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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 2018 ==
 


== Spring 2017  ==
 
{| cellpadding="8"
{| cellpadding="8"
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!align="left" | date   
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!align="left" | host(s)
!align="left" | host(s)
|-
|-
|'''Monday, January 9, 9th floor'''
|Sep 12    '''Room 911'''
| [http://www.stat.berkeley.edu/~racz/ Miklos Racz] (Microsoft)
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
|[[#Monday, January 9: Miklos Racz (Microsoft) |  ''Statistical inference in networks and genomics'' ]]
|[[#Sep 12: Gunther Uhlmann (Univ. of Washington)|  Harry Potter's Cloak via Transformation Optics ]]
| Valko
| Li
|
|
|-
|-
|January 13, B239
|Sep 14    '''Room 911'''
| [https://math.berkeley.edu/people/faculty/mihaela-ifrim/ Mihaela Ifrim] (Berkeley)
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series
|[[#Friday, January 13: Mihaela Ifrim (Berkeley) |  ''Two dimensional water waves'' ]]
|[[#Sep 14: Gunther Uhlmann (Univ. of Washington) |  Journey to the Center of the Earth ]]
| Angenent
| Li
|
|
|-
|-
|'''Tuesday, January 17, B139'''
|Sep 21    '''Room 911'''
| [https://web.math.princeton.edu/~fabiop/ Fabio Pusateri] (Princeton)
| [http://stuart.caltech.edu/ Andrew Stuart] (Caltech) LAA lecture
|[[#Tuesday, January 17: Fabio Pusateri (Princeton) |  ''The Water Waves problem'' ]]
|[[#Sep 21: Andrew Stuart (Caltech) |  The Legacy of Rudolph Kalman ]]
| Angenent
| Jin
|
|
|-
|-
|January 20, B239
|Sep 28
| [http://math.mit.edu/~sraskin/ Sam Raskin] (MIT)
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
|[[#Friday, January 20: Sam Raskin (MIT) |   Tempered local geometric Langlands  ]]
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
| Arinkin
| Thiffeault
|
|
|-
|-
|'''Monday, January 23, B239'''
|Oct 5
| [http://www.math.umd.edu/~tdarvas/ Tamas Darvas] (Maryland)
| [http://www.personal.psu.edu/eus25/ Eyal Subag] (Penn State)
|[[#Monday, January 23: Tamas Darvas (Maryland) |  Geometry on the space of Kahler metrics and applications to canonical metrics ]]
|[[#Oct 5: Eyal Subag (Penn State)|  Symmetries of the hydrogen atom and algebraic families  ]]
| Viaclovsky
| Gurevich
|
|
|-
|-
|January 27
|Oct 12
|Reserved for possible job talks
| Arie Levit (Yale)
|[[# |   ]]
|[[# TBA| TBA  ]]
|
| Gurevich
|
|
|-
|-
|February 3
|Oct 19
| Melanie Matchett Wood (UW-Madison)
| Jeremy Teitelbaum (U Connecticut)
|[[#Friday, February 3: Melanie Matchett Wood (UW-Madison) | Random groups from generators and relations ]]
|[[# TBATBA ]]
|
|
|-
|Monday, February 6 (Wasow lecture)
| Benoit Perthame (University of Paris VI)
|[[#Monday, February 6: Benoit Perthame (University of Paris VI)| Models for neural networks; analysis, simulations and behaviour ]]
| Jin
|   
|-
|February 10 (WIMAW lecture)
| Alina Chertock (NC State Univ.)
|[[# |  ]]
| WIMAW
|
|-
|February 17
| [http://web.math.ucsb.edu/~ponce/ Gustavo Ponce] (UCSB)
|[[#  |    ]]
| Minh-Binh Tran
|
|-
|February 24
| [http://acms.nd.edu/people/faculty/jonathan-hauenstein/ Jonathan Hauenstein] (Notre Dame)
|[[# |    ]]
| Boston
| Boston
|
|
|-
|-
|March 3
|Oct 26
| [http://www.math.utah.edu/~bromberg/ Ken Bromberg] (University of Utah)
| Douglas Ulmer (Arizona)
|[[# |   ]]
|[[# TBA| TBA  ]]
|Dymarz
| Yang
|
|
|-
|-
|Tuesday, March 7, 4PM (Distinguished Lecture)
|Nov 2
| [http://pages.iu.edu/~temam/  Roger Temam] (Indiana University)
| Reserved for job talk
|[[#  |    ]]
|[[# TBA|  TBA ]]
|Smith
| hosting faculty
|
|
|-
|-
|'''Wednesday, March 8, 2:25PM '''
|Nov 9
| [http://pages.iu.edu/~temam/  Roger Temam] (Indiana University)
| Reserved for job talk
|[[#  |    ]]
|[[# TBA|  TBA ]]
|Smith
| hosting faculty
|
|
|-
|-
|March 10
|Nov 16
| '''No Colloquium'''
| Reserved for job talk
|[[# |   ]]
|[[# TBA| TBA  ]]
|
| hosting faculty
|
|
|-
|-
|'''Wednesday, March 15, 4PM '''
|Nov 30
| [http://verso.mat.uam.es/web/ezuazua/zuazua.html Enrique Zuazua] (Universidad Autónoma de Madrid)
| Reserved for job talk
|[[#   TBA|   TBA  ]]
|[[# TBA| TBA  ]]
| Jin & Minh-Binh Tran
| hosting faculty
|
|
|-
|-
|March 17
|Dec 7
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
| Reserved for job talk
| TBA
| M. Matchett Wood
|
|-
|March 24
|  '''Spring Break'''
|[[# |    ]]
|
|
|-
|'''Wednesday, March 29  at 3:30PM (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  ]]
|[[# TBA|  TBA  ]]
|Li
| hosting faculty
|}
 
==Fall 2017==
 
{| cellpadding="8"
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!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|
|
|-
|DATE
| [LINK PERSON] 
|[[# 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
=== Sep 12: Gunther Uhlmann (Univ. of Washington) ===
paths in graphs, we discover a series of rich and surprising connections
Harry Potter's Cloak via Transformation Optics
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) ===
Can we make objects invisible? This has been a subject of human
Title: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture
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.


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.
=== Sep 14: Gunther Uhlmann (Univ. of Washington) ===
Journey to the Center of the Earth


The conjecture was formulated by Fritz Horn in the early 1970s, and is strongly related to Boltzmann's H-theorem.
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.


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.  
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.


=== September 30: Akos Magyar (University of Georgia) ===
We will also describe some recent results, join with Plamen Stefanov
Title: Geometric Ramsey theory
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.


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.
=== Sep 21: Andrew Stuart (Caltech) ===


=== October 14: Ling Long (LSU) ===
The Legacy of Rudolph Kalman
Title: Hypergeometric functions over finite fields


Abstract: Hypergeometric functions are special functions with lot of
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.
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) ===
=== Sep 28: Gautam Iyer (CMU) ===
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).
Stirring and Mixing


=== October 28: Linda Reichl (UT Austin) ===
Mixing is something one encounters often in everyday life (e.g. stirring cream into coffee). I will talk about two mathematical
Title: Microscopic hydrodynamic modes in a binary mixture
aspects of mixing that arise in the context of fluid dynamics:


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.
1. How efficiently can stirring "mix"?


===Monday, October 31: Kathryn Mann (Berkeley) ===
2. What is the interaction between diffusion and mixing.
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.
Both these aspects are rich in open problems whose resolution involves tools from various different areas. I present a brief survey of existing
results, and talk about a few open problems.


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


===November 7: Gaven Martin (New Zealand Institute for Advanced Study) ===
Symmetries of the hydrogen atom and algebraic families
Title: Siegel's problem on small volume lattices


Abstract: We outline in very general terms the history and the proof of the identification
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) symmetryIn 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 modulesI 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.
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 1945Siegel 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) ===
== Past Colloquia ==
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/Blank|Blank]]


===November 18: Andrew Snowden (University of Michigan)===
[[Colloquia/Spring2018|Spring 2018]]
Title: Recent progress in representation stability


Abstract:  Representation stability is a relatively new field that studies
[[Colloquia/Fall2017|Fall 2017]]
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)===
[[Colloquia/Spring2017|Spring 2017]]
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.
 
=== Monday, January 9: Miklos Racz (Microsoft) ===
''Statistical inference in networks and genomics''
 
Abstract: From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas.
 
I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are root-finding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying high-dimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a high-density, durable, and easy-to-manipulate storage medium of digital data.
 
=== Friday, January 13: Mihaela Ifrim (Berkeley) ===
''Two dimensional water waves''
 
The classical water-wave problem consists of solving the Euler equations in the presence of a free fluid surface (e.g the water-air interface). This talk will provide an overview of recent developments concerning the motion of a two dimensional incompressible fluid with a free surface. There is a wide range of problems that fall under the heading of water waves, depending on a number of assumptions that can be applied: surface tension, gravity, finite bottom, infinite bottom, rough bottom, etc., and combinations thereof. We will present the physical motivation for studying such problems, followed by the discussion of several interesting mathematical questions related to them. The first step in the analysis is the choice of coordinates, where multiple choices are available. Once the equations are derived we will discuss the main issues arising when analysing local well-posedness, as well as the long time behaviour of solutions with small, or small and localized data. In the last part of the talk we will introduce a new, very robust method which allows one to obtain enhanced lifespan bounds for the solutions. If time permits we will also introduce an alternative method to the scattering theory, which in some cases yields a straightforward route to proving global existence results and obtaining an asymptotic description of solutions. This is joint work with Daniel Tataru, and in part with John Hunter.
 
=== Tuesday, January 17:  Fabio Pusateri (Princeton) ===
''The Water Waves problem''
 
We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas.
 
=== Friday, January 20: Sam Raskin (MIT) === 
''Tempered local geometric Langlands ''
 
The (arithmetic) Langlands program is a cornerstone of modern representation theory and number theory. It has two incarnations: local and global. The former conjectures the existence of certain "local terms," and the latter predicts remarkable interactions between these local terms. By necessity, the global story is predicated on the local.
 
Geometric Langlands attempts to find similar patterns in the geometry of curves. However, the scope of the subject has been limited by a meager local theory, which has not been adequately explored.
 
The subject of this talk is a part of a larger investigation into local geometric Langlands. We will give an elementary overview of the expectations of this theory, discuss a certain concrete conjecture in the area (on "temperedness"), and provide evidence for this conjecture.
 
=== Monday, January 23: Tamas Darvas (Maryland) ===
''Geometry on the space of Kahler metrics and applications to canonical metrics''
 
A basic problem in Kahler geometry, going back to Calabi in the 50's, is to find Kahler
metrics with the best curvature properties, e.g., Einstein metrics. Such special metrics are
minimizers of well known functionals on the space of all Kahler metrics H. However these
functionals become convex only if an adequate geometry is chosen on H. One such choice of
Riemannian geometry was proposed by Mabuchi in the 80's, and was used to address a number of
uniqueness questions in the theory. In this talk I will present more general Finsler geometries on
H, that still enjoy many of the  properties that Mabuchi's geometry has, and I will give
applications related to existence of special Kahler metrics, including the recent resolution of
Tian's related properness conjectures. 
 
 
=== Friday, February 3: Melanie Matchett Wood (UW-Madison) ===
''Random groups from generators and relations''
 
We consider a model of random groups that starts with a free group on n generators and takes the quotient by n random relations.  We discuss this model in the case of abelian groups (starting with a free abelian group), and its relationship to the Cohen-Lenstra heuristics, which predict the distribution of class groups of number fields.  We will explain a universality theorem, an analog of the central limit theorem for random groups, that says the resulting distribution of random groups is largely insensitive to the distribution from which the relations are chosen.  Finally, we discuss joint work with Yuan Liu on the non-abelian random groups built in this way, including the existence of a limit of the random groups as n goes to infinity. 
 
=== Monday, February 6: Benoit Perthame (University of Paris VI) ===
''Models for neural networks; analysis, simulations and behaviour''
 
Neurons exchange informations via discharges, propagated
by membrane potential,  which trigger firing of the many connected
neurons. How to describe large networks of such neurons? What are the properties of these mean-field equations?
How can such a network generate a spontaneous activity?
Such questions can be tackled using nonlinear integro-differential
equations. These are now classically used in the neuroscience community to describe
neuronal networks or neural assemblies. Among them, the best known is certainly
Wilson-Cowan's equation which
describe spiking rates arising in different brain locations.
 
Another classical model is the integrate-and-fire equation that describes
neurons through their voltage using a particular type of Fokker-Planck equations. Several mathematical results will be presented concerning existence, blow-up, convergence to steady state,
for the excitatory and inhibitory neurons, with or without refractory states. Conditions for the transition to spontaneous activity (periodic solutions) will be discussed.
 
One can also describe directly the spike time
distribution which seems to encode more directly the neuronal information. 
This leads to a structured population equation that describes
at time $t$ the probability to find a neuron with time $s$
elapsed since its last discharge.  Here, we can 
show that small or large connectivity
leads to desynchronization. For intermediate regimes, sustained
periodic activity occurs.
A common mathematical tool is the use of the relative entropy method.
 
This talk is based on works with K. Pakdaman and D. Salort, M. Caceres, J. A. Carrillo and D. Smets.
 
 
== Past Colloquia ==


[[Archived Fall 2016 Colloquia|Fall 2016]]
[[Archived Fall 2016 Colloquia|Fall 2016]]

Revision as of 10:23, 19 September 2018

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.

Fall 2018

date speaker title host(s)
Sep 12 Room 911 Gunther Uhlmann (Univ. of Washington) Distinguished Lecture series Harry Potter's Cloak via Transformation Optics Li
Sep 14 Room 911 Gunther Uhlmann (Univ. of Washington) Distinguished Lecture series Journey to the Center of the Earth Li
Sep 21 Room 911 Andrew Stuart (Caltech) LAA lecture The Legacy of Rudolph Kalman Jin
Sep 28 Gautam Iyer (CMU) Stirring and Mixing Thiffeault
Oct 5 Eyal Subag (Penn State) Symmetries of the hydrogen atom and algebraic families Gurevich
Oct 12 Arie Levit (Yale) TBA Gurevich
Oct 19 Jeremy Teitelbaum (U Connecticut) TBA Boston
Oct 26 Douglas Ulmer (Arizona) TBA Yang
Nov 2 Reserved for job talk TBA hosting faculty
Nov 9 Reserved for job talk TBA hosting faculty
Nov 16 Reserved for job talk TBA hosting faculty
Nov 30 Reserved for job talk TBA hosting faculty
Dec 7 Reserved for job talk TBA hosting faculty

Abstracts

Sep 12: Gunther Uhlmann (Univ. of Washington)

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 results, and talk about a few open problems.

Oct 5: Eyal Subag (Penn State)

Symmetries of the hydrogen atom and algebraic families

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.

Past Colloquia

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Spring 2018

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Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012