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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]] == -->
+
== Spring 2018 ==
  
== Spring 2017  ==
 
 
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
Line 15: Line 11:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|'''Monday, January 9, 9th floor'''
+
|January 29 (Monday)
| [http://www.stat.berkeley.edu/~racz/ Miklos Racz] (Microsoft)
+
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
|[[#Monday, January 9:  Miklos Racz (Microsoft) |  ''Statistical inference in networks and genomics''  ]]
+
|[[#January 29 Li Chao (Columbia)|  Elliptic curves and Goldfeld's conjecture ]]
| Valko
+
| Jordan Ellenberg
 
|
 
|
 
|-
 
|-
|January 13, B239
+
|February 2 (Room: 911)
| [https://math.berkeley.edu/people/faculty/mihaela-ifrim/ Mihaela Ifrim] (Berkeley)
+
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)
|[[#Friday, January 13: Mihaela Ifrim (Berkeley) |  ''Two dimensional water waves''  ]]
+
|[[#February 2 Thomas Fai (Harvard)|  The Lubricated Immersed Boundary Method ]]
| Angenent
+
| Spagnolie, Smith
 
|
 
|
 
|-
 
|-
|'''Tuesday, January 17, B139'''
+
|February 5 (Monday, Room: 911)
| [https://web.math.princeton.edu/~fabiop/ Fabio Pusateri] (Princeton)
+
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
|[[#Tuesday, January 17:  Fabio Pusateri (Princeton) | ''The Water Waves problem''  ]]
+
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]
| Angenent
+
| Ellenberg, Gurevitch
 
|
 
|
 
|-
 
|-
|January 20, B239
+
|February 6 (Tuesday 2 pm, Room 911)
| [http://math.mit.edu/~sraskin/ Sam Raskin] (MIT)
+
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
|[[#Friday, January 20: Sam Raskin (MIT) |   Tempered local geometric Langlands ]]
+
|[[#February 6 Alex Lubotzky (Hebrew University)|  Groups' approximation, stability and high dimensional expanders ]]
| Arinkin
+
| Ellenberg, Gurevitch
 
|
 
|
 
|-
 
|-
|'''Monday, January 23, B239'''
+
|February 9
| [http://www.math.umd.edu/~tdarvas/ Tamas Darvas] (Maryland)
+
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)
|[[#Monday, January 23: Tamas Darvas (Maryland) |  Geometry on the space of Kahler metrics and applications to canonical metrics ]]
+
|[[#February 9 Wes Pegden (CMU)|  The fractal nature of the Abelian Sandpile ]]
| Viaclovsky
+
| Roch
 
|
 
|
 
|-
 
|-
|January 27
+
|March 2
|Reserved for possible job talks
+
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)
|[[# |   ]]
+
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]
|
+
| Caldararu
 
|
 
|
 
|-
 
|-
|February 3, 9th floor
+
| March 16  (Room: 911)
| Melanie Matchett Wood (UW-, Madison)
+
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)
|[[#Friday, February 3: Melanie Matchett Wood (UW-Madison) | Random groups from generators and relations ]]
+
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]
|
+
|
+
|-
+
|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
 
| WIMAW
 
|
 
|
 
|-
 
|-
|February 17, 9th floor
+
|April 5 (Thursday, Room: 911)
| [http://web.math.ucsb.edu/~ponce/ Gustavo Ponce] (UCSB)
+
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)
| [[#Friday, February 17: Gustavo Ponce(UCSB) The Korteweg-de Vries equation vs. the Benjamin-Ono equation ]]  
+
|[[#April 5 John Baez (UC Riverside)|  Monoidal categories of networks  ]]
| Minh-Binh Tran
+
| Craciun
 
|
 
|
 
|-
 
|-
|February 24
+
| April 6
| [http://acms.nd.edu/people/faculty/jonathan-hauenstein/ Jonathan Hauenstein] (Notre Dame)
+
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)
|[[#February 24: Jonathan Hauenstein (Notre Dame) Real solutions of polynomial equations ]]
+
|[[# Edray GoinsToroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups ]]
| Boston
+
| Melanie
 
|
 
|
 
|-
 
|-
|March 3
+
| April 13 (911 Van Vleck)
| [http://www.math.utah.edu/~bromberg/ Ken Bromberg] (University of Utah)
+
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)
|[[# |   ]]
+
|[[#April 13, Jill Pipher, Brown University| Mathematical ideas in cryptography  ]]
|Dymarz
+
| WIMAW
 
|
 
|
 
|-
 
|-
|Tuesday, March 7, 4PM (Distinguished Lecture)
+
| April 16 (Monday)
| [http://pages.iu.edu/~temam/ Roger Temam] (Indiana University)  
+
| [http://www-users.math.umn.edu/~cberkesc/ Christine Berkesch Zamaere ] (University of Minnesota)
|[[# |   ]]
+
|[[#April 16, Christine Berkesch Zamaere (University of Minnesota)| Free complexes on smooth toric varieties  ]]
|Smith
+
| Erman, Sam
 
|
 
|
 
|-
 
|-
|'''Wednesday, March 8, 4PM, B239 '''
+
| April 25 (Wednesday, Room: 911)
| [http://pages.iu.edu/~temam/ Roger Temam] (Indiana University)  
+
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Tsuda University) Wasow lecture
|[[# |   ]]
+
|[[#April 25, Hitoshi Ishii (Tsuda University)| Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory  ]]
|Smith
+
| Tran
 
|
 
|
 
|-
 
|-
|March 10
+
| May 1 (Tuesday, 4:30pm, Room: B102 VV)
| '''No Colloquium'''
+
| [https://math.uchicago.edu/~aneves/ Andre Neves] (University Chicago and Imperial College London) Distinguished lecture
|[[# |   ]]
+
|[[#May 1, Andre Neves (University Chicago and Imperial College London)| Wow, so many minimal surfaces! (Part I)]]
|
+
| Lu Wang
 
|
 
|
 
|-
 
|-
|'''Wednesday, March 15, 4PM '''
+
| May 2 (Wednesday, 3pm, Room: B325 VV)
| [http://verso.mat.uam.es/web/ezuazua/zuazua.html Enrique Zuazua] (Universidad Autónoma de Madrid)
+
| [https://math.uchicago.edu/~aneves/ Andre Neves] (University of Chicago and Imperial College London) Distinguished lecture
|[[#   TBA|   TBA ]]
+
|[[#May 2, Andre Neves (University Chicago and Imperial College London)| Wow, so many minimal surfaces! (Part II) ]]
| Jin & Minh-Binh Tran
+
| Lu Wang
 
|
 
|
 
|-
 
|-
|March 17
+
| May 4
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
+
| [http://math.mit.edu/~cohn/ Henry Cohn] (Microsoft Research and MIT)
| TBA
+
| M. Matchett Wood
+
|
+
|-
+
|March 24
+
|  '''Spring Break'''
+
|[[# |    ]]
+
|
+
|
+
|-
+
|'''Wednesday, March 29  at 3:30PM (Wasow)'''
+
| [https://math.nyu.edu/faculty/serfaty/ Sylvia Serfaty] (NYU)
+
|[[# TBA|  TBA]]
+
|Tran
+
|
+
|-
+
|March 31
+
| '''No Colloquium'''
+
|[[# |    ]]
+
|
+
|
+
|-
+
|April 7
+
| [http://www.math.uiuc.edu/~schenck/ Hal Schenck]
+
|[[# |    ]]
+
|Erman
+
|
+
|-
+
|April 14
+
|  Wilfrid Gangbo
+
|[[# |    ]]
+
|Feldman & Tran
+
|
+
|-
+
|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  ]]
 
|[[# TBA|  TBA  ]]
|Li
+
| Ellenberg
|}
+
 
+
==Fall 2017==
+
 
+
{| cellpadding="8"
+
!align="left" | date 
+
!align="left" | speaker
+
!align="left" | title
+
!align="left" | host(s)
+
|-
+
|September 8
+
|
+
|[[# TBA|  TBA  ]]
+
|
+
 
|
 
|
 
|-
 
|-
|September 15
+
|date
|
+
| person (institution)
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
|
+
| hosting faculty
 
|
 
|
 
|-
 
|-
| '''Wednesday, September 20, LAA lecture
+
|date
| Andrew Stuart (Caltech)
+
| person (institution)
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
| Jin
+
| hosting faculty
 
|
 
|
 
|-
 
|-
|September 22
+
|date
|
+
| person (institution)
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
|
+
| hosting faculty
 
|
 
|
 
|-
 
|-
|September 29
+
|date
|
+
| person (institution)
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
|
+
| hosting faculty
 
|
 
|
 
|-
 
|-
|October 6
+
|date
|
+
| person (institution)
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
|
+
| hosting faculty
 
|
 
|
 
|-
 
|-
|October 13
+
|date
|
+
| person (institution)
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
|
+
| hosting faculty
 
|
 
|
 
|-
 
|-
|October 20
+
|date
|
+
| person (institution)
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
|
+
| hosting faculty
 
|
 
|
 
|-
 
|-
|October 27
+
|date
|
+
| person (institution)
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|
 
|-
 
|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  ]]
 
|
 
|
 
|-
 
 
 
|}
 
|}
  
== Abstracts ==
+
== Spring Abstracts ==
=== September 16: Po-Shen Loh (CMU) ===
+
Title: Directed paths: from Ramsey to Pseudorandomness
+
  
Abstract: Starting from an innocent Ramsey-theoretic question regarding directed
 
paths in graphs, we discover a series of rich and surprising connections
 
that lead into the theory around a fundamental result in Combinatorics:
 
Szemeredi's Regularity Lemma, which roughly states that every graph (no
 
matter how large) can be well-approximated by a bounded-complexity
 
pseudorandom object.  Using these relationships, we prove that every
 
coloring of the edges of the transitive N-vertex tournament using three
 
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) ===
+
===January 29 Li Chao (Columbia)===
Title: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture
+
  
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.
+
Title: Elliptic curves and Goldfeld's conjecture
  
The conjecture was formulated by Fritz Horn in the early 1970s, and is strongly related to Boltzmann's H-theorem.
+
Abstract:
 +
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.
  
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.
+
=== February 2 Thomas Fai (Harvard) ===
  
=== September 30: Akos Magyar (University of Georgia) ===
+
Title: The Lubricated Immersed Boundary Method
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.
+
Abstract:
 +
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.
  
=== October 14: Ling Long (LSU) ===  
+
===February 5 Alex Lubotzky (Hebrew University)===
Title: Hypergeometric functions over finite fields
+
  
Abstract: Hypergeometric functions are special functions with lot of
+
TitleHigh dimensional expanders: From Ramanujan graphs to Ramanujan complexes
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) ===
+
Abstract:  
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).
+
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.  
  
=== October 28: Linda Reichl (UT Austin) ===
+
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.
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.
+
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.  
  
===Monday, October 31: Kathryn Mann (Berkeley) ===
 
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. 
+
===February 6 Alex Lubotzky (Hebrew University)===
  
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. 
+
Title: Groups' approximation, stability and high dimensional expanders
  
===November 7: Gaven Martin (New Zealand Institute for Advanced Study) ===
+
Abstract:  
Title: Siegel's problem on small volume lattices
+
  
Abstract: We outline in very general terms the history and the proof of the identification
+
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 normsWe 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.
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) ===
+
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.
Title: Shapes of Julia Sets
+
  
Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded. William 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.
+
All notions will be explained.       Joint work with M, De Chiffre, L. Glebsky and A. Thom.
  
===November 18: Andrew Snowden (University of Michigan)===
+
===February 9 Wes Pegden (CMU)===
Title: Recent progress in representation stability
+
  
Abstract: Representation stability is a relatively new field that studies
+
Title: The fractal nature of the Abelian Sandpile
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)===
+
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.
Title: Definability in degree structures
+
  
Abstract:  Some incomputable sets are more incomputable than others. We use
+
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.
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)===
+
===March 2 Aaron Bertram (Utah)===
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.
+
Title: Stability in Algebraic Geometry
  
===Monday, December 5: Botong Wang (UW-Madison)===
+
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.
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.
+
===March 16 Anne Gelb (Dartmouth)===
  
=== Friday, December 9: Aaron Brown (U Chicago) ===
+
Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity
''Lattice actions and recent progress in the Zimmer program''
+
  
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.
+
Abstract: We introduce the variance based joint sparsity (VBJS) method for sparse signal recovery and image reconstruction from multiple measurement vectors. Joint sparsity techniques employing $\ell_{2,1}$ minimization are typically used, but the algorithm is computationally intensive and requires fine tuning of parameters. The VBJS method uses a weighted $\ell_1$ joint sparsity algorithm, where the weights depend on the pixel-wise variance. The VBJS method is accurate, robust, cost efficient and also reduces the effects of false data.
  
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:
 
(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) ===
 
''Metric spaces of non-positive curvature and applications in several complex variables''
 
  
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.
 
  
=== Monday, January 9: Miklos Racz (Microsoft) ===
+
===April 5 John Baez (UC Riverside)===
''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.
+
Title: Monoidal categories of networks
  
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.
+
Abstract: Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.
  
=== Friday, January 13: Mihaela Ifrim (Berkeley) ===
 
''Two dimensional water waves''
 
  
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.
 
  
=== Tuesday, January 17:  Fabio Pusateri (Princeton) ===
 
''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.
 
  
=== Friday, January 20: Sam Raskin (MIT) ===  
+
===April 6 Edray Goins (Purdue)===
''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.
+
Title: Toroidal Bely&#301;  Pairs, Toroidal Graphs, and their Monodromy Groups
  
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.
+
Abstract: A Bely&#301; map <math> \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) </math> is a rational function with at most three critical values; we may assume these values are <math> \{ 0, \, 1, \, \infty \}. </math>  A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1.  Such graphs can be drawn on the sphere by composing with stereographic projection: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). </math>  Replacing <math> \mathbb P^1 </math> with an elliptic curve <math>E </math>, there is a similar definition of a Bely&#301; map <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). </math>  Since <math> E(\mathbb C) \simeq \mathbb T^2(\mathbb R) </math> is a torus, we call <math> (E, \beta) </math> a toroidal Bely&#301; pair.  The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R). </math>
  
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.
+
This project seeks to create a database of such Bely&#301; pairs, their corresponding Dessins d'Enfant, and their monodromy groups.  For each positive integer <math> N </math>, there are only finitely many toroidal Bely&#301; pairs <math> (E, \beta) </math> with <math> \deg \, \beta = N. </math>  Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences <math> \mathcal D </math> on the ramification indices as multisets on three partitions of N. For each degree sequence, we compute all possible monodromy groups <math> G = \text{im} \, \bigl[ \pi_1 \bigl( \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \} \bigr) \to S_N \bigr]; </math> they are the ``Galois closure'' of the group of automorphisms of the graph.  Finally, for each possible monodromy group, we compute explicit formulas for Bely&#301; maps <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) </math> associated to some elliptic curve <math> E: \ y^2 = x^3 + A \, x + B. </math>  We will discuss some of the challenges of determining the structure of these groups, and present visualizations of group actions on the torus.  
  
=== Monday, January 23: Tamas Darvas (Maryland) ===
+
This work is part of PRiME (Purdue Research in Mathematics Experience) with Chineze Christopher, Robert Dicks, Gina Ferolito, Joseph Sauder, and Danika Van Niel with assistance by Edray Goins and Abhishek Parab.
''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
+
===April 13, Jill Pipher, Brown University===
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. 
+
  
 +
Title:  Mathematical ideas in cryptography
  
=== Friday, February 3: Melanie Matchett Wood (UW-Madison) ===
+
Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,
''Random groups from generators and relations''
+
including homomorphic encryption.
  
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. 
+
===April 16, Christine Berkesch Zamaere (University of Minnesota)===
 +
Title: Free complexes on smooth toric varieties
  
=== Monday, February 6: Benoit Perthame (University of Paris VI) ===
+
Abstract: Free resolutions have been a key part of using homological algebra to compute and characterize geometric invariants over projective space. Over more general smooth toric varieties, this is not the case. We will discuss the another family of complexes, called virtual resolutions, which appear to play the role of free resolutions in this setting. This is joint work with Daniel Erman and Gregory G. Smith.
''Models for neural networks; analysis, simulations and behaviour''
+
  
Neurons exchange informations via discharges, propagated
 
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
+
===April 25, Hitoshi Ishii (Tsuda University)===
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,
+
Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory
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  
+
Abstract:  In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equationsI explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.
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.
+
===May 1 and 2, Andre Neves (University of Chicago and Imperial College London)===
 +
Title: Wow, so many minimal surfaces!
  
 +
Abstract: Minimal surfaces are ubiquitous in geometry and  applied science but their existence theory is rather mysterious.  For instance, Yau in 1982 conjectured that any 3-manifold admits infinitely many closed minimal surfaces but the best one knows is the existence of at least three.
  
 +
After a brief historical account, I will talk about my ongoing work with Marques  and  the progress we made on this question jointly with Irie and Song: we showed that for generic metrics, minimal hypersurfaces are dense and equidistributed. In particular, this settles Yau’s conjecture for generic metrics.
  
=== February 17: Gustavo Ponce, UCSB===
+
The first talk will be more general and the second talk will contain proofs of the denseness and equidistribution results. This part is join work with Irie, Marques, and Song.
  
''The Korteweg-de Vries equation vs. the Benjamin-Ono equation''
+
== Future Colloquia ==
 +
[[Colloquia/Blank|Fall 2018]]
  
In this talk we shall study the <math>k</math>-generalized Korteweg-de Vries <math>(k-KdV)</math> equation
+
== Past Colloquia ==
  
<math>\partial_t u + \partial_x^3u +u^k\,\partial_xu=0,\;\;\;\;\;\;\;x,t\in\Bbb R,\, k\in \Bbb Z^+, </math>
+
[[Colloquia/Blank|Blank]]
  
and the <math>k</math>-generalized Benjamin-Ono (<math>k</math>-BO) equation
+
[[Colloquia/Fall2017|Fall 2017]]
  
<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>
+
[[Colloquia/Spring2017|Spring 2017]]
 
+
where <math>\mathcal {H}</math> denotes the Hilbert transform,
+
 
+
<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>
+
 
+
The goal is to review and analyze results concerning solutions of the initial value properties associated to these equations.
+
+
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) ===
+
''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. 
+
 
+
 
+
== Past Colloquia ==
+
  
 
[[Archived Fall 2016 Colloquia|Fall 2016]]
 
[[Archived Fall 2016 Colloquia|Fall 2016]]

Latest revision as of 17:54, 24 April 2018

Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Spring 2018

date speaker title host(s)
January 29 (Monday) Li Chao (Columbia) Elliptic curves and Goldfeld's conjecture Jordan Ellenberg
February 2 (Room: 911) Thomas Fai (Harvard) The Lubricated Immersed Boundary Method Spagnolie, Smith
February 5 (Monday, Room: 911) Alex Lubotzky (Hebrew University) High dimensional expanders: From Ramanujan graphs to Ramanujan complexes Ellenberg, Gurevitch
February 6 (Tuesday 2 pm, Room 911) Alex Lubotzky (Hebrew University) Groups' approximation, stability and high dimensional expanders Ellenberg, Gurevitch
February 9 Wes Pegden (CMU) The fractal nature of the Abelian Sandpile Roch
March 2 Aaron Bertram (University of Utah) Stability in Algebraic Geometry Caldararu
March 16 (Room: 911) Anne Gelb (Dartmouth) Reducing the effects of bad data measurements using variance based weighted joint sparsity WIMAW
April 5 (Thursday, Room: 911) John Baez (UC Riverside) Monoidal categories of networks Craciun
April 6 Edray Goins (Purdue) Toroidal Belyĭ Pairs, Toroidal Graphs, and their Monodromy Groups Melanie
April 13 (911 Van Vleck) Jill Pipher (Brown) Mathematical ideas in cryptography WIMAW
April 16 (Monday) Christine Berkesch Zamaere (University of Minnesota) Free complexes on smooth toric varieties Erman, Sam
April 25 (Wednesday, Room: 911) Hitoshi Ishii (Tsuda University) Wasow lecture Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory Tran
May 1 (Tuesday, 4:30pm, Room: B102 VV) Andre Neves (University Chicago and Imperial College London) Distinguished lecture Wow, so many minimal surfaces! (Part I) Lu Wang
May 2 (Wednesday, 3pm, Room: B325 VV) Andre Neves (University of Chicago and Imperial College London) Distinguished lecture Wow, so many minimal surfaces! (Part II) Lu Wang
May 4 Henry Cohn (Microsoft Research and MIT) TBA Ellenberg
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty

Spring Abstracts

January 29 Li Chao (Columbia)

Title: Elliptic curves and Goldfeld's conjecture

Abstract: 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.

February 2 Thomas Fai (Harvard)

Title: The Lubricated Immersed Boundary Method

Abstract: 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)

Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes

Abstract:

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.

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.

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.


February 6 Alex Lubotzky (Hebrew University)

Title: Groups' approximation, stability and high dimensional expanders

Abstract:

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 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.

All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.

February 9 Wes Pegden (CMU)

Title: The fractal nature of the Abelian Sandpile

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.

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.

March 2 Aaron Bertram (Utah)

Title: Stability in Algebraic Geometry

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.

March 16 Anne Gelb (Dartmouth)

Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity

Abstract: We introduce the variance based joint sparsity (VBJS) method for sparse signal recovery and image reconstruction from multiple measurement vectors. Joint sparsity techniques employing $\ell_{2,1}$ minimization are typically used, but the algorithm is computationally intensive and requires fine tuning of parameters. The VBJS method uses a weighted $\ell_1$ joint sparsity algorithm, where the weights depend on the pixel-wise variance. The VBJS method is accurate, robust, cost efficient and also reduces the effects of false data.



April 5 John Baez (UC Riverside)

Title: Monoidal categories of networks

Abstract: Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.



April 6 Edray Goins (Purdue)

Title: Toroidal Belyĭ Pairs, Toroidal Graphs, and their Monodromy Groups

Abstract: A Belyĭ map  \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) is a rational function with at most three critical values; we may assume these values are  \{ 0, \, 1, \, \infty \}. A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection:  \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). Replacing  \mathbb P^1 with an elliptic curve E , there is a similar definition of a Belyĭ map  \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). Since  E(\mathbb C) \simeq \mathbb T^2(\mathbb R) is a torus, we call  (E, \beta) a toroidal Belyĭ pair. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm:  \beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R).

This project seeks to create a database of such Belyĭ pairs, their corresponding Dessins d'Enfant, and their monodromy groups. For each positive integer  N , there are only finitely many toroidal Belyĭ pairs  (E, \beta) with  \deg \, \beta = N. Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences  \mathcal D on the ramification indices as multisets on three partitions of N. For each degree sequence, we compute all possible monodromy groups  G = \text{im} \, \bigl[ \pi_1 \bigl( \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \} \bigr) \to S_N \bigr]; they are the ``Galois closure of the group of automorphisms of the graph. Finally, for each possible monodromy group, we compute explicit formulas for Belyĭ maps  \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) associated to some elliptic curve  E: \ y^2 = x^3 + A \, x + B. We will discuss some of the challenges of determining the structure of these groups, and present visualizations of group actions on the torus.

This work is part of PRiME (Purdue Research in Mathematics Experience) with Chineze Christopher, Robert Dicks, Gina Ferolito, Joseph Sauder, and Danika Van Niel with assistance by Edray Goins and Abhishek Parab.

April 13, Jill Pipher, Brown University

Title: Mathematical ideas in cryptography

Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research, including homomorphic encryption.

April 16, Christine Berkesch Zamaere (University of Minnesota)

Title: Free complexes on smooth toric varieties

Abstract: Free resolutions have been a key part of using homological algebra to compute and characterize geometric invariants over projective space. Over more general smooth toric varieties, this is not the case. We will discuss the another family of complexes, called virtual resolutions, which appear to play the role of free resolutions in this setting. This is joint work with Daniel Erman and Gregory G. Smith.


April 25, Hitoshi Ishii (Tsuda University)

Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory

Abstract: In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations. I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.

May 1 and 2, Andre Neves (University of Chicago and Imperial College London)

Title: Wow, so many minimal surfaces!

Abstract: Minimal surfaces are ubiquitous in geometry and applied science but their existence theory is rather mysterious. For instance, Yau in 1982 conjectured that any 3-manifold admits infinitely many closed minimal surfaces but the best one knows is the existence of at least three.

After a brief historical account, I will talk about my ongoing work with Marques and the progress we made on this question jointly with Irie and Song: we showed that for generic metrics, minimal hypersurfaces are dense and equidistributed. In particular, this settles Yau’s conjecture for generic metrics.

The first talk will be more general and the second talk will contain proofs of the denseness and equidistribution results. This part is join work with Irie, Marques, and Song.

Future Colloquia

Fall 2018

Past Colloquia

Blank

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012