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


==Fall 2017==
==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)
|-
|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  ]]
| Yang
|
|-
|September 15
|
|[[#|  ]]
|
|
|
|-
|September 22, '''9th floor'''
| Jaeyoung Byeon (KAIST)
|[[#September 22: Jaeyoung Byeon (KAIST) |  Patterns formation for elliptic systems with large interaction forces  ]]
| Rabinowitz & 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 ]]
| 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 ]]
| Max
|
|-
|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  ]]
| Stovall, Seeger
|
|-
|We, November 1,
|[http://pages.iu.edu/~shaoguo/  Shaoming Guo] (Indiana)
|[[#November 1: Shaoming Guo (Indiana)|  Parsell-Vinogradov systems in higher dimensions  ]]
|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,
|Jan 25
| [http://www.math.harvard.edu/~tcollins/homepage.html  Tristan Collins] (Harvard)
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW
|[[#November 27:Tristan Collins (Harvard)|  The J-equation and stability ]]
|[[#Beata Randrianantoanina (Miami University Ohio) |  Some nonlinear problems in the geometry of Banach spaces and their applications ]]
|Sean Paul
| Tullia Dymarz
|
|
|
|-
|-
|December 5 (Tuesday)
|Jan 30 '''Wednesday'''
| [http://web.sas.upenn.edu/rhynd/ Ryan Hynd] (U Penn)
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
|[[#December 5: Ryan Hynd (U Penn)|  TBA  ]]
|[[#Lillian Pierce (Duke University) |  Short character sums  ]]
|Sigurd Angenent
| Boston and Street
|
|
|
|-
|-
|December 11 (Monday)
|Jan 31 '''Thursday'''
| [https://people.math.ethz.ch/~mooneyc/ Connor Mooney] (ETH Zurich)
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)
|[[#December 11: Connor Mooney (ETH Zurich)|  Regularity vs. Singularity for Elliptic and Parabolic Systems]]
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]
|Sigurd Angenent
| Street
|
|
|-
|-
|December 13 (Wednesday)
|Feb 1
| [http://math.mit.edu/~blwilson/ Bobby Wilson] (MIT)
| [https://services.math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke University)
|[[#December 13: Bobby Wilson (MIT) | TBA ]]
|Andreas Seeger
|
|-
|December 15 (Friday)
| [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)
| [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)
| [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 ==
=== 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.
 
 
=== September 22: Jaeyoung Byeon (KAIST) ===
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.
 
===October 6: Jonathan Hauenstein (Notre Dame) ===
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.
 
===October 13: Tomoko Kitagawa (Berkeley) ===
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?
 
 
 
===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).
 
===October 27: Stefanie Petermichl (Toulouse)===
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
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:
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)===
Title: Recent progress in Min-Max Theory
 
Abstract:
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)===
Title: The equations defining curves and moduli spaces
 
Abstract:
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.
 
 
===November 27:Tristan Collins (Harvard)===
Titile: 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 point.  The critical points of the J-functional are described by a fully-nonlinear PDE called the J-equation.  I will discuss some recent progress on this problem, and indicate the role of algebraic geometry in proving estimates for the J-equation.
 
===December 5: Ryan Hynd (U Penn)===
Title: TBA.
 
===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 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
 
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.
 
== Spring 2018 ==
 
{| cellpadding="8"
!align="left" | date 
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
| March 16
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| WIMAW
| Qin
|
|
|-
|-
|April 4 (Wednesday)
|Feb 5 '''Tuesday'''
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| Craciun
| Denisov
|
|
|-
|-
| April 6
|Feb 8
| Reserved
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)
|[[# TBA| TBA ]]
|[[#Aaron Naber (Northwestern) |   A structure theory for spaces with lower Ricci curvature bounds ]]
| Melanie
| Street
|
|
|-
|-
| April 13
|Feb 15
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)
|  
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| WIMAW
|  
|
|
|-
|-
| April 25 (Wednesday)
|Feb 22
| Hitoshi Ishii (Waseda University) Wasow lecture
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| Tran
| Erman and Corey
|
|
|-
|-
|date
|March 4
| person (institution)
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture
|[[# TBA| TBA ]]
|[[# TBA| TBA ]]
| hosting faculty
| 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 ==


=== <DATE>: <PERSON> (INSTITUTION) ===
===Beata Randrianantoanina (Miami University Ohio)===
Title: <TITLE>


Abstract: <ABSTRACT>
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 ==
== Past Colloquia ==


[[Colloquia/Blank|Blank Colloquia]]
[[Colloquia/Blank|Blank]]
 
[[Colloquia/Fall2018|Fall 2018]]
 
[[Colloquia/Spring2018|Spring 2018]]
 
[[Colloquia/Fall2017|Fall 2017]]


[[Colloquia/Spring2017|Spring 2017]]
[[Colloquia/Spring2017|Spring 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

Retrieved from "https://wiki.math.wisc.edu/index.php?title=Colloquia/Fall18&oldid=16709"