Difference between revisions of "Colloquia"

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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 2018 ==
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!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|January 30
+
|January 29 (Monday)
 
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
 
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
|[[# TBATBA ]]
+
|[[#January 29 Li Chao (Columbia)Elliptic curves and Goldfeld's conjecture ]]
 
| Jordan Ellenberg
 
| Jordan Ellenberg
 
|
 
|
 
|-
 
|-
|February 2
+
|February 2 (Room: 911)
 
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)
 
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)
|[[# TBATBA  ]]
+
|[[#February 2 Thomas Fai (Harvard)The Lubricated Immersed Boundary Method ]]
 
| Spagnolie, Smith
 
| Spagnolie, Smith
 +
|
 +
|-
 +
|February 5 (Monday, Room: 911)
 +
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)
 +
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]
 +
| Ellenberg, Gurevitch
 +
|
 +
|-
 +
|February 6 (Tuesday 2 pm, Room 911)
 +
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)
 +
|[[#February 6 Alex Lubotzky (Hebrew University)|  Groups' approximation, stability and high dimensional expanders ]]
 +
| Ellenberg, Gurevitch
 +
|
 +
|-
 +
|February 9
 +
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)
 +
|[[#February 9 Wes Pegden (CMU)|  The fractal nature of the Abelian Sandpile ]]
 +
| Roch
 +
|
 +
|-
 +
|March 2
 +
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)
 +
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]
 +
| Caldararu
 
|
 
|
 
|-
 
|-
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|-
 
|-
 
| April 6
 
| April 6
| Reserved
+
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)
|[[# TBATBA ]]
+
|[[# Edray GoinsToroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups ]]
 
| Melanie
 
| Melanie
 
|
 
|
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|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
 
| WIMAW
 
| WIMAW
 +
|
 +
|-
 +
| April 20
 +
| Xiuxiong Chen(Stony Brook University)
 +
|[[# Xiuxiong Chen|  TBA  ]]
 +
| Bing Wang
 
|
 
|
 
|-
 
|-
 
| April 25 (Wednesday)
 
| April 25 (Wednesday)
| Hitoshi Ishii (Waseda University) Wasow lecture
+
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
 
| Tran
 
| Tran
Line 112: Line 138:
 
|}
 
|}
  
 +
== Spring Abstracts ==
  
==Fall 2017==
 
  
{| cellpadding="8"
+
===January 29 Li Chao (Columbia)===
!align="left" | Date 
+
!align="left" | Speaker
+
!align="left" | Title
+
!align="left" | Host(s)
+
|-
+
|September 8
+
| [https://sites.google.com/a/wisc.edu/theresa-c-anderson/home/ Tess Anderson] (Madison)
+
|[[#September 8: Tess Anderson (Madison) |  A Spherical Maximal Function along the Primes  ]]
+
| Tonghai Yang
+
|
+
|-
+
|September 15
+
|
+
|[[#|  ]]
+
|
+
|
+
|
+
|-
+
|September 22, '''9th floor'''
+
| Jaeyoung Byeon (KAIST)
+
|[[#September 22: Jaeyoung Byeon (KAIST) |  Patterns formation for elliptic systems with large interaction forces  ]]
+
| Paul Rabinowitz & Chanwoo Kim
+
|
+
|
+
|
+
|-
+
|October 6,  '''9th floor'''
+
| [http://www3.nd.edu/~jhauenst/ Jonathan Hauenstein] (Notre Dame)
+
|[[#October 6: Jonathan Hauenstein (Notre Dame) |  Real solutions of polynomial equations ]]
+
| Nigel Boston
+
|
+
|-
+
|October 13, '''9th floor'''
+
| [http://www.tomokokitagawa.com/ Tomoko L. Kitagawa] (Berkeley)
+
|[[#October 13: Tomoko Kitagawa (Berkeley) |  A Global History of Mathematics from 1650 to 2017 ]]
+
| Laurentiu Maxim
+
|
+
|-
+
|October 20
+
|  [http://cims.nyu.edu/~pgermain/ Pierre Germain] (Courant, NYU)
+
|[[#October 13: Pierre Germain (Courant, NYU) |  Stability of the Couette flow in the Euler and Navier-Stokes equations ]]
+
|  Minh-Binh Tran
+
|
+
|-
+
|October 27
+
|Stefanie Petermichl (Toulouse)
+
|[[#October 27: Stefanie Petermichl (Toulouse)  |  Higher order Journé commutators  ]]
+
| Betsy Stovall, Andreas Seeger
+
|
+
|-
+
|November 1 (Wednesday)
+
|[http://pages.iu.edu/~shaoguo/  Shaoming Guo] (Indiana)
+
|[[#November 1: Shaoming Guo (Indiana)|  Parsell-Vinogradov systems in higher dimensions  ]]
+
|Andreas Seeger
+
|
+
|
+
|
+
|
+
|
+
|-
+
|November 17
+
| [http://math.mit.edu/~ylio/  Yevgeny Liokumovich] (MIT)
+
|[[#November 17:Yevgeny Liokumovich (MIT)|  Recent progress in Min-Max Theory  ]]
+
|Sean Paul
+
|-
+
|November 21, '''9th floor'''
+
| [https://web.stanford.edu/~mkemeny/homepage.html  Michael Kemeny] (Stanford)
+
|[[#November 21:Michael Kemeny (Stanford)|  The equations defining curves and moduli spaces  ]]
+
|Jordan Ellenberg
+
|
+
|-
+
|November 24
+
|'''Thanksgiving break'''
+
|
+
|
+
|-
+
|November 27,
+
| [http://www.math.harvard.edu/~tcollins/homepage.html  Tristan Collins] (Harvard)
+
|[[#November 27:Tristan Collins (Harvard)|  The J-equation and stability  ]]
+
|Sean Paul
+
|
+
|
+
|-
+
|December 5 (Tuesday)
+
| [http://web.sas.upenn.edu/rhynd/  Ryan Hynd] (U Penn)
+
|[[#December 5: Ryan Hynd (U Penn)|  Adhesion dynamics and the sticky particle system]]
+
|Sigurd Angenent
+
|
+
|-
+
|December 8 (Friday)
+
| [https://cims.nyu.edu/~chennan/  Nan Chen] (Courant, NYU)
+
|[[#December 8: Nan Chen (Courant, NYU)|  A Conditional Gaussian Framework for Uncertainty Quantification, Data Assimilation and Prediction of Complex Turbulent Dynamical Systems  ]]
+
|Leslie Smith
+
|
+
|
+
|-
+
|December 11 (Monday)
+
| [https://people.math.ethz.ch/~mooneyc/  Connor Mooney] (ETH Zurich)
+
|[[#December 11: Connor Mooney (ETH Zurich)|  Regularity vs. Singularity for Elliptic and Parabolic Systems]]
+
|Sigurd Angenent
+
|
+
|-
+
|December 13 (Wednesday)
+
| [http://math.mit.edu/~blwilson/ Bobby Wilson] (MIT)
+
|[[#December 13: Bobby Wilson (MIT) | Projections in Banach Spaces and Harmonic Analysis ]]
+
|Andreas Seeger
+
|
+
|-
+
|December 15 (Friday) '''9th floor'''
+
| [http://roy.lederman.name/ Roy Lederman] (Princeton)
+
|[[#December 15: Roy Lederman (Princeton) | Inverse Problems and Unsupervised Learning with applications to Cryo-Electron Microscopy (cryo-EM) ]]
+
|Leslie Smith
+
|
+
|-
+
|December 18 (Monday) '''B115'''
+
| [https://web.stanford.edu/~jchw/ Jenny Wilson] (Stanford)
+
|[[#December 18: Jenny Wilson (Stanford)|  Stability in the homology of configuration spaces]]
+
|Jordan Ellenberg
+
|
+
|-
+
|December 19 (Tuesday) '''9th floor'''
+
| [https://web.stanford.edu/~amwright/  Alex Wright] (Stanford)
+
|[[#December 19: Alex Wright (Stanford)|  Dynamics, geometry, and the moduli space of Riemann surfaces]]
+
|Jordan Ellenberg
+
|}
+
  
== Fall Abstracts ==
+
Title: Elliptic curves and Goldfeld's conjecture
=== September 8: Tess Anderson (Madison) ===
+
Title: A Spherical Maximal Function along the Primes
+
  
Abstract: Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior.  The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example. In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to. We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory.  This is joint work with Cook, Hughes, and Kumchev.
+
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) ===
  
=== September 22: Jaeyoung Byeon (KAIST) ===
+
Title: The Lubricated Immersed Boundary Method
Title: Patterns formation for elliptic systems with large interaction forces
+
  
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions.  The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.
+
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 6: Jonathan Hauenstein (Notre Dame) ===
+
===February 5 Alex Lubotzky (Hebrew University)===
Title: Real solutions of polynomial equations
+
  
Abstract: Systems of nonlinear polynomial equations arise frequently in applications with the set of real solutions typically corresponding to physically meaningful solutions. Efficient algorithms for computing real solutions are designed by exploiting structure arising from the application.  This talk will highlight some of these algorithms for various applications such as solving steady-state problems of hyperbolic conservation laws, solving semidefinite programs, and computing all steady-state solutions of the Kuramoto model.
+
TitleHigh dimensional expanders: From Ramanujan graphs to Ramanujan complexes
  
===October 13: Tomoko Kitagawa (Berkeley) ===
+
Abstract:  
Title: A Global History of Mathematics from 1650 to 2017
+
  
Abstract: This is a talk on the global history of mathematics. We will first focus on France by revisiting some of the conversations between Blaise Pascal (1623–1662) and Pierre de Fermat (1607–1665). These two “mathematicians” discussed ways of calculating the possibility of winning a gamble and exchanged their opinions on geometry. However, what about the rest of the world? We will embark on a long oceanic voyage to get to East Asia and uncover the unexpected consequences of blending foreign mathematical knowledge into domestic intelligence, which was occurring concurrently in Beijing and Kyoto. How did mathematicians and scientists contribute to the expansion of knowledge? What lessons do we learn from their experiences?
+
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.
  
===October 20: Pierre Germain (Courant, NYU) ===
 
Title: Stability of the Couette flow in the Euler and Navier-Stokes equations
 
  
Abstract: I will discuss the question of the (asymptotic) stability of the Couette flow in Euler and Navier-Stokes. The Couette flow is the simplest nontrivial stationary flow, and the first one for which this question can be fully answered. The answer involves the mathematical understanding of important physical phenomena such as inviscid damping and enhanced dissipation. I will present recent results in dimension 2 (Bedrossian-Masmoudi) and dimension 3 (Bedrossian-Germain-Masmoudi).
+
===February 6 Alex Lubotzky (Hebrew University)===
  
===October 27: Stefanie Petermichl (Toulouse)===
+
Title: Groups' approximation, stability and high dimensional expanders
Title: Higher order Journé commutators
+
 
+
Abstract: We consider questions that stem from operator theory via Hankel and
+
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
+
more basic terms, let us consider a function on the unit circle in its
+
Fourier representation. Let P_+ denote the projection onto
+
non-negative and P_- onto negative frequencies. Let b denote
+
multiplication by the symbol function b. It is a classical theorem by
+
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
+
only if b is in an appropriate space of functions of bounded mean
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oscillation. The necessity makes use of a classical factorisation
+
theorem of complex function theory on the disk. This type of question
+
can be reformulated in terms of commutators [b,H]=bH-Hb with the
+
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
+
as in the real variable setting, in the multi-parameter setting or
+
other, these classifications can be very difficult.
+
 
+
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
+
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of
+
spaces of bounded mean oscillation via L^p boundedness of commutators.
+
We present here an endpoint to this theory, bringing all such
+
characterisation results under one roof.
+
 
+
The tools used go deep into modern advances in dyadic harmonic
+
analysis, while preserving the Ansatz from classical operator theory.
+
 
+
===November 1: Shaoming Guo (Indiana) ===
+
Title: Parsell-Vinogradov systems in higher dimensions
+
  
 
Abstract:  
 
Abstract:  
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
 
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
 
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.
 
  
===November 17:Yevgeny Liokumovich (MIT)===
+
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.
Title: Recent progress in Min-Max Theory
+
  
Abstract:
+
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.
Almgren-Pitts Min-Max Theory is a method of constructing minimal hypersurfaces in Riemannian manifolds. In the last few years a number of long-standing open problems in Geometry, Geometric Analysis and 3-manifold Topology have been solved using this method. I will explain the main ideas and challenges in Min-Max Theory with an emphasis on its quantitative aspect: what quantitative information about the geometry and topology of minimal hypersurfaces can be extracted from the theory?
+
  
===November 21:Michael Kemeny (Stanford)===
+
All notions will be explained.      Joint work with M, De Chiffre, L. Glebsky and A. Thom.
Title: The equations defining curves and moduli spaces
+
  
Abstract:
+
===February 9 Wes Pegden (CMU)===
A projective variety is a subset of projective space defined by polynomial equations. One of the oldest problems in algebraic geometry is to give a qualitative description of the equations defining a variety, together with
+
the relations amongst them. When the variety is an algebraic curve (or Riemann surface), several conjectures
+
made since the 80s give a fairly good picture of what we should expect. I will describe a new variational approach to these conjectures,
+
which reduces the problem to studying cycles on Hurwitz space or on the moduli space of curves.
+
  
 +
Title: The fractal nature of the Abelian Sandpile
  
===November 27:Tristan Collins (Harvard)===
+
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: The J-equation and stability
+
  
Abstract: Donaldson and Chen introduced the J-functional in '99, and explained its importance in the existence problem for constant scalar curvature metrics on compact Kahler manifolds. An important open problem is to find algebro-geometric conditions under which the J-functional has a critical pointThe critical points of the J-functional are described by a fully-nonlinear PDE called the J-equationI will discuss some recent progress on this problem, and indicate the role of algebraic geometry in proving estimates for the J-equation.
+
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 packingsIn this talk, we will survey our work in this area, and discuss avenues of current and future research.
  
===December 5: Ryan Hynd (U Penn)===
+
===March 2 Aaron Bertram (Utah)===
Title: Adhesion dynamics and the sticky particle system.
+
  
Abstract: The sticky particle system expresses the conservation of mass and
+
Title: Stability in Algebraic Geometry
momentum for a collection of particles that only interact via perfectly inelastic collisions. 
+
The equations were first considered in astronomy in a model for the expansion of
+
matter without pressure. These equations also play a central role in the theory of optimal
+
transport.  Namely, the geodesics in an appropriately metrized space of probability
+
measures correspond to solutions of the sticky particle system.  We will survey what is
+
known about solutions and discuss connections with Hamilton-Jacobi equations.
+
  
===December 8: Nan Chen (Courant, NYU)===
+
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: A Conditional Gaussian Framework for Uncertainty Quantification, Data Assimilation and Prediction of Complex Turbulent Dynamical Systems
+
  
Abstract:
+
===April 6 Edray Goins (Purdue)===
A conditional Gaussian framework for uncertainty quantification, data assimilation and prediction of nonlinear turbulent dynamical systems will be introduced in this talk. Despite the conditional Gaussianity, the dynamics remain highly nonlinear and are able to capture strongly non-Gaussian features such as intermittency and extreme events. The conditional Gaussian structure allows efficient and analytically solvable conditional statistics that facilitates the real-time data assimilation and prediction.
+
  
The talk will include three applications of such conditional Gaussian framework. In the first part, a physics-constrained nonlinear stochastic model is developed, and is applied to predicting the Madden-Julian oscillation indices with strongly non-Gaussian intermittent features. The second part regards the state estimation and data assimilation of multiscale and turbulent ocean flows using noisy Lagrangian tracers. Rigorous analysis shows that an exponential increase in the number of tracers is required for reducing the uncertainty by a fixed amount. This indicates a practical information barrier. In the last part of the talk, an efficient statistically accurate algorithm is developed that is able to solve a rich class of high dimensional Fokker-Planck equation with strong non-Gaussian features and beat the curse of dimensions.
+
Title: Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups
 
+
===December 11: Connor Mooney (ETH Zurich)===
+
Title: Regularity vs. Singularity for Elliptic and Parabolic Systems
+
 
+
Abstract:
+
Hilbert's 19th problem asks if minimizers of &ldquo;natural&rdquo; variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDEs. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, as well as outstanding open problems. Some of this is joint work with O. Savin.
+
 
+
===December 13: Bobby Wilson (MIT)===
+
Title:  Projections in Banach Spaces and Harmonic Analysis
+
 
+
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.
+
 
+
===December 15: Roy Lederman (Princeton)===
+
Title: Inverse Problems and Unsupervised Learning with applications to Cryo-Electron Microscopy (cryo-EM)
+
 
+
Abstract:
+
Cryo-EM is an imaging technology that is revolutionizing structural biology; the Nobel Prize in Chemistry 2017 was recently awarded to Jacques Dubochet, Joachim Frank and Richard Henderson “for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution".
+
+
Cryo-electron microscopes produce a large number of very noisy two-dimensional projection images of individual frozen molecules. Unlike related methods, such as computed tomography (CT), the viewing direction of each image is unknown. The unknown directions, together with extreme levels of noise and additional technical factors, make the determination of the structure of molecules challenging.
+
+
While other methods for structure determination, such as x-ray crystallography and nuclear magnetic resonance (NMR), measure ensembles of molecules together, cryo-EM produces measurements of individual molecules. Therefore, cryo-EM could potentially be used to study mixtures of different conformations of molecules. Indeed, current algorithms have been very successful at analyzing homogeneous samples, and can recover some distinct conformations mixed in solutions, but, the determination of multiple conformations, and in particular, continuums of similar conformations (continuous heterogeneity), remains one of the open problems in cryo-EM.
+
+
I will discuss a one-dimensional discrete model problem, Heterogeneous Multireference Alignment, which captures many of the group properties and other mathematical properties of the cryo-EM problem. I will then discuss different components which we are introducing in order to address the problem of continuous heterogeneity in cryo-EM: 1. “hyper-molecules,” the mathematical formulation of truly continuously heterogeneous molecules, 2. computational and numerical tools for formulating associated priors, and 3. Bayesian algorithms for inverse problems with an unsupervised-learning component for recovering such hyper-molecules in cryo-EM.
+
 
+
===December 18: Jenny Wilson (Stanford)===
+
Title: Stability in the homology of configuration spaces
+
 
+
Abstract:
+
This talk will illustrate some patterns in the homology of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena -- relationships between unstable homology classes in different degrees -- established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius--Kupers--Randal-Williams.
+
 
+
===December 19: Alex Wright (Stanford)===
+
Title: Dynamics, geometry, and the moduli space of Riemann surfaces
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Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.
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== Spring 2018 ==
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|January 30
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| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
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|[[# TBA|  TBA  ]]
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| Jordan Ellenberg
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|-
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|February 2
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| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)
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|[[# TBA|  TBA  ]]
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| Spagnolie, Smith
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|-
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| March 16
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|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)
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|[[# TBA|  TBA  ]]
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| WIMAW
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|April 4 (Wednesday)
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| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)
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|[[# TBA|  TBA  ]]
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| Craciun
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| April 6
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| Reserved
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|[[# TBA|  TBA  ]]
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| Melanie
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| April 13
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| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)
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|[[# TBA|  TBA  ]]
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| WIMAW
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|-
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| April 25 (Wednesday)
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| Hitoshi Ishii (Waseda University) Wasow lecture
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|[[# TBA|  TBA  ]]
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| Tran
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== Spring Abstracts ==
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=== <DATE>: <PERSON> (INSTITUTION) ===
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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>
Title: <TITLE>
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Abstract: <ABSTRACT>
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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.
  
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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.
  
 
== Past Colloquia ==
 
== Past Colloquia ==

Revision as of 09:33, 5 March 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 Anne Gelb (Dartmouth) TBA WIMAW
April 4 (Wednesday) John Baez (UC Riverside) TBA Craciun
April 6 Edray Goins (Purdue) Toroidal Belyĭ Pairs, Toroidal Graphs, and their Monodromy Groups Melanie
April 13 Jill Pipher (Brown) TBA WIMAW
April 20 Xiuxiong Chen(Stony Brook University) TBA Bing Wang
April 25 (Wednesday) Hitoshi Ishii (Waseda University) Wasow lecture TBA Tran
date person (institution) TBA hosting faculty
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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.

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.

Past Colloquia

Blank Colloquia

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012