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


== Fall 2018 ==
==Spring 2019==
 


{| cellpadding="8"
{| cellpadding="8"
Line 14: Line 13:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|Sep 12    '''Room 911'''
|Jan 25
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW
|[[#Sep 12: Gunther Uhlmann (Univ. of Washington)|  Harry Potter's Cloak via Transformation Optics ]]
|[[#Beata Randrianantoanina (Miami University Ohio) |  Some nonlinear problems in the geometry of Banach spaces and their applications ]]
| Li
| Tullia Dymarz
|
|
|-
|-
|Sep 14    '''Room 911'''
|Jan 30 '''Wednesday'''
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
|[[#Sep 14: Gunther Uhlmann (Univ. of Washington) |  Journey to the Center of the Earth  ]]
|[[#Lillian Pierce (Duke University) |  Short character sums  ]]
| Li
| Boston and Street
|
|
|-
|-
|Sep 21    '''Room 911'''
|Jan 31 '''Thursday'''
| [http://stuart.caltech.edu/ Andrew Stuart] (Caltech) LAA lecture
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)
|[[#Sep 21: Andrew Stuart (Caltech) | The Legacy of Rudolph Kalman ]]
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]
| Jin
| Street
|
|
|-
|-
|Sep 28
|Feb 1
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
| [https://services.math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke University)
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
|[[# TBA| TBA  ]]
| Thiffeault
| Qin
|
|
|-
|-
|Oct 5
|Feb 5 '''Tuesday'''
| [http://www.personal.psu.edu/eus25/ Eyal Subag] (Penn State)
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)
|[[#Oct 5: Eyal Subag (Penn State)|  Symmetries of the hydrogen atom and algebraic families ]]
|[[# TBA|  TBA ]]
| Gurevich
| Denisov
|
|
|-
|-
|Oct 12
|Feb 8
| [https://www.math.wisc.edu/~andreic/ Andrei Caldararu] (Madison)
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)
|[[#Oct 12: Andrei Caldararu (Madison) | Mirror symmetry and derived categories ]]
|[[#Aaron Naber (Northwestern) |   A structure theory for spaces with lower Ricci curvature bounds ]]
| ...
| Street
|
|
|-
|-
|Oct 19
|Feb 15
| [https://teitelbaum.math.uconn.edu/# Jeremy Teitelbaum] (U Connecticut)
|  
|[[#Oct 19:  Jeremy Teitelbaum (U Connecticut)|  Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist  ]]
|[[# TBA|  TBA ]]
| Boston
|
|-
|Oct 26
| [http://math.arizona.edu/~ulmer/index.html Douglas Ulmer] (Arizona)
|[[#Oct 26: Douglas Ulmer (Arizona) | Rational numbers, rational functions, and rational points ]]
| Yang
|
|-
|Nov 2  '''Room 911'''
| [https://sites.google.com/view/ruixiang-zhang/home?authuser=0# Ruixiang Zhang] (Madison)
|[[#Nov 2: Ruixiang Zhang (Madison) |  The Fourier extension operator ]]
|  
|  
|
|
|-
|-
|Nov 7  '''Wednesday'''
|Feb 22
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)
|[[#Nov 7: Luca Spolaor (MIT) |  (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]
|[[# TBA|  TBA ]]
| Feldman
| Erman and Corey
|
|
|-
|-
|Nov 12  '''Monday'''
|March 4
| [http://www.math.tamu.edu/~annejls/ Anne Shiu] (Texas A&M)
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture
|[[#Nov 9: Anne Shiu (Texas A&M) | Dynamics of biochemical reaction systems  ]]
|[[# TBA| TBA ]]
| Craciun, Stechmann
| Kim
|
|
|-
|-
|Nov 19 '''Monday'''
|March 8
| [https://sites.google.com/site/ayomdin/ Alexander Yom Din] (Caltech)  
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)
|[[#Nov 19: Alexander Yom Din (Caltech) | From analysis to algebra to geometry - an example in representation theory of real groups ]]
|[[# TBA| TBA ]]
| Boston, Gurevitch
| Erman
|
|
|-
|-
|Nov 20 '''Tuesday, Room 911'''
|March 15
| [http://http://www.math.uchicago.edu/~drh/ Denis Hirschfeldt] (University of Chicago)
| Maksym Radziwill (Caltech)
|[[#Nov 20: Denis Hirschfeldt (University of Chicago)|  Computability and Ramsey Theory ]]
|[[# TBA|  TBA ]]
| Andrews
| Marshall
|
|
|-
|-
|Nov 26 '''Monday, Room 911'''
|March 29
| [http://math.mit.edu/directory/profile.php?pid=1415 Vadim Gorin] (MIT)
| Jennifer Park (OSU)
|[[#Nov 26: Vadim Gorin (MIT)|  Macroscopic fluctuations through Schur generating functions ]]
|[[# TBA|  TBA ]]
| Anderson
| Marshall
|
|
|-
|-
|Nov 28 '''Wednesday'''
|April 5
| [http://www.math.ias.edu/~gchen/ Gao Chen](IAS)
| Ju-Lee Kim (MIT)
|[[#Nov 28: Gao Chen(IAS) | A Torelli type theorem ]]
|[[# TBA| TBA  ]]
| Paul
| Gurevich
|
|
|-
|-
|Nov 30
|April 12
| [https://math.indiana.edu/about/faculty/fisher-david.html David Fisher](Indiana U.)
| Evitar Procaccia (TAMU)
|[[#Nov 30: David Fisher (Indiana U.) | New Techniques for Zimmer's Conjecture ]]
|[[# TBA|  TBA  ]]
| Kent
| Gurevich
|-
|Dec 3 '''Monday'''
| [http://bena-tshishiku.squarespace.com/ Bena Tshishiku](Harvard)
|[[#Dec 3: Bena Tshishiku (Harvard)|  TBA  ]]
| Paul
|
|
|-
|-
|Dec 5 '''Wednesday'''
|April 19
| [http://www.mit.edu/~ssen90/ Subhabrata Sen](MIT and Microsoft Research New England)
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)
|[[#Dec 5: Subhabrata Sen | TBA ]]
|[[# TBA| TBA ]]
| Anderson
| Jean-Luc
|
|
|-
|-
|Dec 7
|April 26
| Leonardo Zepeda-Nunez
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)
|[[#Dec 7: Leonardo Zepeda-Nunez | TBA ]]
|[[# TBA| TBA ]]
| Stechmann
| WIMAW
|
|
|-
|-
|Dec 10 '''Monday'''
|May 3
| [http://math.mit.edu/~maxe/ Max Engelstein](MIT)
| Tomasz Przebinda (Oklahoma)
|[[#Dec 10: Max Engelstein (MIT)|  The role of Energy in Regularity ]]
|[[# TBA|  TBA ]]
| Feldman
| Gurevich
|
|
|}
|}
Line 136: Line 118:
== Abstracts ==
== Abstracts ==


=== Sep 12: Gunther Uhlmann (Univ. of Washington) ===
===Beata Randrianantoanina (Miami University Ohio)===
Harry Potter's Cloak via Transformation Optics
 
Can we make objects invisible? This has been a subject of human
fascination for millennia in Greek mythology, movies, science fiction,
etc. including the legend of Perseus versus Medusa and the more recent
Star Trek and Harry Potter. In the last fifteen years or so there have been
several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion
one of them, the so-called "traansformation optics"
in a non-technical fashion n the so-called that has received the most attention in the
scientific literature.
 
=== Sep 14: Gunther Uhlmann (Univ. of Washington) ===
Journey to the Center of the Earth
 
We will consider the inverse problem of determining the sound
speed or index of refraction of a medium by measuring the travel times of
waves going through the medium. This problem arises in global seismology
in an attempt to determine the inner structure of the Earth by measuring
travel times of earthquakes. It has also several applications in optics
and medical imaging among others.
 
The problem can be recast as a geometric problem: Can one determine the
Riemannian metric of a Riemannian manifold with boundary by measuring
the distance function between boundary points? This is the boundary
rigidity problem. We will also consider the problem of determining
the metric from the scattering relation, the so-called lens rigidity
problem. The linearization of these problems involve the integration
of a tensor along geodesics, similar to the X-ray transform.
 
We will also describe some recent results, join with Plamen Stefanov
and Andras Vasy, on the partial data case, where you are making
measurements on a subset of the boundary. No previous knowledge of
Riemannian geometry will be assumed.
 
=== Sep 21: Andrew Stuart (Caltech) ===
 
The Legacy of Rudolph Kalman
 
In 1960 Rudolph Kalman published what is arguably the first paper to develop a systematic, principled approach to the use of data to improve the predictive capability of mathematical models. As our ability to gather data grows at an enormous rate, the importance of this work continues to grow too. The lecture will describe this paper, and developments that have stemmed from it, revolutionizing fields such space-craft control, weather prediction, oceanography and oil recovery, and with potential for use in new fields such as medical imaging and artificial intelligence. Some mathematical details will be also provided, but limited to simple concepts such as optimization, and iteration; the talk is designed to be broadly accessible to anyone with an interest in quantitative science.
 
=== Sep 28: Gautam Iyer (CMU) ===
 
Stirring and Mixing
 
Mixing is something one encounters often in everyday life (e.g. stirring cream into coffee). I will talk about two mathematical
aspects of mixing that arise in the context of fluid dynamics:
 
1. How efficiently can stirring "mix"?
 
2. What is the interaction between diffusion and mixing.


Both these aspects are rich in open problems whose resolution involves tools from various different areas. I present a brief survey of existing
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.
results, and talk about a few open problems.


=== Oct 5: Eyal Subag (Penn State)===
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.


Symmetries of the hydrogen atom and algebraic families
===Lillian Pierce (Duke University)===


The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry.  In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden,  give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules.  I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system.  No prior knowledge on quantum mechanics or representation theory will be assumed.
Title: Short character sums


=== Oct 12: Andrei Caldararu (Madison)===
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.


Mirror symmetry and derived categories
===Dean Baskin (Texas A&M)===


Mirror symmetry is a remarkable phenomenon, first discovered in physics. It relates two seemingly disparate areas of mathematics, symplectic and algebraic geometry. Its initial formulation was rather narrow, as a technique for computing enumerative invariants (so-called Gromov-Witten invariants) of symplectic varieties by solving certain differential equations describing the variation of Hodge structure of “mirror" varieties. Over the past 25 years this narrow view has expanded considerably, largely due to insights of M. Kontsevich who introduced techniques from derived categories into the subject. Nowadays mirror symmetry encompasses wide areas of mathematics, touching on subjects like birational geometry, number theory, homological algebra, etc.
Title: Radiation fields for wave equations


In my talk I shall survey some of the recent developments in mirror symmetry, and I will explain how my work fits in the general picture. In particular I will describe an example of derived equivalent but not birational Calabi-Yau three folds (joint work with Lev Borisov); and a recent computation of a categorical Gromov-Witten invariant of positive genus (work with my former student Junwu Tu).
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.


=== Oct 19:  Jeremy Teitelbaum (U Connecticut)===
===Aaron Naber (Northwestern)===
Lessons Learned and New Perspectives:
From Dean and Provost to aspiring Data Scientist


After more than 10 years in administration, including 9 as Dean of
Title: A structure theory for spaces with lower Ricci curvature bounds.
Arts and Sciences and 1 as interim Provost at UConn, I have returned
to my faculty position. I am spending a year as a visiting scientist
at the Jackson Laboratory for Genomic Medicine (JAX-GM) in Farmington,
Connecticut, trying to get a grip on some of the mathematical problems
of interest to researchers in cancer genomics.  In this talk, I will offer some personal
observations about being a mathematician and a high-level administrator, talk a bit about
the research environment at an independent research institute like JAX-GM, outline
a few problems that I've begun to learn about, and conclude with a
discussion of how these experiences have shaped my view of graduate training in mathematics.


=== Oct 26: Douglas Ulmer (Arizona)===
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.


Rational numbers, rational functions, and rational points
One of the central concerns of arithmetic geometry is the study of
solutions of systems of polynomial equations where the solutions are
required to lie in a "small" field such as the rational numbers.  I
will explain the landscape of expectations and conjectures in this
area, focusing on curves and their Jacobians over global fields
(number fields and function fields), and then survey the progress made
over the last decade in the function field case.  The talk is intended
to be accessible to a wide audience.
=== Nov 2: Ruixiang Zhang (Madison)===
The Fourier extension operator
I will present an integral operator that originated in the study of the Euclidean Fourier transform and is closely related to many problems in PDE, spectral theory, analytic number theory, and combinatorics. I will then introduce some recent developments in harmonic analysis concerning this operator. I will mainly focus on various new ways to "induct on scales" that played an important role in the recent solution in all dimensions to Carleson's a.e. convergence problem on free Schrödinger solutions.
=== Nov 7: Luca Spolaor (MIT)===
(Log)-Epiperimetric Inequality and the Regularity of Variational Problems
In this talk I will present a new method for studying the regularity of minimizers to variational problems. I will start by introducing the notion of blow-up, using as a model case the so-called Obstacle problem. Then I will state the (Log)-epiperimetric inequality and explain how it is used to prove uniqueness of the blow-up and regularity results for the solution near its singular set. I will then show the flexibility of this method by describing how it can be applied to other free-boundary problems and to (almost)-area minimizing currents.
Finally I will describe some future applications of this method both in regularity theory and in other settings.
=== Nov 9: Anne Shiu (Texas A&M)===
Dynamics of biochemical reaction systems
Reaction networks taken with mass-action kinetics arise in many settings,
from epidemiology to population biology to systems of chemical reactions.
This talk focuses on certain biological signaling networks, namely,
phosphorylation networks, and their resulting dynamical systems. For many
of these systems, the set of steady states admits a rational
parametrization (that is, the set is the image of a map with
rational-function coordinates). We describe how such a parametrization
allows us to investigate the dynamics, including the emergence of
bistability in a network underlying ERK regulation, and the capacity for
oscillations in a mixed processive/distributive phosphorylation network.
=== Nov 19: Alexander Yom Din (Caltech)===
 
From analysis to algebra to geometry - an example in representation theory of real groups
Representation theory of non-compact real groups, such as SL(2,R), is a fundamental discipline with uses in harmonic analysis, number theory, physics, and more. This theory is analytical in nature, but in the course of the 20th century it was algebraized and geometrized (the key contributions are by Harish-Chandra for the former and by Beilinson-Bernstein for the latter). Roughly and generally speaking, algebraization strips layers from the objects of study until we are left with a bare skeleton, amenable to symbolic manipulation. Geometrization, again very roughly, reveals how algebraic objects have secret lives over spaces - thus more amenable to human intuition. In this talk, I will try to motivate and present one example - the calculation of the Casselman-Jacquet module of a principal series representation (I will explain the terms in the talk).
=== Nov 20: Denis Hirschfeldt (University of Chicago)===
Computability and Ramsey Theory
Computability theory can be seen as the study of the fine
structure of definability. Much of its power relies on the deep
connections between definability and computation. These connections can be seen in fundamental results such as Post's Theorem, which establishes a connection between the complexity of formulas needed to define a given set of natural numbers and its computability-theoretic strength. As has become increasingly clear, they can also be seen in the computability-theoretic analysis of objects whose definitions come from notions that arise naturally in combinatorics. The heuristic here is that
computability-theoretically natural notions tend to be combinatorially
natural, and vice-versa. I will discuss some results and open questions in
the computability-theoretic analysis of combinatorial principles, in
particular Ramsey-theoretic ones such as versions of Ramsey's Theorem for colorings of countably infinite sets, and versions of Hindman's Theorem, which states that for every coloring of the natural numbers with finitely many colors, there is an infinite set of numbers such that all nonempty sums of distinct elements of this set have the same color.
=== Nov 26: Vadim Gorin (MIT)===
Macroscopic fluctuations through Schur generating functions
I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.
It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.
=== Nov 28: Gao Chen (IAS) ===
A Torelli type theorem
The length of a circle determines the shape of it. In this talk, we will discuss non-trivial generalizations of this fact for Riemann surfaces, hyperkähler 4-manifolds, Calabi-Yau threefolds and G_2, Spin(7) manifolds.
=== Nov 30: David Fisher (Indiana U.) ===
New Techniques for Zimmer's Conjecture
Lattices in higher rank simple Lie groups are known to be
extremely rigid. Examples of this are Margulis' superrigidity theorem,
which shows they have very few linear represenations, and Margulis'
arithmeticity theorem, which shows they are all constructed via number
theory. Motivated by these and other results, in 1983 Zimmer made a
number of conjectures about actions of these groups on compact
manifolds. After providing some history and motivation, I will discuss
a recent result that makes dramatic progress on the conjecture in all
cases and proves it in many of them. I will place some emphasis on
surprising connections to other areas of mathematics that arise in the
proof.
=== Dec 10: Max Engelstein (MIT) ===
The role of Energy in Regularity
The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes.  A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field. 
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.


== Past Colloquia ==
== Past Colloquia ==


[[Colloquia/Blank|Blank]]
[[Colloquia/Blank|Blank]]
[[Colloquia/Fall2018|Fall 2018]]


[[Colloquia/Spring2018|Spring 2018]]
[[Colloquia/Spring2018|Spring 2018]]

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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