
 
|
 
|-
 
|-
|date
+
|Dec 5 '''Wednesday, Room 911'''
| person (institution)
+
| [http://www.mit.edu/~ssen90/ Subhabrata Sen](MIT and Microsoft Research New England)
|[[# TBA| TBA  ]]
+
|[[#Dec 5: Subhabrata Sen (MIT and Microsoft Research New England) | Random graphs, Optimization, and Spin glasses ]]
| hosting faculty
+
| Anderson
 
|
 
|
 
|-
 
|-
|date
+
|Dec 7 '''Room 911'''
| person (institution)
+
| [https://math.berkeley.edu/people/faculty/leonardo-zepeda-n-ez Leonardo Zepeda-Nunez](Berkeley)
|[[# TBA| TBA  ]]
+
|[[#Dec 7: Leonardo Zepeda-Nunez (Berkeley) | Accelerating ab-initio molecular dynamics via multi-scale neural networks ]]
| hosting faculty
+
| Stechmann
 
|
 
|
 
|-
 
|-
|date
+
|Dec 10 '''Monday'''
| person (institution)
+
| [http://math.mit.edu/~maxe/ Max Engelstein](MIT)
|[[# TBA|  TBA  ]]
+
|[[#Dec 10: Max Engelstein (MIT)|  The role of Energy in Regularity ]]
| hosting faculty
+
| Feldman
|
+
|-
+
|date
+
| person (institution)
+
|[[# TBATBA ]]
+
| hosting faculty
+
 
|
 
|
 
|}
 
|}
  
== Spring Abstracts ==
+
== Abstracts ==
  
 +
=== Sep 12: Gunther Uhlmann (Univ. of Washington) ===
 +
Harry Potter's Cloak via Transformation Optics
  
===January 29 Li Chao (Columbia)===
+
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.
  
Title: Elliptic curves and Goldfeld's conjecture
+
=== Sep 14: Gunther Uhlmann (Univ. of Washington) ===
 +
Journey to the Center of the Earth
  
Abstract:
+
We will consider the inverse problem of determining the sound
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.
+
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.
  
=== February 2 Thomas Fai (Harvard) ===
+
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.
  
Title: The Lubricated Immersed Boundary Method
+
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.
  
Abstract:
+
=== Sep 21: Andrew Stuart (Caltech) ===
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.
+
  
===February 5 Alex Lubotzky (Hebrew University)===
+
The Legacy of Rudolph Kalman
  
Title:  High dimensional expanders: From Ramanujan graphs to Ramanujan complexes
+
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.
  
Abstract:  
+
=== Sep 28: Gautam Iyer (CMU) ===
  
Expander graphs in general, and Ramanujan graphs , in particular,  have played a major role in  computer science in the last 5 decades  and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders.
+
Stirring and Mixing
  
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1.
+
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:
  
This question was answered recently affirmatively (by  T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders.  
+
1. How efficiently can stirring "mix"?
  
 +
2. What is the interaction between diffusion and mixing.
  
===February 6 Alex Lubotzky (Hebrew University)===
+
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.
  
Title: Groups' approximation, stability and high dimensional expanders
+
=== Oct 5: Eyal Subag (Penn State)===
  
Abstract:
+
Symmetries of the hydrogen atom and algebraic families
  
Several well-known open questions, such as: are all groups sofic or hyperlinear?,  have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the  unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms.  We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.
+
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.
  
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability  and using  high dimensional expanders, it is shown that  some non-residually finite groups  (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated.
+
=== Oct 12: Andrei Caldararu (Madison)===
  
All notions will be explained.      Joint work with M, De Chiffre, L. Glebsky and A. Thom.
+
Mirror symmetry and derived categories
  
===February 9 Wes Pegden (CMU)===
+
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 fractal nature of the Abelian Sandpile
+
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).
  
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor.
+
===  Oct 19:   Jeremy Teitelbaum (U Connecticut)===
 +
Lessons Learned and New Perspectives:
 +
From Dean and Provost to aspiring Data Scientist
  
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation)We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings.  In this talk, we will survey our work in this area, and discuss avenues of current and future research.
+
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 positionI 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.
  
===March 2 Aaron Bertram (Utah)===
+
=== Oct 26: Douglas Ulmer (Arizona)===
  
Title: Stability in Algebraic Geometry
+
Rational numbers, rational functions, and rational points
  
Abstract: Stability was originally introduced in algebraic geometry in the context of finding a projective quotient space for the action of an algebraic group on a projective manifold. This, in turn, led in the 1960s to a notion of slope-stability for vector bundles on a Riemann surface, which was an important tool in the classification of vector bundles. In the 1990s, mirror symmetry considerations led Michael Douglas to notions of stability for "D-branes" (on a higher-dimensional manifold) that corresponded to no previously known mathematical definition. We now understand each of these notions of stability as a distinct point of a complex "stability manifold" that is an important invariant of the (derived) category of complexes of vector bundles of a projective manifold. In this talk I want to give some examples to illustrate the various stabilities, and also to describe some current work in the area.
+
One of the central concerns of arithmetic geometry is the study of
 +
solutions of systems of polynomial equations where the solutions are
 +
required to lie in a "small" field such as the rational numbers.  I
 +
will explain the landscape of expectations and conjectures in this
 +
area, focusing on curves and their Jacobians over global fields
 +
(number fields and function fields), and then survey the progress made
 +
over the last decade in the function field case.  The talk is intended
 +
to be accessible to a wide audience.
 +
 
 +
=== Nov 2: Ruixiang Zhang (Madison)===
 +
 
 +
The Fourier extension operator
 +
 
 +
I will present an integral operator that originated in the study of the Euclidean Fourier transform and is closely related to many problems in PDE, spectral theory, analytic number theory, and combinatorics. I will then introduce some recent developments in harmonic analysis concerning this operator. I will mainly focus on various new ways to "induct on scales" that played an important role in the recent solution in all dimensions to Carleson's a.e. convergence problem on free Schrödinger solutions.
 +
 
 +
=== Nov 7: Luca Spolaor (MIT)===
 +
 
 +
(Log)-Epiperimetric Inequality and the Regularity of Variational Problems
 +
 
 +
In this talk I will present a new method for studying the regularity of minimizers to variational problems. I will start by introducing the notion of blow-up, using as a model case the so-called Obstacle problem. Then I will state the (Log)-epiperimetric inequality and explain how it is used to prove uniqueness of the blow-up and regularity results for the solution near its singular set. I will then show the flexibility of this method by describing how it can be applied to other free-boundary problems and to (almost)-area minimizing currents.
 +
Finally I will describe some future applications of this method both in regularity theory and in other settings.
 +
 
 +
=== Nov 9: Anne Shiu (Texas A&M)===
 +
 
 +
Dynamics of biochemical reaction systems
 +
 
 +
Reaction networks taken with mass-action kinetics arise in many settings,
 +
from epidemiology to population biology to systems of chemical reactions.
 +
This talk focuses on certain biological signaling networks, namely,
 +
phosphorylation networks, and their resulting dynamical systems. For many
 +
of these systems, the set of steady states admits a rational
 +
parametrization (that is, the set is the image of a map with
 +
rational-function coordinates). We describe how such a parametrization
 +
allows us to investigate the dynamics, including the emergence of
 +
bistability in a network underlying ERK regulation, and the capacity for
 +
oscillations in a mixed processive/distributive phosphorylation network.
 +
 
 +
=== Nov 19: Alexander Yom Din (Caltech)===
 +
 
 +
From analysis to algebra to geometry - an example in representation theory of real groups
 +
 
 +
Representation theory of non-compact real groups, such as SL(2,R), is a fundamental discipline with uses in harmonic analysis, number theory, physics, and more. This theory is analytical in nature, but in the course of the 20th century it was algebraized and geometrized (the key contributions are by Harish-Chandra for the former and by Beilinson-Bernstein for the latter). Roughly and generally speaking, algebraization strips layers from the objects of study until we are left with a bare skeleton, amenable to symbolic manipulation. Geometrization, again very roughly, reveals how algebraic objects have secret lives over spaces - thus more amenable to human intuition. In this talk, I will try to motivate and present one example - the calculation of the Casselman-Jacquet module of a principal series representation (I will explain the terms in the talk).
 +
 
 +
=== Nov 20: Denis Hirschfeldt (University of Chicago)===
 +
 
 +
Computability and Ramsey Theory
 +
 
 +
Computability theory can be seen as the study of the fine
 +
structure of definability. Much of its power relies on the deep
 +
connections between definability and computation. These connections can be seen in fundamental results such as Post's Theorem, which establishes a connection between the complexity of formulas needed to define a given set of natural numbers and its computability-theoretic strength. As has become increasingly clear, they can also be seen in the computability-theoretic analysis of objects whose definitions come from notions that arise naturally in combinatorics. The heuristic here is that
 +
computability-theoretically natural notions tend to be combinatorially
 +
natural, and vice-versa. I will discuss some results and open questions in
 +
the computability-theoretic analysis of combinatorial principles, in
 +
particular Ramsey-theoretic ones such as versions of Ramsey's Theorem for colorings of countably infinite sets, and versions of Hindman's Theorem, which states that for every coloring of the natural numbers with finitely many colors, there is an infinite set of numbers such that all nonempty sums of distinct elements of this set have the same color.
 +
 
 +
=== Nov 26: Vadim Gorin (MIT)===
 +
 
 +
Macroscopic fluctuations through Schur generating functions
 +
 
 +
I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.
 +
 
 +
It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.
 +
 
 +
=== Nov 28: Gao Chen (IAS) ===
 +
 
 +
A Torelli type theorem
 +
 
 +
The length of a circle determines the shape of it. In this talk, we will discuss non-trivial generalizations of this fact for Riemann surfaces, hyperkähler 4-manifolds, Calabi-Yau threefolds and G_2, Spin(7) manifolds.
 +
 
 +
=== Nov 30: David Fisher (Indiana U.) ===
 +
 
 +
New Techniques for Zimmer's Conjecture
 +
 
 +
Lattices in higher rank simple Lie groups are known to be
 +
extremely rigid. Examples of this are Margulis' superrigidity theorem,
 +
which shows they have very few linear represenations, and Margulis'
 +
arithmeticity theorem, which shows they are all constructed via number
 +
theory. Motivated by these and other results, in 1983 Zimmer made a
 +
number of conjectures about actions of these groups on compact
 +
manifolds. After providing some history and motivation, I will discuss
 +
a recent result that makes dramatic progress on the conjecture in all
 +
cases and proves it in many of them. I will place some emphasis on
 +
surprising connections to other areas of mathematics that arise in the
 +
proof.
 +
 
 +
=== Dec 3: Bena Tshishiku (Harvard) ===
 +
 
 +
Surface bundles, monodromy, and arithmetic groups
 +
 
 +
Fiber bundles with fiber a surface arise in many areas including hyperbolic geometry, symplectic geometry, and algebraic geometry. Up to isomorphism, a surface bundle is completely determined by its monodromy representation, which is a homomorphism to a mapping class group. This allows one to use algebra to study the topology of surface bundles. Unfortunately, the monodromy representation is typically difficult to ``compute" (e.g. determine its image). In this talk, I will discuss some recent work toward computing monodromy groups for holomorphic surface bundles, including certain examples of Atiyah and Kodaira. This can be applied to the problem of counting the number of ways that certain 4-manifolds fiber over a surface. This is joint work with Nick Salter.
 +
 
 +
=== Dec 5: Subhabrata Sen (MIT and Microsoft Research New England) ===
 +
 
 +
Random graphs, Optimization, and Spin glasses
 +
 
 +
Combinatorial optimization problems are ubiquitous in diverse mathematical applications. The desire to understand their ``typical" behavior motivates a study of these problems on random instances. In spite of a long and rich history, many natural questions in this domain are still intractable to rigorous mathematical analysis.  Graph cut problems such as Max-Cut and Min-bisection are canonical examples in this class. On the other hand, physicists study these questions using the non-rigorous ``replica" and ``cavity" methods, and predict complex, intriguing features. In this talk, I will describe some recent progress in our understanding of their typical properties on random graphs, obtained via connections to the theory of mean-field spin glasses. The new techniques are broadly applicable, and lead to novel algorithmic and statistical consequences.
 +
 
 +
=== Dec 7: Leonardo Zepeda-Nunez (Berkeley) ===
 +
 
 +
Accelerating ab-initio molecular dynamics via multi-scale neural networks
 +
 
 +
Deep learning has rapidly become a large field with an ever-growing range of applications; however, its intersection with scientific computing remains in its infancy, mainly due to the high accuracy that scientific computing problems require, which depends greatly on the architecture of the neural network.
 +
 
 +
In this talk we present a novel deep neural network with a multi-scale architecture inspired in H-matrices (and H2-matrices) to efficiently approximate, within 3-4 digits, several challenging non-linear maps arising from the discretization of PDEs, whose evaluation would otherwise require computationally intensive iterative methods.
 +
 
 +
In particular, we focus on the notoriously difficult Kohn-Sham map arising from Density Functional Theory (DFT). We show that the proposed multiscale-neural network can efficiently learn this map, thus bypassing an expensive self-consistent field iteration. In addition, we show the application of this methodology to ab-initio molecular dynamics, for which we provide examples for 1D problems and small, albeit realistic, 3D systems.
 +
 
 +
Joint work with Y. Fan, J. Feliu-Faaba, L. Lin,  W. Jia, and L. Ying
 +
 
 +
=== Dec 10: Max Engelstein (MIT) ===
 +
 
 +
The role of Energy in Regularity
 +
 
 +
The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes.  A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.
 +
 
 +
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field. 
 +
 
 +
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.  
  
 
== Past Colloquia ==
 
== Past Colloquia ==
  
[[Colloquia/Blank|Blank Colloquia]]
+
[[Colloquia/Blank|Blank]]
 +
 
 +
[[Colloquia/Spring2018|Spring 2018]]
  
 
[[Colloquia/Fall2017|Fall 2017]]
 
[[Colloquia/Fall2017|Fall 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.

Past Colloquia

Blank

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012