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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]].
 
==Spring 2019==


== Fall 2016  ==
 
{| cellpadding="8"
{| cellpadding="8"
!align="left" | date   
!align="left" | date   
Line 15: Line 13:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 9
|Jan 25
|  
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW
|[[#  |    ]]
|[[#Beata Randrianantoanina (Miami University Ohio) |  Some nonlinear problems in the geometry of Banach spaces and their applications ]]
|
| Tullia Dymarz
|
|
|-
|-
|September 16
|Jan 30 '''Wednesday'''
|[http://www.math.cmu.edu/~ploh/ Po-Shen Loh] (CMU)
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
|Directed paths: from Ramsey to Pseudorandomness
|[[#Lillian Pierce (Duke University) |  Short character sums  ]]
|Ellenberg
| Boston and Street
|
|
|-
|-
|September 23
|Jan 31 '''Thursday'''
| [http://www.math.wisc.edu/~craciun/ Gheorghe Craciun] (UW-Madison)
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)
|Toric Differential Inclusions and a Proof of the Global Attractor Conjecture
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations  ]]
| Street
| Street
| 
|[[#  |    ]]
| 
|-
|September 30
|[http://math.uga.edu/~magyar/ Akos Magyar]  (University of Georgia)
|Geometric Ramsey theory
| Cook
|
|
|-
|-
|October 7
|Feb 1
|  
| [https://services.math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke University)
|[[# |   ]]
|[[# TBA| TBA  ]]
|
| Qin
|
|
|-
|-
|October 14
|Feb 5 '''Tuesday'''
| [https://www.math.lsu.edu/~llong/ Ling Long] (LSU)
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)
|Hypergeometric functions over finite fields
|[[# TBA|  TBA  ]]
| Yang
| Denisov
|
|
|-
|-
|October 21
|Feb 8
|'''No colloquium this week'''
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)
|[[#  |    ]]
|[[#Aaron Naber (Northwestern) |  A structure theory for spaces with lower Ricci curvature bounds ]]
|
| Street
|
|
|-
|-
|'''Tuesday, October 25, 9th floor'''
|Feb 15
|[http://users.math.yale.edu/users/steinerberger/ Stefan Steinerberger] (Yale)
|  
|Three Miracles in Analysis
|[[# TBA|  TBA  ]]
|Seeger
|  
|
|
|-
|-
|October 28, 9th floor
|Feb 22
| [http://order.ph.utexas.edu/people/Reichl.htm Linda Reichl] (UT Austin)
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)
|Microscopic hydrodynamic modes in a binary mixture
|[[# TBA|  TBA  ]]
|Minh-Binh Tran
| Erman and Corey
|
|
|-
|-
|'''Monday, October 31, B239'''
|March 4
| [https://math.berkeley.edu/~kpmann/ Kathryn Mann] (Berkeley)
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture
|Groups acting on the circle
|[[# TBA| TBA ]]
|Smith
| Kim
|
|
|-
|-
|November 4
|March 8
|
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)
|
|[[# TBA|  TBA  ]]
|  
| Erman
|
|
|-
|-
|'''Monday, November 7 at 4:30, 9th floor''' ([http://www.ams.org/meetings/lectures/maclaurin-lectures AMS Maclaurin lecture])
|March 15
| [http://www.massey.ac.nz/massey/expertise/profile.cfm?stref=339830 Gaven Martin] (New Zealand Institute for Advanced Study)
| Maksym Radziwill (Caltech)
|Siegel's problem on small volume lattices
|[[# TBA|  TBA  ]]
| Marshall
| Marshall
|
|
|-
|-
|November 11
|March 29
| Reserved for possible job talks
| Jennifer Park (OSU)
|[[# |   ]]
|[[# TBA| TBA  ]]
|
| Marshall
|
|
|-
|-
|'''Wednesday, November 16, 9th floor'''
|April 5
| [http://math.uchicago.edu/~klindsey/ Kathryn Lindsey] (U Chicago)
| Ju-Lee Kim (MIT)
|Shapes of Julia Sets
|[[# TBA|  TBA  ]]
|Michell
| Gurevich
|
|
|-
|-
|November 18, B239
|April 12
|[http://www-personal.umich.edu/~asnowden/ Andrew Snowden] (University of Michigan)
| Evitar Procaccia (TAMU)
|Recent progress in representation stability
|[[# TBA|  TBA  ]]
|Ellenberg
| Gurevich
|
|
|-
|-
|'''Monday, November 21, 9th floor'''
|April 19
|[https://www.fmi.uni-sofia.bg/fmi/logic/msoskova/index.html Mariya Soskova] (University of Wisconsin-Madison)
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)
|Definability in degree structures
|[[# TBA|  TBA  ]]
|Smith
| Jean-Luc
|
|
|-
|-
|November 25
|April 26
|  '''Thanksgiving break'''
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)
|[[# |    ]]
|[[# TBA|  TBA  ]]
|
|
|-
|December 2, 9th floor
|  [http://math.columbia.edu/~hshen/ Hao Shen] (Columbia)
|[[#Friday, December 2:  Hao Shen (Columbia) | ''Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?'']]
|Roch
|
|-
|'''Monday, December 5, B239'''
|  [https://www.math.wisc.edu/~wang/ Botong Wang] (UW Madison)
|[[#Monday, December 5: Botong Wang  (UW-Madison) | ''Enumeration of points, lines, planes, etc.'']]
|Maxim
|
|-
|December 9, B239
|  [http://math.uchicago.edu/~awbrown/ Aaron Brown] (U Chicago)
| [[#Friday, December 9: Aaron Brown (U Chicago) | ''Lattice actions and recent progress in the Zimmer program'']]
|Kent
|-
|'''Monday, December 19, B115'''
|  [http://math.uchicago.edu/~andrew.zimmer/ Andrew Zimmer] (U Chicago)
| Metric spaces of non-positive curvature and applications in several complex variables
|Gong
|}
 
== Spring 2017  ==
 
{| cellpadding="8"
!align="left" | date 
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|January 20
|  [http://math.mit.edu/~sraskin/ Sam Raskin] (MIT)
|[[# TBA |  TBA  ]]
| 
|
|-
|January 27
|Reserved for possible job talks
|[[# |    ]]
|
|
|-
|February 3
|
|[[#  |    ]]
|
|
|-
|February 6 (Wasow lecture)
| Benoit Perthame (University of Paris VI)
|[[# TBA|  TBA ]]
| Jin
|  
|-
|February 10 (WIMAW lecture)
| Alina Chertock (NC State Univ.)
|[[# |  ]]  
| WIMAW
| WIMAW
|
|
|-
|-
|February 17
|May 3
| [http://web.math.ucsb.edu/~ponce/ Gustavo Ponce] (UCSB)
| Tomasz Przebinda (Oklahoma)
|[[#  |    ]]
|[[# TBA|  TBA  ]]
| Minh-Binh Tran
| Gurevich
|
|-
|February 24
|
|[[#  |    ]]
|
|
|-
|March 3
| [http://www.math.utah.edu/~bromberg/ Ken Bromberg] (University of Utah)
|[[# |   ]]
|Dymarz
|
|-
|Tuesday, March 7, 4PM (Distinguished Lecture)
| [http://pages.iu.edu/~temam/  Roger Temam] (Indiana University)
|[[#  |    ]]
|Smith
|
|-
|Wednesday, March 8, 2:25PM
| [http://pages.iu.edu/~temam/  Roger Temam] (Indiana University)
|[[# |    ]]
|Smith
|
|-
|March 10
| '''No Colloquium'''
|[[# |    ]]
|
|
|-
|March 17
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
| TBA
| M. Matchett Wood
|
|-
|March 24
| '''Spring Break'''
|[[# |    ]]
|
|
|-
|Wednesday, March 29 (Wasow)
| [https://math.nyu.edu/faculty/serfaty/ Sylvia Serfaty] (NYU)
|[[# TBA|  TBA]]
|Tran
|
|-
|March 31
| '''No Colloquium'''
|[[# |    ]]
|
|
|-
|April 7
| [http://www.math.uiuc.edu/~schenck/ Hal Schenck]
|[[# |    ]]
|Erman
|
|
|-
|April 14
|  Wilfrid Gangbo
|[[# |    ]]
|Feldman & Tran
|
|-
|April 21
|  [http://www.math.stonybrook.edu/~mde/ Mark Andrea de Cataldo]  (Stony Brook)
|TBA
| Maxim
|
|-
|April 28
| [http://users.cms.caltech.edu/~hou/ Thomas Yizhao Hou] 
|[[# TBA|  TBA  ]]
|Li
|}
|}


== Abstracts ==
== Abstracts ==
=== September 16: Po-Shen Loh (CMU) ===
Title: Directed paths: from Ramsey to Pseudorandomness


Abstract: Starting from an innocent Ramsey-theoretic question regarding directed
===Beata Randrianantoanina (Miami University Ohio)===
paths in graphs, we discover a series of rich and surprising connections
that lead into the theory around a fundamental result in Combinatorics:
Szemeredi's Regularity Lemma, which roughly states that every graph (no
matter how large) can be well-approximated by a bounded-complexity
pseudorandom object.  Using these relationships, we prove that every
coloring of the edges of the transitive N-vertex tournament using three
colors contains a directed path of length at least sqrt(N) e^{log^* N}
which entirely avoids some color.  The unusual function log^* is the
inverse function of the tower function (iterated exponentiation).


=== September 23: Gheorghe Craciun (UW-Madison) ===
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.
Title: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture


Abstract: The Global Attractor Conjecture says that a large class of polynomial dynamical systems, called toric dynamical systems, have a globally attracting point within each linear invariant space. In particular, these polynomial dynamical systems never exhibit multistability, oscillations or chaotic dynamics.  
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.


The conjecture was formulated by Fritz Horn in the early 1970s, and is strongly related to Boltzmann's H-theorem.
===Lillian Pierce (Duke University)===


We discuss the history of this problem, including the connection between this conjecture and the Boltzmann equation. Then, we introduce toric differential inclusions, and describe how they can be used to prove this conjecture in full generality.
Title: Short character sums


=== September 30: Akos Magyar (University of Georgia) ===
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.
Title: Geometric Ramsey theory


Abstract: Initiated by Erdos, Graham, Montgomery and others in the 1970's, geometric Ramsey theory studies geometric configurations, determined up to translations, rotations and possibly dilations, which cannot be destroyed by finite partitions of Euclidean spaces. Later it was shown by ergodic and Fourier analytic methods that such results are also possible in the context of sets of positive upper density in Euclidean spaces or the integer lattice. We present a new approach, motivated by developments in arithmetic combinatorics, which provide new results as well new proofs of some classical results in this area.
===Dean Baskin (Texas A&M)===


=== October 14: Ling Long (LSU) ===
Title: Radiation fields for wave equations
Title: Hypergeometric functions over finite fields


Abstract: Hypergeometric functions are special functions with lot of
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.
symmetries. In this talk, we will introduce hypergeometric functions over finite
fields, originally due to Greene, Katz and McCarthy, in a way that is
parallel to the classical hypergeometric functions, and discuss their
properties and applications to character sums and the arithmetic of
hypergeometric abelian varieties.
This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher, and Fang-Ting Tu.


=== Tuesday, October 25, 9th floor: Stefan Steinerberger (Yale) ===
===Aaron Naber (Northwestern)===
Title: Three Miracles in Analysis


Abstract: I plan to tell three stories: all deal with new points of view on very classical objects and have in common that there is a miracle somewhere. Miracles are nice but difficult to reproduce, so in all three cases the full extent of the underlying theory is not clear and many interesting open problems await. (1) An improvement of the Poincare inequality on the Torus that encodes a lot of classical Number Theory. (2) If the Hardy-Littlewood maximal function is easy to compute, then the function is sin(x). (Here, the miracle is both in the statement and in the proof). (3) Bounding classical integral operators (Hilbert/Laplace/Fourier-transforms) in L^2 -- but this time from below (this problem originally arose in medical imaging). Here, the miracle is also known as 'Slepian's miracle' (this part is joint work with Rima Alaifari, Lillian Pierce and Roy Lederman).
Title: A structure theory for spaces with lower Ricci curvature bounds.


=== October 28: Linda Reichl (UT Austin) ===
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.
Title: Microscopic hydrodynamic modes in a binary mixture


Abstract: Expressions for propagation speeds and decay rates of hydrodynamic modes in a binary mixture can be obtained directly from spectral properties of the Boltzmann equations describing the mixture. The derivation of hydrodynamic behavior from the spectral properties of the kinetic equation provides an alternative to Chapman-Enskog theory, and removes the need for lengthy calculations of transport coefficients in the mixture. It also provides a sensitive test of the completeness of kinetic equations describing the mixture. We apply the method to a hard-sphere binary mixture and show that it gives excellent agreement with light scattering experiments on noble gas mixtures.


===Monday, October 31: Kathryn Mann (Berkeley) ===
== Past Colloquia ==
Title: Groups acting on the circle
 
Abstract:  Given a group G and a manifold M, can one describe all the actions of G on M?  This is a basic and natural question from geometric topology, but also a very difficult one -- even in the case where M is the circle, and G is a familiar, finitely generated group. 


In this talk, I’ll introduce you to the theory of groups acting on the circle, building on the perspectives of Ghys, Calegari, Goldman and others.  We'll see some tools, old and new, some open problems, and some connections between this theory and themes in topology (like foliated bundles) and dynamics. 
[[Colloquia/Blank|Blank]]


===November 7: Gaven Martin (New Zealand Institute for Advanced Study) ===
[[Colloquia/Fall2018|Fall 2018]]
Title: Siegel's problem on small volume lattices


Abstract: We outline in very general terms the history and the proof of the identification
[[Colloquia/Spring2018|Spring 2018]]
of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3
Coxeter group extended by the involution preserving the symmetry of this
diagram. This gives us the smallest regular tessellation of hyperbolic 3-space.
This solves (in three dimensions) a problem posed by Siegel in 1945.  Siegel solved this problem in two dimensions by deriving the
signature formula identifying the (2,3,7)-triangle group as having minimal
co-area.
There are strong connections with arithmetic hyperbolic geometry in
the proof, and the result has applications in the maximal symmetry groups
of hyperbolic 3-manifolds in much the same way that Hurwitz's 84g-84 theorem
and Siegel's result do.


===Wednesday, November 16 (9th floor): Kathryn Lindsey (U Chicago) ===
[[Colloquia/Fall2017|Fall 2017]]
Title: Shapes of Julia Sets


Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded.  William Thurston asked "What are the possible shapes of polynomial Julia sets?"  For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name?  It turns out the answer to all of these is "yes!"  I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes.
[[Colloquia/Spring2017|Spring 2017]]


===November 18: Andrew Snowden (University of Michigan)===
[[Archived Fall 2016 Colloquia|Fall 2016]]
Title: Recent progress in representation stability
 
Abstract:  Representation stability is a relatively new field that studies
somewhat exotic algebraic structures and exploits their properties to
prove results (often asymptotic in nature) about objects of interest.
I will describe some of the algebraic structures that appear (and
state some important results about them), give a sampling of some
notable applications (in group theory, topology, and algebraic
geometry), and mention some open problems in the area.
 
===Monday, November 21:  Mariya Soskova (University of Wisconsin-Madison)===
Title:  Definability in degree structures
 
Abstract:  Some incomputable sets are more incomputable than others. We use
Turing reducibility and enumeration reducibility to measure the
relative complexity of incomputable sets. By identifying sets of the
same complexity, we can associate to each reducibility a degree
structure: the partial order of the Turing degrees and the partial
order of the enumeration degrees. The two structures are related in
nontrivial ways. The first has an isomorphic copy in the second and
this isomorphic copy is an automorphism base. In 1969, Rogers asked a
series of questions about the two degree structures with a common
theme: definability. In this talk I will introduce the main concepts
and describe the work that was motivated by these questions.
 
===Friday, December 2:  Hao Shen (Columbia)===
Title:  Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?
 
Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.
 
===Monday, December 5:  Botong Wang (UW-Madison)===
Title:  Enumeration of points, lines, planes, etc.
 
Abstract: It is a theorem of de Bruijn and Erdos that n points in the plane determine at least n lines, unless all the points lie on a line. This is one of the earliest results in enumerative combinatorial geometry. We will present a higher dimensional generalization of this theorem, which confirms a “top-heavy” conjecture of Dowling and Wilson in 1975. I will give a sketch of the key ideas of the proof, which are the hard Lefschetz theorem and the decomposition theorem in algebraic geometry. I will also talk about a log-concave conjecture on the number of independent sets. These are joint works with June Huh.
 
=== Friday, December 9: Aaron Brown (U Chicago) ===
''Lattice actions and recent progress in the Zimmer program''
 
Abstract: The Zimmer Program is a collection of conjectures and questions regarding actions of lattices in higher-rank simple Lie groups on compact manifolds.  For instance, it is conjectured that all non-trivial volume-preserving actions are built from algebraic examples using standard constructions.  In particular—on manifolds whose dimension is below the dimension of all algebraic examples—Zimmer’s conjecture asserts that every action is finite. 
 
I will present some background, motivation, and selected previous results in the Zimmer program.  I will then explain two of my results within the Zimmer program:
(1) a solution to Zimmer’s conjecture for actions of cocompact lattices in SL(n,R) (joint with D. Fisher and S. Hurtado);
(2) a classification (up to topological semiconjugacy) of all actions on tori whose induced action on homology satisfies certain criteria (joint with F. Rodriguez Hertz and Z. Wang).
 
=== Monday, December 19: Andrew Zimmer (U Chicago) ===
''Metric spaces of non-positive curvature and applications in several complex variables''
 
Abstract:  In this talk I will discuss how to use ideas from the theory of metric spaces of non-positive curvature to understand the behavior of holomorphic maps between bounded domains in complex Euclidean space. Every bounded domain has an metric, called the Kobayashi metric, which is distance non-increasing with respect to holomorphic maps. Moreover, this metric often satisfies well-known non-positive curvature type conditions (for instance, Gromov hyperbolicity or visibility) and one can then use these conditions to understand the behavior of holomorphic maps. Some of what I will talk about is joint work with Gautam Bharali.
 
== Past Colloquia ==


[[Colloquia/Spring2016|Spring 2016]]
[[Colloquia/Spring2016|Spring 2016]]

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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