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= Mathematics Colloquium =
 
= Mathematics Colloquium =
  
 
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.
 
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.
  
== Spring 2015  ==
+
The calendar for spring 2019 can be found [[Colloquia/Spring2019|here]].
  
Go to next semester, [[Colloquia/Fall2015|Fall 2015]].
+
==Spring 2019==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
!align="left" | date
+
!align="left" | date  
 
!align="left" | speaker
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | title
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
| '''January 12''' (special time: '''3PM''')
+
|Jan 25
| [http://math.nd.edu/people/visiting-faculty/botong-wang/ Botong Wang] (Notre Dame)
+
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW
| [[Colloquia#January 12:  Botong Wang (Notre Dame) | Cohomology jump loci of algebraic varieties]]
+
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications  ]]
| Maxim
+
| Tullia Dymarz
 +
|
 
|-
 
|-
| '''January 14''' (special time: '''11AM''')
+
|Jan 30 '''Wednesday'''
| [http://www.math.uiuc.edu/~jathreya/ Jayadev Athreya] (UIUC)
+
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
| [[Colloquia#January 14:  Jayadev Athreya (UIUC) | Counting points for random (and not-so-random) geometric structures]]
+
|[[#Lillian Pierce (Duke University) | Short character sums  ]]
| Ellenberg
+
| Boston and Street
 +
|
 
|-
 
|-
| '''January 15''' (special time: '''3PM''')
+
|Jan 31 '''Thursday'''
| [http://www.math.sunysb.edu/~chili/ Chi Li] (Stony Brook)
+
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)
| [[Colloquia#January 15:  Chi Li (Stony Brook) | On Kahler-Einstein metrics and K-stability]]
+
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations  ]]
| Sean Paul
+
| Street
 +
|
 
|-
 
|-
| '''January 21'''
+
|Feb 1
| [http://www.math.utoronto.ca/cms/kitagawa-jun/ Jun Kitagawa] (Toronto)
+
| [https://services.math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke University)
| [[Colloquia#January 21: Jun Kitagawa (Toronto) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics]]
+
|[[# TBA|  TBA ]]
| Feldman
+
| Qin
 +
|
 
|-
 
|-
| '''January 23''' (special room/time: '''B135, 2:30PM''')
+
|Feb 5 '''Tuesday'''
| [http://math.duke.edu/~adding/ Nicolas Addington] (Duke)
+
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)
| [[Colloquia#January 23: Nicolas Addington (Duke) | Recent developments in rationality of cubic 4-folds]]
+
|[[# TBA|  TBA ]]
| Ellenberg
+
| Denisov
 +
|
 
|-
 
|-
| '''Monday January 26 4pm'''
+
|Feb 8
| [http://www.bcamath.org/en/people/minh-binh Minh Binh Tran] (CAM)
+
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)
| [[Colloquia#January 26:  Minh Binh Tran (CAM) | Nonlinear approximation theory for the homogeneous Boltzmann
+
|[[#Aaron Naber (Northwestern) |   A structure theory for spaces with lower Ricci curvature bounds  ]]
equation]]
+
| Street
| Jin
+
|
 
|-
 
|-
| January 30
+
|Feb 15
| Tentatively reserved for possible interview
+
|
 +
|[[# TBA|  TBA  ]]
 +
|  
 
|
 
|
 +
|-
 +
|Feb 22
 +
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)
 +
|[[# TBA|  TBA  ]]
 +
| Erman and Corey
 
|
 
|
 
|-
 
|-
| '''Monday, February 2 4pm'''
+
|March 4
| [https://web.math.princeton.edu/~ajsb/ Afonso Bandeira] (Princeton)
+
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture
| [[Colloquia#February 2:  Afonso Bandeira (Princeton) | Tightness of convex relaxations for certain inverse problems on graphs]]
+
|[[# TBA| TBA ]]
| Ellenberg
+
| Kim
 +
|
 
|-
 
|-
| February 6
+
|March 8
| Morris Hirsch (UC Berkeley and UW Madison)
+
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)
| [[Colloquia#February 6:  Morris Hirsch (UC Berkeley and UW Madison) | Fixed points of Lie transformation group, and zeros of Lie algebras of vector fields]]
+
|[[# TBATBA ]]
| Stovall
+
| Erman
 +
|
 
|-
 
|-
| February 13
+
|March 15
| [http://www.math.ucsb.edu/~mputinar/ Mihai Putinar] (UC Santa Barbara, Newcastle University)
+
| Maksym Radziwill (Caltech)
| [[Colloquia#February 13: Mihai Putinar (UC Santa Barbara) | Quillen’s property of real algebraic varieties]]
+
|[[# TBA| TBA  ]]
| Budišić
+
| Marshall
 +
|
 
|-
 
|-
| February 20
+
|March 29
| [http://www.mathcs.emory.edu/~dzb/ David Zureick-Brown] (Emory University)
+
| Jennifer Park (OSU)
| [[Colloquia#February 20: David Zureick-Brown (Emory University) | Diophantine and tropical geometry]]
+
|[[# TBA| TBA  ]]
| Ellenberg
+
| Marshall
 +
|
 
|-
 
|-
| '''Monday, February 23, 4pm'''
+
|April 5
| [http://www.math.uiuc.edu/~jathreya/ Jayadev Athreya] (UIUC)
+
| Ju-Lee Kim (MIT)
| [[Colloquia#'''Monday''' February 23: Jayadev Athreya (UIUC) | The Erdos-Szusz-Turan distribution for equivariant point processes]]
+
|[[# TBA|  TBA ]]
| Mari-Beffa
+
| Gurevich
 +
|
 
|-
 
|-
| February 27
+
|April 12
| [http://www.math.rochester.edu/people/faculty/allan/ Allan Greenleaf] (University of Rochester)
+
| Evitar Procaccia (TAMU)
| [[Colloquia#February 27: Allan Greenleaf (University of Rochester) | Erdos-Falconer Configuration problems]]
+
|[[# TBA| TBA  ]]
| Seeger
+
| Gurevich
 
|
 
|
 
|-
 
|-
| March 6
+
|April 19
| [http://math.mit.edu/~lguth/ Larry Guth] (MIT)
+
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)
| [[Colloquia#March 6: Larry Guth (MIT) | Introduction to incidence geometry]]
+
|[[# TBA|  TBA ]]
| Stovall
+
| Jean-Luc
 +
|
 
|-
 
|-
| March 13
+
|April 26
|[http://www.ma.utexas.edu/text/webpages/gordon.html Cameron Gordon] (UT-Austin)
+
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)
| [[Colloquia#March 13: Cameron Gordon (UT-Austin) | Left-orderability and 3-manifold groups]]
+
|[[# TBA|  TBA ]]
| Maxim
+
| WIMAW
 +
|
 
|-
 
|-
| March 20
+
|May 3
| [http://www.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)
+
| Tomasz Przebinda (Oklahoma)
| [[Colloquia#March 20:  Aaron Naber (Northwestern) | Regularity and New Directions in Einstein Manifolds]]
+
|[[# TBATBA ]]
| Paul
+
| Gurevich
|-
 
| '''March 27 11am B239'''
 
|  [http://www.univie.ac.at/complexanalysis/people/IlyaKossovskiy.php Ilya Kossovskiy] (University of Vienna)
 
| [[Colloquia#March 27 11am B239: Ilya Kossovskiy (University of Vienna) | On Poincare's "Probleme local"]]
 
| Gong
 
|-
 
| March 27
 
|[http://php.indiana.edu/~korr/ Kent Orr] (Indiana University at Bloomigton)
 
| [[Colloquia#March 27: Kent Orr (Indiana University) | The Isomorphism Problem for metabelian groups]]
 
| Maxim
 
|-
 
| April 3
 
| University holiday
 
 
|
 
|
|
 
|-
 
| April 10
 
| [http://www-users.math.umn.edu/~jyfoo/ Jasmine Foo] (University of Minnesota)
 
| [[Colloquia#April 10:  Jasmine Foo (University of Minnesota) | Evolutionary dynamics in spatially structured populations]]
 
| Roch, WIMAW
 
|-
 
| April 17
 
| [http://www.math.uiuc.edu/~kkirkpat/ Kay Kirkpatrick] (University of Illinois-Urbana Champaign)
 
| [[Colloquia#April 17:  Kay Kirkpatrick (UIUC) | Non-normal asymptotics of the mean-field Heisenberg model]]
 
| Stovall
 
|-
 
| April 24
 
| Marianna Csornyei (University of Chicago)
 
| [[Colloquia#April 24:  Marianna Csornyei (University of Chicago) | Tangents of curves and differentiability of functions]]
 
| Seeger, Stovall
 
|-
 
| May 1
 
| [http://www.math.washington.edu/~bviray/ Bianca Viray] (University of Washington)
 
| [[Colloquia#May 1:  Bianca Viray (University of Washington) | Cryptography and abelian varieties with extra endomorphisms]]
 
| Erman
 
|-
 
| May 8
 
| [http://www.math.ucla.edu/~mroper/www/Home.html Marcus Roper] (UCLA)
 
| [[Colloquia#May 8:  Marcus Roper (UCLA) | Nonlinear flow in micro- and myco-fluidics]]
 
| Roch
 
 
|}
 
|}
  
 
== Abstracts ==
 
== Abstracts ==
  
===January 12:  Botong Wang (Notre Dame)===
+
===Beata Randrianantoanina (Miami University Ohio)===
  
====Cohomology jump loci of algebraic varieties====
+
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.
  
In the moduli spaces of vector bundles (or local systems), cohomology jump loci are the algebraic sets where certain cohomology group has prescribed dimension. We will discuss some arithmetic and deformation theoretic aspects of cohomology jump loci. If time permits, we will also talk about some applications in algebraic statistics.
+
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.
  
===January 14:  Jayadev Athreya (UIUC)===
+
===Lillian Pierce (Duke University)===
  
====Counting points for random (and not-so-random) geometric structures====
+
Title: Short character sums
  
We describe a philosophy of how certain counting problems can be studied by methods of probability theory and dynamics on appropriate moduli spaces. We focus on two particular cases:
+
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.
  
(1) Counting for Right-Angled Billiards: understanding the dynamics on and volumes of moduli spaces of meromorphic quadratic differentials yields interesting universality phenomenon for billiards in polygons with interior angles integer multiples of 90 degrees. This is joint work with A. Eskin and A. Zorich
+
===Dean Baskin (Texas A&M)===
  
(2) Counting for almost every quadratic form: understanding the geometry of a random lattice allows yields striking diophantine and counting results for typical (in the sense of measure) quadratic (and other) forms. This is joint work with G. A. Margulis.
+
Title: Radiation fields for wave equations
  
===January 15: Chi Li (Stony Brook)===
+
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.
 
====On Kahler-Einstein metrics and K-stability====
 
  
The existence of Kahler-Einstein metrics on Kahler manifolds is a basic problem in complex differential geometry. This problem has connections to other fields: complex algebraic geometry, partial differential equations and several complex variables. I will discuss the existence of Kahler-Einstein metrics on Fano manifolds and its relation to K-stability. I will mainly focus on the analytic part of the theory, discuss how to solve the related complex Monge-Ampere equations and provide concrete examples in both smooth and conical settings. If time permits, I will also say something about the algebraic part of the theory, including the study of K-stability using the Minimal Model Program (joint with Chenyang Xu) and the existence of proper moduli space of smoothable K-polystable Fano varieties (joint with Xiaowei Wang and Chenyang Xu).
+
===Aaron Naber (Northwestern)===
  
===January 21Jun Kitagawa (Toronto)===
+
TitleA structure theory for spaces with lower Ricci curvature bounds.
  
====Regularity theory for generated Jacobian equations: from optimal transport to geometric optics====
+
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated.  It thus becomes a natural question, how well behaved or badly behaved can such spaces be?  This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like.  In this talk we give an essentially sharp answer to this question.  The talk will require little background, and our time will be spent on understanding the basic statements and examples.  The work discussed is joint with Cheeger, Jiang and with Li.
  
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.
 
  
===January 23:  Nicolas Addington (Duke)===
+
== Past Colloquia ==
 
 
====Recent developments in rationality of cubic 4-folds====
 
 
 
The question of which cubic 4-folds are rational is one of the foremost open problems in algebraic geometry.  I'll start by explaining what this means and why it's interesting; then I'll discuss three approaches to solving it (including one developed in the last year), my own work relating the three approaches to one another, and the troubles that have befallen each approach.
 
 
 
===January 26:  Minh Binh Tran (CAM)===
 
 
 
====Nonlinear approximation theory for the homogeneous Boltzmann equation====
 
 
 
A challenging problem in solving the Boltzmann equation
 
numerically is that the velocity space is approximated by a finite region.
 
Therefore, most methods are based on a truncation technique and the
 
computational cost is then very high if the velocity domain is large.
 
Moreover, sometimes, non-physical conditions have to be imposed on the
 
equation in order to keep the velocity domain bounded. In this talk, we
 
introduce the first nonlinear approximation theory for the Boltzmann
 
equation. Our nonlinear wavelet approximation is non-truncated and based on
 
a nonlinear, adaptive spectral method associated with a new wavelet
 
filtering technique and a new formulation of the equation. The
 
approximation is proved to converge and perfectly preserve most of the
 
properties of the homogeneous Boltzmann equation. It could also be
 
considered as a general framework for approximating kinetic integral
 
equations.
 
 
 
===February 2:  Afonso Bandeira (Princeton)===
 
 
 
====Tightness of convex relaxations for certain inverse problems on graphs====
 
 
 
Many maximum likelihood estimation problems are known to be
 
intractable in the worst case. A common approach is to consider convex
 
relaxations of the maximum likelihood estimator (MLE), and relaxations
 
based on semidefinite programming (SDP) are among the most popular. We
 
will focus our attention on a certain class of graph-based inverse
 
problems and show a couple of remarkable phenomena.
 
 
 
In some instances of these problems (such as community detection under
 
the stochastic block model) the solution to the SDP matches the ground
 
truth parameters (i.e. achieves exact recovery) for information
 
theoretically optimal regimes. This is established using new
 
nonasymptotic bounds for the spectral norm of random matrices with
 
independent entries.
 
 
 
On other instances of these problems (such as angular
 
synchronization), the MLE itself tends to not coincide with the ground
 
truth (although maintaining favorable statistical properties).
 
Remarkably, these relaxations are often still tight (meaning that the
 
solution of the SDP matches the MLE). For angular synchronization we
 
can understand this behavior by analyzing the solutions of certain
 
randomized Grothendieck problems. However, for many other problems,
 
such as the multireference alignment problem in signal processing,
 
this remains a fascinating open problem.
 
 
 
===February 6:  Morris Hirsch (UC Berkeley and UW Madison)===
 
 
 
====Fixed points of  Lie transformation group,  and zeros of Lie algebras of vector fields====
 
 
 
The following questions will be considered:
 
 
When  a connected Lie group G acts effectively on a manifold M,  what  general conditions on G,  M and the action  ensure that the action has a fixed point? 
 
 
If  g is a Lie algebra of  vector fields on M, what general conditions on g and M  ensure that g has a zero?
 
 
Old and new results will be discussed.  For example:
 
 
Theorem: If G is nilpotent and M is a  compact surface of nonzero Euler characteristic, there is a fixed point.
 
 
Theorem:  Suppose G is supersoluble and M is as above.  Then every analytic action of G on M has a fixed point, but this is false for continuous actions, and for groups that are merely solvable.
 
 
Theorem:  Suppose M is a real or complex manifold that is 2-dimensional over the ground field, and g is a Lie algebra of analytic vector fields on M.  Assume  some element X in g spans a 1-dimensional ideal.  If  the zero set K of X is compact and the Poincar'e-Hopf index of X at K is nonzero,  then g vanishes at some point of K.
 
 
No special knowledge of Lie groups will be assumed.
 
 
 
===February 13:  Mihai Putinar (UC Santa Barbara)===
 
 
 
====Quillen’s property of real algebraic varieties====
 
 
 
A famous observation discovered by Fejer and Riesz a century ago
 
is the quintessential algebraic component of every spectral decomposition
 
result. It asserts that every non-negative polynomial on the unit circle is a
 
hermitian square. About half a century ago, Quillen proved that a positive polynomial
 
on an odd dimensional sphere is a sum of hermitian squares. Fact independently
 
rediscovered much later by D’Angelo and Catlin, respectively Athavale. The main subject of
 
the talk will be: on which real algebraic sub varieties of <math>\mathbb{C}^n</math> is Quillen theorem valid?
 
An interlace between real algebraic geometry, quantization techniques and complex
 
hermitian geometry will provide an answer to the above question, and more.
 
Based a recent work with Claus Scheiderer and John D’Angelo.
 
 
 
===February 20: David Zureick-Brown (Emory University)===
 
 
 
====Diophantine and tropical geometry====
 
 
 
Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers
 
<math>a,b,c \geq 2</math> satisfying <math>\tfrac1a + \tfrac1b + \tfrac1c > 1</math>, Darmon and Granville proved that the individual generalized Fermat equation <math>x^a + y^b = z^c</math> has only finitely many coprime integer solutions. Conjecturally something stronger is true: for <math>a,b,c \geq 3</math> there are no non-trivial solutions.
 
 
 
I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.
 
 
 
==='''Monday''' February 23:  Jayadev Athreya (UIUC)===
 
 
 
====The Erdos-Szusz-Turan distribution for equivariant point processes====
 
 
 
We generalize a problem of Erdos-Szusz-Turan on diophantine approximation to a variety of contexts, and use homogeneous dynamics to compute an associated probability distribution on the integers.
 
 
 
 
 
===February 27: Allan Greenleaf (University of Rochester)===
 
 
 
====Erdos-Falconer Configuration problems====
 
 
 
In discrete geometry, there is a large collection of problems due
 
to Erdos and various coauthors starting in the 1940s, which have the
 
following general form: Given a large finite set P of N points
 
in d-dimensional Euclidean space, and a geometric configuration (a line
 
segment of a given length, a triangle with given angles or a given area,
 
etc.), is there a lower bound on how many times that configuration must
 
occur among the points of P? Relatedly, is there an upper bound
 
on the number of times any single configuration can occur? One of the most
 
celebrated problems of this type, the Erdos distinct distances problem
 
in the plane, was essentially solved in 2010 by Guth and Katz, but for many
 
problems of this type only partial results are known.
 
 
 
In continuous geometry, there are analogous problems due to Falconer and
 
others. Here, one looks for results that say that if a set A is large enough (in
 
terms of a lower bound on its Hausdorff dimension, say), then the set of
 
configurations of a given type generated by the points of A  is large (has
 
positive measure, say).
 
I will describe work on Falconer-type problems using some techniques from
 
harmonic analysis, including estimate for multilinear operators. In some
 
cases, these results can be discretized to obtain at least partial results
 
on Erdos-type problems.
 
 
 
 
 
 
 
===March 6:  Larry Guth (MIT)===
 
  
====Introduction to incidence geometry====
+
[[Colloquia/Blank|Blank]]
  
Incidence geometry is a branch of combinatorics that studies the possible intersection patterns of lines, circles, and other simple shapes.  For example, suppose that we have a set of L lines in the plane.  An r-rich point is a point that lies in at least r of these lines.  For a given L, r, how many r-rich points can we make?  This is a typical question in the field, and there are many variations.  What if we replace lines with circles?  What happens in higher dimensions?  We will give an introduction to this field, describing some of the important results, tools, and open problems.
+
[[Colloquia/Fall2018|Fall 2018]]
  
We will discuss two important tools used in the area.  One tool is to apply topology to the problem.  This tool allows us to prove results in R^2 that are stronger than what happens over finite fields.  The second tool is to look for algebraic structure in the problem by studying low-degree polynomials that vanish on the points we are studying.  We will also discuss some of the (many) open problems in the field and try to describe the nature of the difficulties in approaching them.
+
[[Colloquia/Spring2018|Spring 2018]]
  
 +
[[Colloquia/Fall2017|Fall 2017]]
  
 +
[[Colloquia/Spring2017|Spring 2017]]
  
===March 13:  Cameron Gordon (UT-Austin)===
+
[[Archived Fall 2016 Colloquia|Fall 2016]]
  
====Left-orderability and 3-manifold groups====
+
[[Colloquia/Spring2016|Spring 2016]]
  
The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic property of this group being left-orderable is related to two other aspects of 3-dimensional topology, one geometric-topological and the other essentially analytic.
+
[[Colloquia/Fall2015|Fall 2015]]
 
 
===March 20:  Aaron Naber (Northwestern)===
 
 
 
====Regularity and New Directions in Einstein Manifolds====
 
 
 
In this talk we give an overview of recent developments and new directions of manifolds which satisfy the Einstein equation Rc=cg, or more generally just manifolds with bounded Ricci curvature |Rc|<C.  We will discuss the solution of the codimension four conjecture, which roughly says that Gromov-Hausdorff limits (M^n_i,g_i)->(X,d) of manifolds with bounded Ricci curvature are smooth away from a set of codimension four.  In a very different direction, in this lecture we will also explain how Einstein manifolds may be characterized by the behavior of the analysis on path space P(M) of the manifold.  That is, we will see a Riemannian manifold is Einstein if and only if certain gradient estimates for functions on P(M) hold.  One can view this as an infinite dimensional generalization of the Bakry-Emery estimates.
 
 
 
===March 27 11am B239:  Ilya Kossovskiy (University of Vienna)===
 
 
 
====On Poincare's "Probleme local"====
 
 
 
In this talk, we describe a result giving a complete solution to the
 
old question of Poincare on the possible dimensions of the
 
automorphism group of a real-analytic hypersurface in two-dimensional
 
complex space.
 
As the main tool, we introduce the so-called CR (Cauchy-Riemann
 
manifolds) - DS (Dynamical Systems) technique.
 
This technique suggests to replace a real hypersurface with certain
 
degeneracies of the CR-structure by an appropriate dynamical system,
 
and then study mappings and symmetries of the initial real
 
hypersurface accordingly.
 
It turns out that symmetries of the singular differential equation
 
associated with the initial real hypersurface are much easier to study
 
than that of the real hypersurface, and in this way we obtain the
 
solution for the problem of Poincare.
 
 
 
This work is joint with Rasul Shafikov.
 
 
 
===March 27:  Kent Orr (Indiana University)===
 
 
 
====The Isomorphism Problem for metabelian groups====
 
 
 
Perhaps the most fundamental outstanding problem in algorithmic group theory, the Isomorphism Problem for metabelian groups remains a mystery.
 
 
 
I present an introduction to this problem intended to be accessible to graduate students.  In collaboration with Gilbert Baumslag and Roman Mikhailov, I present a new approach to this ancient problem which potentially connects to algebraic geometry, cohomology of groups, number theory, Gromov's view of groups as geometric objects, and a fundamental algebraic construction developed for and motivated by the topology of knots and links.
 
 
 
===April 10:  Jasmine Foo (University of Minnesota)===
 
 
 
====Evolutionary dynamics in spatially structured populations====
 
 
 
In this talk I will present results on a model of spatial evolution on a lattice, motivated by the process of carcinogenesis from healthy epithelial tissue.  Cancer often arises through a sequence of genetic alterations.  Each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion.  To model this we will consider a generalization of the biased voter process which incorporates successive mutations modulating fitness, which is interpreted as the bias in the classical process.  Under this model we will investigate questions regarding the rate of spread and accumulation of mutations, and study the dynamics of spatial heterogeneity in these evolving populations.
 
 
 
===April 17:  Kay Kirkpatrick (UIUC)===
 
 
 
====Non-normal asymptotics of the mean-field Heisenberg model====
 
 
 
I will discuss spin models of magnets and superconductors, with spins in the circle (XY model) and in the sphere (Heisenberg model), that exhibit interesting phase transitions. I will discuss work with Elizabeth Meckes on the mean-field Heisenberg model and its non-normal behavior at the phase transition. There is much that is still mysterious about these models: I’ll mention work in progress with Tayyab Nawaz and Leslie Ross.
 
 
 
===April 24:  Marianna Csornyei (University of Chicago)===
 
 
 
====Tangents of curves and differentiability of functions====
 
 
 
One of the classical theorems of Lebesgue tells us that Lipschitz
 
functions on the real line are differentiable almost everywhere. We study
 
possible generalisations of this theorem and some interesting geometric
 
corollaries.
 
 
 
===May 1:  Bianca Viray (University of Washington)===
 
 
 
====Cryptography and abelian varieties with extra endomorphisms====
 
 
 
Cryptosystems that are based on the discrete logarithm problem require an instantiation of a group of large prime order. For example, for certain primes p, one could use a subgroup of the group of units in F_p, the F_p-points on an elliptic curve, or the F_p-points on the Jacobian of a genus 2 curve. (The group of integers modulo p results in an insecure cryptosystem.) This raises the question of how to construct a curve C of genus 1 or 2 over F_p whose Jacobian has N F_p-points, where N is a large prime. Perhaps surprisingly, this problem is related to curves over the complex numbers whose Jacobian has "extra" endomorphisms, i.e., Jacobians with complex multiplication. This in turn relates to a different counting problem for curves over F_p. We will give an overview of this connection and describe joint work with Lauter which answers this counting problem. This talk will be suitable for a general mathematical audience.
 
 
 
===May 8:  Marcus Roper (UCLA)===
 
 
 
====Nonlinear flow in micro- and myco-fluidics====
 
 
 
Simulations and asymptotic analysis of particle movement in microfluidic channels is made easier by the fact that the equations of motion for particles moving in a microfluidic pipe are linear and reversible. We discuss two examples of interesting phenomena that that emerge from the breakdown of linearity: (1) Inertial microfluidic devices use high speed flow to focus and segregate particles of different sizes. However, there is no consensus on how focusing forces vary with particle size and flow speed, or even on the number of focusing positions that exist within channels having different shapes. I’ll explain how simulations and experiments may have been misinterpreted and develop a partly numerical, partly asymptotic theory for particle focusing. (2) Fungi form microfluidic networks to transport nutrients, fluid and nuclei. Surprisingly these networks are anticongestive – the greater the density of traffic on a fungal freeway, the faster the traffic travels. I’ll explain how this anticongestive property arises using a kinetic model of the motion of nuclei on a fungal freeway, and speculate about why the fungus might benefit from organizing its flow in this way.
 
 
 
== Past Colloquia ==
 
  
 
[[Colloquia/Spring2014|Spring 2015]]
 
[[Colloquia/Spring2014|Spring 2015]]

Latest revision as of 09:43, 24 January 2019

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.

Spring 2019

date speaker title host(s)
Jan 25 Beata Randrianantoanina (Miami University Ohio) WIMAW Some nonlinear problems in the geometry of Banach spaces and their applications Tullia Dymarz
Jan 30 Wednesday Lillian Pierce (Duke University) Short character sums Boston and Street
Jan 31 Thursday Dean Baskin (Texas A&M) Radiation fields for wave equations Street
Feb 1 Jianfeng Lu (Duke University) TBA Qin
Feb 5 Tuesday Alexei Poltoratski (Texas A&M University) TBA Denisov
Feb 8 Aaron Naber (Northwestern) A structure theory for spaces with lower Ricci curvature bounds Street
Feb 15 TBA
Feb 22 Angelica Cueto (Ohio State) TBA Erman and Corey
March 4 Vladimir Sverak (Minnesota) Wasow lecture TBA Kim
March 8 Jason McCullough (Iowa State) TBA Erman
March 15 Maksym Radziwill (Caltech) TBA Marshall
March 29 Jennifer Park (OSU) TBA Marshall
April 5 Ju-Lee Kim (MIT) TBA Gurevich
April 12 Evitar Procaccia (TAMU) TBA Gurevich
April 19 Jo Nelson (Rice University) TBA Jean-Luc
April 26 Kavita Ramanan (Brown University) TBA WIMAW
May 3 Tomasz Przebinda (Oklahoma) TBA Gurevich

Abstracts

Beata Randrianantoanina (Miami University Ohio)

Title: Some nonlinear problems in the geometry of Banach spaces and their applications.

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

Lillian Pierce (Duke University)

Title: Short character sums

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

Dean Baskin (Texas A&M)

Title: Radiation fields for wave equations

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

Aaron Naber (Northwestern)

Title: A structure theory for spaces with lower Ricci curvature bounds.

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


Past Colloquia

Blank

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012