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= Mathematics Colloquium =
= Mathematics Colloquium =


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


[[Colloquia/Fall 2014 | Tentative schedule for Fall 2014]]
The calendar for spring 2019 can be found [[Colloquia/Spring2019|here]].


== Spring 2014 ==
==Spring 2019==


{| cellpadding="8"
{| cellpadding="8"
!align="left" | date
!align="left" | date  
!align="left" | speaker
!align="left" | speaker
!align="left" | title
!align="left" | title
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|'''Mon, Jan 6, 4PM'''
|Jan 25
|[http://www-bcf.usc.edu/~lauda/Aaron_Laudas_Page/Home.html Aaron Lauda] (USC)  
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW
|[[Colloquia#January 6: Aaron Lauda (USC) | An introduction to diagrammatic categorification]]
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications  ]]
|Caldararu
| Tullia Dymarz
|
|-
|-
|'''Wed, Jan 8, 4PM'''
|Jan 30 '''Wednesday'''
|[http://www2.math.umd.edu/~kmelnick/ Karin Melnick] (Maryland)  
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
|[[Colloquia#January 8: Karin Melnick (Maryland) | Normal forms for local flows on parabolic geometries]]
|[[#Lillian Pierce (Duke University) | Short character sums  ]]
|Kent
| Boston and Street
|
|-
|-
|Jan 10, 4PM
|Jan 31 '''Thursday'''
|[http://users.math.yale.edu/~yd82/ Yen Do] (Yale)  
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)
|Convergence of Fourier series and multilinear analysis
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations  ]]
|Denissov
| Street
|
|-
|-
|'''Mon, Jan 13, 4pm'''
|Feb 1
|[http://math.stanford.edu/~wangyi/ Yi Wang] (Stanford)
| [https://services.math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke University)
|Isoperimetric Inequality and Q-curvature
|[[# TBA|  TBA  ]]
|Viaclovsky
| Qin
|
|-
|-
|'''Wen, Jan 15, 4pm'''
|Feb 5 '''Tuesday'''
|[http://www.maths.ox.ac.uk/people/profiles/wei.xiang Wei Xiang] (University of Oxford)
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)
|[[Colloquia#January 15: Wei Xiang (University of Oxford) |Conservation Laws and Shock Waves]]
|[[# TBA| TBA  ]]
|Bolotin
| Denisov
|
|-
|-
|'''Fri, Jan 17, 2:25PM, VV901'''
|Feb 8
|[http://www.math.dartmouth.edu/~gillmana/ Adrianna Gillman] (Dartmouth)  
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)
|Fast direct solvers for linear partial differential equations
|[[#Aaron Naber (Northwestern) |  A structure theory for spaces with lower Ricci curvature bounds  ]]
|Thiffeault
| Street
|
|-
|-
|'''Thu, Jan 23, 2:25, VV901'''
|Feb 15
|[http://www.stat.berkeley.edu/~mshkolni/ Mykhaylo Shkolnikov] (Berkeley)
|  
|[[Colloquia#Thur, Jan 23: Mykhaylo Shkolnikov (Berkeley) | Intertwinings, wave equations and growth models]]
|[[# TBA| TBA  ]]
|Seppalainen
|  
|
|-
|-
|Jan 24
|Feb 22
|[http://www.yanivplan.com/ Yaniv Plan] (Michigan)
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)
|[http://www.math.wisc.edu/wiki/index.php/Applied/ACMS/absS14#Yaniv_Plan_.28Michigan.29 Low-dimensionality in mathematical signal processing]
|[[# TBA|  TBA  ]]
|Thiffeault
| Erman and Corey
|
|-
|-
|Jan 31
|March 4
|[http://csi.usc.edu/~ubli/ubli.html Urbashi Mitra] (USC)
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture
|Underwater Networks: A Convergence of  Communications, Control and Sensing
|[[# TBA| TBA ]]
|Gurevich
| Kim
|
|-
|-
|Feb 7
|March 8
|David Treumann (Boston College)
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)
|Functoriality, Smith theory, and the Brauer homomorphism
|[[# TBA|  TBA  ]]
|Street
| Erman
|
|-
|-
|Feb 14
|March 15
|[http://www.tc.columbia.edu/academics/index.htm?facid=apk16 Alexander Karp] (Columbia Teacher's College)
| Maksym Radziwill (Caltech)
|History of Mathematics Education as a Research Field and as Magistra Vitae
|[[# TBA|  TBA  ]]
|Kiselev
| Marshall
|
|-
|-
|Feb 21
|March 29
|[http://www.math.uci.edu/~szhitomi/ Svetlana Jitomirskaya] (UC-Irvine)
| Jennifer Park (OSU)
|Analytic quasiperiodic cocycles
|[[# TBA|  TBA  ]]
|Kiselev
| Marshall
|
|-
|-
|Feb 28
|April 5
|[http://math.nyu.edu/faculty/shelley/ Michael Shelley] (Courant)
| Ju-Lee Kim (MIT)
|[[Colloquia#February 28: Michael Shelley (Courant) | Mathematical models of soft active materials]]
|[[# TBA| TBA  ]]
|Spagnolie
| Gurevich
|
|-
|-
|March 7
|April 12
|[http://www.math.northwestern.edu/people/facultyProfiles/steve.zelditch.html Steve Zelditch] (Northwestern)
| Evitar Procaccia (TAMU)
|[[Colloquia#March 7: Steve Zelditch (Northwestern) | Shapes and sizes of eigenfunctions]]
|[[# TBA| TBA  ]]
|Seeger
| Gurevich
|
|-
|-
|March 14
|April 19
|[http://www.math.brown.edu/~res/ Richard Schwartz] (Brown)
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)
|[[Colloquia#March 14: Richard Schwartz (Brown) |The projective heat map on pentagons]]
|[[# TBA| TBA  ]]
|Mari-Beffa
| Jean-Luc
|
|-
|-
|<strike>March 21</strike>
|April 26
|'''Spring Break'''
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)
|No Colloquium
|[[# TBA|  TBA  ]]
| WIMAW
|
|
|-
|-
|'''March 26, 7pm, WID'''
|May 3
|[https://www.dpmms.cam.ac.uk/people/t.tokieda/ Tadashi Tokieda] (Cambridge)
| Tomasz Przebinda (Oklahoma)
|[http://c4.discovery.wisc.edu/events/lectures/toymodels/ Toy models]
|[[# TBA|  TBA  ]]
|Thiffeault (C4 von Neumann Public Lecture)
| Gurevich
|-
|March 28
|Cancelled
|
|
|
|-
|April 4
|[http://matthewkahle.org/ Matthew Kahle] (OSU)
|Recent progress in random topology
|Dymarz
|-
|April 11
|[http://www.cs.uchicago.edu/people/risi Risi Kondor] (Chicago)
|Multiresolution Matrix Factorization
|Gurevich
|-
|April 18 (Wasow Lecture)
|[http://mathnt.mat.jhu.edu/sogge/ Christopher Sogge] (Johns Hopkins)
|[[Colloquia#April 18: Christopher Sogge (Johns Hopkins) |Focal points and sup-norms of eigenfunctions]]
|Seeger
|-
|April 25
|[http://www.charlesdoran.net Charles Doran](University of Alberta)
|[[Colloquia#April 25: Charles Doran (University of Alberta) |The Mathematics of Supersymmetry: Graphs, Codes, and Super-Curves]]
|Song
|-
|'''Tuesday, April 29''' (Distinguished Lecture)
|[http://www.msri.org/people/staff/de/ David Eisenbud](Berkeley)
|Matrix factorizations old and new, I
|Erman
|-
|'''Wednesday, April 30''' (Distinguished Lecture)
|[http://www.msri.org/people/staff/de/ David Eisenbud](Berkeley)
|Matrix factorizations old and new, II
|Erman
|-
|May 2
|[http://www.stat.uchicago.edu/~lekheng/ Lek-Heng Lim] (Chicago)
|Hypermatrices
|Boston
|-
|May 9
|[http://www.ma.utexas.edu/users/rward/ Rachel Ward] (UT Austin)
|Sampling theorems for efficient dimensionality reduction and sparse recovery
|WIMAW
|}
|}


== Abstracts ==
== Abstracts ==


===January 6: Aaron Lauda (USC)===
===Beata Randrianantoanina (Miami University Ohio)===


''An introduction to diagrammatic categorification''
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.


Categorification seeks to reveal a hidden layer in mathematical
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.
structures. Often the resulting structures can be combinatorially
complex objects making them difficult to study. One method of
overcoming this difficulty, that has proven very successful, is to
encode the categorification into a diagrammatic calculus that makes
computations simple and intuitive.


In this talk I will review some of the original considerations that
===Lillian Pierce (Duke University)===
led to the categorification philosophy.  We will examine how the
diagrammatic perspective has helped to produce new categorifications
having  profound applications to algebra, representation theory, and
low-dimensional topology.


===January 8: Karin Melnick (Maryland)===
Title: Short character sums


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


The exponential map in Riemannian geometry conjugates the differential of an isometry at a point with the action of the isometry near the point. It thus provides a linear normal form for all isometries fixing a point. Conformal transformations are not linearizable in general. I will discuss a suite of normal forms theorems in conformal geometry and, more generally, for parabolic geometries, a rich family of geometric structures of which conformal, projective, and CR structures are examples.
===Dean Baskin (Texas A&M)===


===January 10, 4PM: Yen Do (Yale)===
Title: Radiation fields for wave equations


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


Almost everywhere convergence of the Fourier series of square
===Aaron Naber (Northwestern)===
integrable functions was first proved by Lennart Carleson in 1966, and
the proof has lead to deep developments in various multilinear settings.
In this talk I would like to introduce a brief history of the subject
and sketch some recent developments, some of these involve my joint
works with collaborators.


===Mon, January 13: Yi Wang (Stanford)===
Title: A structure theory for spaces with lower Ricci curvature bounds.


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


A well-known question in differential geometry is to prove the
isoperimetric inequality under intrinsic curvature conditions. In
dimension 2, the isoperimetric inequality is controlled by the integral of
the positive part of the Gaussian curvature. In my recent work, I prove
that on simply connected conformally flat manifolds of higher dimensions,
the role of the Gaussian curvature can be replaced by the Branson's
Q-curvature. The isoperimetric inequality is valid if the integral of the
Q-curvature is below a sharp threshold. Moreover, the isoperimetric
constant depends only on the integrals of the Q-curvature. The proof
relies on the theory of $A_p$ weights in harmonic analysis.
===January 15: Wei Xiang (University of Oxford)===
''Conservation Laws and Shock Waves''
The study of continuum physics gave birth to the theory of quasilinear
systems in divergence form, commonly called conservation laws. In this
talk, conservation laws, the Euler equations, and the definition of the
corresponding weak solutions will be introduced first. Then a short history
of the studying of conservation laws and shock waves will be given. Finally
I would like to present two of our current research projects. One is on the
mathematical analysis of shock diffraction by convex cornered wedges, and
the other one is on the validation of weakly nonlinear geometric optics for
entropy solutions of nonlinear hyperbolic systems of conservation laws.
Fri, Jan 17, 2:25PM, VV901 Adrianna Gillman (Dartmouth) Fast direct solvers for linear partial differential equations
===Fri, Jan 17: Adrianna Gillman (Dartmouth) ===
''Fast direct solvers for linear partial differential equations''
The cost of solving a large linear system often determines what can and cannot be modeled computationally in many areas of science and engineering. Unlike Gaussian elimination which scales cubically with the respect to the number of unknowns, fast direct solvers construct an inverse of a linear in system with a cost that scales linearly or nearly linearly. The fast direct solvers presented in this talk are designed for the linear systems arising from the discretization of linear partial differential equations. These methods are more robust, versatile and stable than iterative schemes. Since an inverse is computed, additional right-hand sides can be processed rapidly. The talk will give the audience a brief introduction to the core ideas, an overview of recent advancements, and it will conclude with a sampling of challenging application examples including the scattering of waves.
===Thur, Jan 23: Mykhaylo Shkolnikov (Berkeley) ===
''Intertwinings, wave equations and growth models''
We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal.
===Jan 24: Yaniv Plan (Michigan) ===
''Low-dimensionality in mathematical signal processing''
Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. This conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from MRI to quantum state tomography. It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? We explore this question under modern models of low-dimensionality and measurement acquisition.
===Thur, Jan 30: Urbashi Mitra (USC) ===
''Underwater Networks: A Convergence of  Communications, Control and Sensing''
The oceans cover 71% of the earth’s surface and represent one of the least explored frontiers, yet the oceans are integral to climate regulation, nutrient production, oil retrieval and transportation. Future scientific and technological efforts to achieve better understanding of oceans and water-related applications will rely heavily on our ability to communicate reliably between instruments, vehicles (manned and unmanned), human operators, platforms and sensors of all types.  Underwater acoustic communication techniques have not reached the same maturity as those for terrestrial radio communications and present some unique opportunities for new developments in information and communication theories.  Key features of underwater acoustic communication channels are examined: slow speed of propagation, significant delay spreads, sparse multi-path, time-variation and range-dependent available bandwidth.  Another unique feature of underwater networks is that the cost of communication, sensing and control are often comparable resulting in new tradeoffs between these activities.    We examine some new results (with implications wider than underwater systems) in  channel identifiability,  communicating over channels with state and cooperative game theory motivated by the underwater network application.
===Feb 7: David Treumann (Boston College) ===
''Functoriality, Smith theory, and the Brauer homomorphism''
Smith theory is a technique for relating the mod p homologies of X and of the fixed points of X by an automorphism of order p.  I will discuss how, in the setting of locally symmetric spaces, it provides an easy method (no trace formula) for lifting mod p automorphic forms from G^{sigma} to G, where G is an arithmetic group and sigma is an automorphism of G of order p.  This lift is compatible with Hecke actions via an analog of the Brauer homomorphism from modular representation theory, and is often compatible with a homomorphism of L-groups on the Galois side.  The talk is based on joint work with Akshay Venkatesh.  I hope understanding the talk will require less number theory background than understanding the abstract.
===Feb 14: Alexander Karp (Columbia Teacher's College) ===
''History of Mathematics Education as a Research Field and as Magistra Vitae''
The presentation will be based on the experience of putting together and editing the Handbook
on the History of Mathematics Education, which will be published by Springer in the near future.
This volume, which was prepared by a large group of researchers from different countries,
contains the first systematic account of the history of the development of mathematics education
in the whole world (and not just in some particular country or region). The editing of such a
book gave rise to thoughts about the methodology of research in this field, and also about what
constitutes an object of such research. These are the thoughts that the presenter intends to share
with his audience. From them, it is natural to pass to an analysis of the current situation and how
it might develop.
===Feb 21: Svetlana Jitomirskaya (UC-Irvine)===
''Analytic quasiperiodic cocycles''
Analytic quasiperiodic matrix cocycles is a simple dynamical
system, where analytic and dynamical properties are related in an
unexpected and remarkable way. We will focus on this relation, leading to
a new approach to the proof of joint continuity of Lyapunov exponents in
frequency and cocycle, at irrational frequency, first proved for SL(2,C)
cocycles in Bourgain-Jitom., 2002. The approach is powerful enough to
handle singular and multidimensional cocycles, thus establishing the above
continuity in full generality. This has important consequences including
a dense open version of Bochi-Viana theorem in this setting, with a
completely different underlying mechanism of the proof. A large part of
the talk  is a report on a joint work with A. Avila and C. Sadel.
===February 28: Michael Shelley (Courant)===
''Mathematical models of soft active materials''
Soft materials that have an "active" microstructure are important examples of so-called active matter.  Examples include suspensions of motile microorganisms or particles, "active gels" made up of actin and myosin, and suspensions of microtubules cross-linked by motile motor-proteins. These nonequilibrium materials can have unique mechanical properties and organization, show spontaneous activity-driven flows, and are part of self-assembled structures such as the cellular cortex and mitotic spindle. I will discuss the nature and modeling of these materials, focusing on fluids driven by "active stresses" generated by swimming, motor-protein activity, and surface tension gradients. Amusingly, the latter reveals a new class of fluid flow singularities and an unexpected connection to the Keller-Segel equation.


== Past Colloquia ==


===March 7: Steve Zelditch (Northwestern)===
[[Colloquia/Blank|Blank]]
''Shapes and sizes of eigenfunction''


Eigenfunctions of the Laplacian (or Schroedinger operators) arise as stationary states in quantum mechanics. They are not apriori geometric
[[Colloquia/Fall2018|Fall 2018]]
objects but we would like to relate the nodal (zero) sets and Lp norms of eigenfunctions to the geometry of geometrics. I will explain what is
known (and unknown) and norms and nodal sets of eigenfunctions. No prior knowledge of quantum mechanics is assumed.


===March 14: Richard Schwartz (Brown)===
[[Colloquia/Spring2018|Spring 2018]]
''The projective heat map on pentagons''


In this talk I'll describe several maps defined on
[[Colloquia/Fall2017|Fall 2017]]
the space of polygons.  These maps are described in terms
of simple straight-line constructions, and are therefore
natural with respect to projective geometry.  One of them,
the pentagram map, is now known to be a discrete completely
integrable system. I'll concentrate on a variant of the
pentagram map, which behaves somewhat like heat flow on convex
polygons but which does crazy things to non-convex polygons.
I'll sketch a computer-assisted analysis of what happens
for pentagons.  I'll illustrate the talk with computer demos.


===April 4: Matthew Kahle (OSU) ===
[[Colloquia/Spring2017|Spring 2017]]
"Recent progress in random topology"


The study of random topological spaces: manifolds, simplicial
[[Archived Fall 2016 Colloquia|Fall 2016]]
complexes, knots, groups, has received a lot of attention in recent
years. This talk will mostly focus on random simplicial complexes, and
especially on a certain kind of topological phase transition, where
the probability that that a certain homology group is trivial passes
from 0 to 1 within a narrow window. The archetypal result in this area
is the Erdős–Rényi theorem, which characterizes the threshold edge
probability where the random graph becomes connected.


One recent breakthrough has been in the application of "Garland's
[[Colloquia/Spring2016|Spring 2016]]
method", which allows one to prove homology-vanishing theorems by
showing that certain Laplacians have large spectral gaps. This reduces
problems in random topology to understanding eigenvalues of certain
random matrices, and the method has been surprisingly successful.


This talk is intended for a broad mathematical audience, and I will
[[Colloquia/Fall2015|Fall 2015]]
not assume any particular prerequisites in probability or topology.
Part of this is joint work with Christopher Hoffman and Elliot
Paquette.
===April 11: Risi Kondor (Chicago) ===
''Multiresolution Matrix Factorization''


Matrices that appear in modern data analysis and machine learning problems often exhibit complex
[[Colloquia/Spring2014|Spring 2015]]
hierarchical structure, which goes beyond what can be uncovered by traditional linear algebra tools,
such as eigendecomposition. In this talk I describe a new notion of matrix factorization inspired by
multiresolution analysis that can capture structure in matrices at multiple different scales.
The resulting Multiresolution Matrix Factorizations (MMFs) not only provide a
wavelet basis for sparse approximation, but can also be used for matrix compression and as a prior
for matrix completion. The work presented in this talk is joint with Nedelina Teneva and Vikas Garg.


 
[[Colloquia/Fall2014|Fall 2014]]
===April 18: Christopher Sogge (Johns Hopkins) ===
''Focal points and sup-norms of eigenfunctions''
 
If (M,g)  is a compact real analytic Riemannian manifold, we give a necessary and
sufficient condition for there to be a sequence of quasimodes saturating sup-norm 
estimates.  The condition is that there exists a self-focal point x_0\in M  for the
geodesic flow at which the associated Perron-Frobenius operator
U: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M)  has a nontrivial invariant function.  The proof is
based on von Neumann's ergodic theorem and stationary phase.  This is joint work with
Steve Zelditch.
 
===April 25: Charles Doran (University of Alberta) ===
''The Mathematics of Supersymmetry: Graphs, Codes, and Super-Curves''
 
In physics, supersymmetry is a pairing between bosons and fermions appearing in theories of subatomic particles. One may study supersymmetry mathematically by using Adinkras, which are graphs with vertices representing the particles in a supersymmetric theory and edges corresponding to the supersymmetry pairings. In combinatorial terms, Adinkras are N-regular, edge N-colored bipartite graphs with signs assigned to the edges and heights assigned to the vertices, subject to certain conditions. We will see how to capture some of the structure of an Adinkra using binary linear error-correcting codes, and all of it using a very special case of a geometric construction due to Grothendieck. The talk is designed to be accessible to an undergraduate audience.
 
===April 29 and 30: David Eisenbud (University of California, Berkeley and MSRI) ===
''Matrix factorizations old and new''
 
You cannot factor f=xy-z^2 nontrivially as a product of power series, but you can factor f times a 2x2 identity matrix as the product of the matrices
<br>
<div align=center>
    <table class="matrix">
        <tr>
            <td>x</td>
            <td>z</td>
            <td>    </td>
            <td>and </td>
            <td>    </td>
            <td>y</td>
            <td>-z</td>
        </tr>
        <tr>
            <td>z</td>
            <td>y</td>
            <td>    </td>
            <td></td>
            <td>    </td>
            <td>-z</td>
            <td>x.</td>
        </tr>
    </table>
</div>
It turns out that any power series of order at least has a "matrix factorization" in this sense, and that this is the key to understanding the simplest infinite free resolutions, as I proved in the 1980s. Such matrix factorizations have since proven useful in many contexts. Recently Irena Peeva and I have discovered what I believe is the natural extension of this
idea to systems of polynomials called complete intersections. I'll explain some of the old theory and sketch the new development.
 
The first talk will be aimed at a general audience and the second talk will cover some of the recent advances.
 
===May 2: Lek-Heng Lim (Chicago)===
'' Hypermatrices.''
 
 
This talk is intended for those who, like the speaker, have at some point
wondered whether there is a theory of three- or higher-dimensional
matrices that parallels matrix theory. We will explain why a d-dimensional
hypermatrix is related to but not quite the same as an order-d tensor.
 
We discuss how notions like rank, norm, determinant, eigen and singular
values may be generalized to hypermatrices. We will see that, far from
being artificial constructs, these notions have appeared naturally in a
wide range of applications and can be enormously useful. We will examine
several examples, highlighting three from the speaker's recent work: (i)
rank of 3-hypermatrices and blind source separation in signal processing,
(ii) positive definiteness of 6-hypermatrices and self-concordance in
convex optimization, (iii) nuclear norm of 3-hypermatrices and bipartite
separability in quantum computing.
 
(i) is joint work with Pierre Comon and (iii) is joint work with Shmuel
Friedland
 
===May 9: Rachel Ward (UT Austin)===
'' Sampling theorems for efficient dimensionality reduction and sparse recovery.''
 
Embedding high-dimensional data sets into subspaces of much lower dimension is important for reducing storage cost and speeding up computation in several applications, including numerical linear algebra, manifold learning, and theoretical computer science. Moreover, central to the relatively new field of compressive sensing, if the original data set is known to be sparsely representable in a given basis, then it is possible to efficiently 'invert’ a random dimension-reducing map to recover the high-dimensional data via e.g. l1-minimization. We will survey recent results in these areas, and then show how near-equivalences between fundamental concepts such as restricted isometries and Johnson-Lindenstrauss embeddings can be used to leverage results in one domain and apply to another. Finally, we discuss how these and other recent results for structured random matrices can be used to derive sampling strategies in various settings, from low-rank matrix completion to function interpolation.
 
== Past Colloquia ==


[[Colloquia/Spring2014|Spring 2014]]
[[Colloquia/Spring2014|Spring 2014]]

Latest revision as of 14:43, 24 January 2019

Mathematics Colloquium

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

The calendar for spring 2019 can be found here.

Spring 2019

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

Abstracts

Beata Randrianantoanina (Miami University Ohio)

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

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

Lillian Pierce (Duke University)

Title: Short character sums

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

Dean Baskin (Texas A&M)

Title: Radiation fields for wave equations

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

Aaron Naber (Northwestern)

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

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


Past Colloquia

Blank

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012

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