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


== Spring 2018 ==
The calendar for spring 2019 can be found [[Colloquia/Spring2019|here]].
 
==Spring 2019==


{| cellpadding="8"
{| cellpadding="8"
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!align="left" | host(s)
!align="left" | host(s)
|-
|-
|January 29 (Monday)
|Jan 25
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW
|[[#January 29 Li Chao (Columbia)|  Elliptic curves and Goldfeld's conjecture ]]
|[[#Beata Randrianantoanina (Miami University Ohio) |  Some nonlinear problems in the geometry of Banach spaces and their applications ]]
| Jordan Ellenberg
| Tullia Dymarz
|
|
|-
|-
|February 2 (Room: 911)
|Jan 30 '''Wednesday'''
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
|[[#February 2 Thomas Fai (Harvard)|  The Lubricated Immersed Boundary Method ]]
|[[#Lillian Pierce (Duke University) |  Short character sums  ]]
| Spagnolie, Smith
| Boston and Street
|
|
|-
|-
|February 5 (Monday, Room: 911)
|Jan 31 '''Thursday'''
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations  ]]
| Ellenberg, Gurevitch
| Street
|
|
|-
|-
|February 6 (Tuesday 2 pm, Room 911)
|Feb 1
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
| [https://services.math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke University)
|[[#February 6 Alex Lubotzky (Hebrew University)|  Groups' approximation, stability and high dimensional expanders ]]
|[[# TBA|  TBA  ]]
| Ellenberg, Gurevitch
| Qin
|
|
|-
|-
|February 9
|Feb 5 '''Tuesday'''
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)
|[[#February 9 Wes Pegden (CMU)|  The fractal nature of the Abelian Sandpile ]]
|[[# TBA|  TBA  ]]
| Roch
| Denisov
|
|
|-
|-
|March 2
|Feb 8
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]
|[[#Aaron Naber (Northwestern) |   A structure theory for spaces with lower Ricci curvature bounds  ]]
| Caldararu
| Street
|
|
|-
|-
| March 16  (Room: 911)
|Feb 15
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)
|  
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]
|[[# TBA| TBA  ]]
| WIMAW
|  
|
|
|-
|-
|April 5 (Thursday, Room: 911)
|Feb 22
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)
|[[#April 5 John Baez (UC Riverside)|  Monoidal categories of networks  ]]
|[[# TBA|  TBA ]]
| Craciun
| Erman and Corey
|
|-
| April 6
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)
|[[# Edray Goins|  Toroidal Belyĭ Pairs, Toroidal Graphs, and their Monodromy Groups  ]]
| Melanie
|
|-
| April 13 (911 Van Vleck)
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)
|[[#April 13, Jill Pipher, Brown University|  Mathematical ideas in cryptography  ]]
| WIMAW
|
|-
| April 16 (Monday)
| [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 ]]
| Erman, Sam
|
|-
| April 25 (Wednesday, Room: 911)
| [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  ]]
| Tran
|
|-
| May 1 (Tuesday, 4:30pm, Room: B102 VV)
| [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
|
|-
| May 2 (Wednesday, 3pm, Room: B325 VV)
| [https://math.uchicago.edu/~aneves/ Andre Neves] (University of Chicago and Imperial College London) Distinguished lecture
|[[#May 2, Andre Neves (University Chicago and Imperial College London)|  Wow, so many minimal surfaces! (Part II)  ]]
| Lu Wang
|
|
|-
|-
| May 4
|March 4
| [http://math.mit.edu/~cohn/ Henry Cohn] (Microsoft Research and MIT)
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture
|[[# TBA| The sphere packing problem in dimensions 8 and 24  ]]
|[[# TBA| TBA ]]
| Ellenberg
| Kim
|
|
|-
|-
|date
|March 8
| person (institution)
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| hosting faculty
| Erman
|
|
|-
|-
|date
|March 15
| person (institution)
| Maksym Radziwill (Caltech)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| hosting faculty
| Marshall
|
|
|-
|-
|date
|March 29
| person (institution)
| Jennifer Park (OSU)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| hosting faculty
| Marshall
|
|
|-
|-
|date
|April 5
| person (institution)
| Ju-Lee Kim (MIT)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| hosting faculty
| Gurevich
|
|
|-
|-
|date
|April 12
| person (institution)
| Evitar Procaccia (TAMU)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| hosting faculty
| Gurevich
|
|
|-
|-
|date
|April 19
| person (institution)
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| hosting faculty
| Jean-Luc
|
|
|-
|-
|date
|April 26
| person (institution)
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| hosting faculty
| WIMAW
|
|
|-
|-
|date
|May 3
| person (institution)
| Tomasz Przebinda (Oklahoma)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| hosting faculty
| Gurevich
|
|
|}
|}


== Spring Abstracts ==
== Abstracts ==


===Beata Randrianantoanina (Miami University Ohio)===


===January 29 Li Chao (Columbia)===
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.


Title: Elliptic curves and Goldfeld's conjecture
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.


Abstract:
===Lillian Pierce (Duke University)===
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: Short character sums


Title: The Lubricated Immersed Boundary Method
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.


Abstract:
===Dean Baskin (Texas A&M)===
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: Radiation fields for wave equations


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


Abstract:
===Aaron Naber (Northwestern)===


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


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.  
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.


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.


== Past Colloquia ==


===February 6 Alex Lubotzky (Hebrew University)===
[[Colloquia/Blank|Blank]]


Title:  Groups' approximation, stability and high dimensional expanders
[[Colloquia/Fall2018|Fall 2018]]


Abstract:
[[Colloquia/Spring2018|Spring 2018]]
 
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 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>
 
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.
 
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 joint work with Irie, Marques, and Song.
 
== Future Colloquia ==
[[Colloquia/Blank|Fall 2018]]
 
== Past Colloquia ==
 
[[Colloquia/Blank|Blank]]


[[Colloquia/Fall2017|Fall 2017]]
[[Colloquia/Fall2017|Fall 2017]]

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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