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We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal. | We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal. | ||
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===Jan 24: Yaniv Plan (Michigan) === | ===Jan 24: Yaniv Plan (Michigan) === | ||
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the talk is a report on a joint work with A. Avila and C. Sadel. | the talk is a report on a joint work with A. Avila and C. Sadel. | ||
+ | ===Feb 28: Michael Shelley (Courant)=== | ||
+ | ''Mathematical models of soft active materials'' | ||
+ | |||
+ | Soft materials that have an "active" microstructure are important examples of so-called active matter. Examples include suspensions of motile microorganisms or particles, "active gels" made up of actin and myosin, and suspensions of microtubules cross-linked by motile motor-proteins. These nonequilibrium materials can have unique mechanical properties and organization, show spontaneous activity-driven flows, and are part of self-assembled structures such as the cellular cortex and mitotic spindle. I will discuss the nature and modeling of these materials, focusing on fluids driven by "active stresses" generated by swimming, motor-protein activity, and surface tension gradients. Amusingly, the latter reveals a new class of fluid flow singularities and an unexpected connection to the Keller-Segel equation. | ||
===March 28: Michael Lacey (GA Tech) === | ===March 28: Michael Lacey (GA Tech) === |
Revision as of 21:11, 18 February 2014
Mathematics Colloquium
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
Spring 2014
date | speaker | title | host(s) |
---|---|---|---|
Mon, Jan 6, 4PM | Aaron Lauda (USC) | An introduction to diagrammatic categorification | Caldararu |
Wed, Jan 8, 4PM | Karin Melnick (Maryland) | Normal forms for local flows on parabolic geometries | Kent |
Jan 10, 4PM | Yen Do (Yale) | Convergence of Fourier series and multilinear analysis | Denissov |
Mon, Jan 13, 4pm | Yi Wang (Stanford) | Isoperimetric Inequality and Q-curvature | Viaclovsky |
Wen, Jan 15, 4pm | Wei Xiang (University of Oxford) | Conservation Laws and Shock Waves | Bolotin |
Fri, Jan 17, 2:25PM, VV901 | Adrianna Gillman (Dartmouth) | Fast direct solvers for linear partial differential equations | Thiffeault |
Thu, Jan 23, 2:25, VV901 | Mykhaylo Shkolnikov (Berkeley) | Intertwinings, wave equations and growth models | Seppalainen |
Jan 24 | Yaniv Plan (Michigan) | Low-dimensionality in mathematical signal processing | Thiffeault |
Jan 31 | Urbashi Mitra (USC) | Underwater Networks: A Convergence of Communications, Control and Sensing | Gurevich |
Feb 7 | David Treumann (Boston College) | Functoriality, Smith theory, and the Brauer homomorphism | Street |
Feb 14 | Alexander Karp (Columbia Teacher's College) | History of Mathematics Education as a Research Field and as Magistra Vitae | Kiselev |
Feb 21 | Svetlana Jitomirskaya (UC-Irvine) | Analytic quasiperiodic cocycles | Kiselev |
Feb 28 | Michael Shelley (Courant) | Mathematical models of soft active materials | Spagnolie |
March 7 | Steve Zelditch (Northwestern) | Seeger | |
March 14 | |||
Spring Break | No Colloquium | ||
March 26, 7pm, WID | Tadashi Tokieda (Cambridge) | Toy models | Thiffeault (C4 von Neumann Public Lecture) |
March 28 | Michael Lacey (GA Tech) | The Two Weight Inequality for the Hilbert Transform | Street |
April 4 | Richard Schwartz (Brown) | Mari-Beffa | |
April 11 | Risi Kondor (Chicago) | Gurevich | |
April 18 (Wasow Lecture) | Christopher Sogge (Johns Hopkins) | Seeger | |
April 25 | Charles Doran(University of Alberta) | Song | |
Monday, April 28 (Distinguished Lecture) | David Eisenbud(Berkeley) | A mystery concerning algebraic plane curves | Maxim |
Tuesday, April 29 (Distinguished Lecture) | David Eisenbud(Berkeley) | Matrix factorizations old and new | Maxim |
Wednesday, April 30 (Distinguished Lecture) | David Eisenbud(Berkeley) | Easy solution of polynomial equations over finite fields | Maxim |
May 2 | Lek-Heng Lim (Chicago) | Boston | |
May 9 | Rachel Ward (UT Austin) | WIMAW |
Abstracts
January 6: Aaron Lauda (USC)
An introduction to diagrammatic categorification
Categorification seeks to reveal a hidden layer in mathematical structures. Often the resulting structures can be combinatorially complex objects making them difficult to study. One method of overcoming this difficulty, that has proven very successful, is to encode the categorification into a diagrammatic calculus that makes computations simple and intuitive.
In this talk I will review some of the original considerations that led to the categorification philosophy. We will examine how the diagrammatic perspective has helped to produce new categorifications having profound applications to algebra, representation theory, and low-dimensional topology.
January 8: Karin Melnick (Maryland)
Normal forms for local flows on parabolic geometries
The exponential map in Riemannian geometry conjugates the differential of an isometry at a point with the action of the isometry near the point. It thus provides a linear normal form for all isometries fixing a point. Conformal transformations are not linearizable in general. I will discuss a suite of normal forms theorems in conformal geometry and, more generally, for parabolic geometries, a rich family of geometric structures of which conformal, projective, and CR structures are examples.
January 10, 4PM: Yen Do (Yale)
Convergence of Fourier series and multilinear analysis
Almost everywhere convergence of the Fourier series of square integrable functions was first proved by Lennart Carleson in 1966, and the proof has lead to deep developments in various multilinear settings. In this talk I would like to introduce a brief history of the subject and sketch some recent developments, some of these involve my joint works with collaborators.
Mon, January 13: Yi Wang (Stanford)
Isoperimetric Inequality and Q-curvature
A well-known question in differential geometry is to prove the isoperimetric inequality under intrinsic curvature conditions. In dimension 2, the isoperimetric inequality is controlled by the integral of the positive part of the Gaussian curvature. In my recent work, I prove that on simply connected conformally flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson's Q-curvature. The isoperimetric inequality is valid if the integral of the Q-curvature is below a sharp threshold. Moreover, the isoperimetric constant depends only on the integrals of the Q-curvature. The proof relies on the theory of $A_p$ weights in harmonic analysis.
January 15: Wei Xiang (University of Oxford)
Conservation Laws and Shock Waves
The study of continuum physics gave birth to the theory of quasilinear systems in divergence form, commonly called conservation laws. In this talk, conservation laws, the Euler equations, and the definition of the corresponding weak solutions will be introduced first. Then a short history of the studying of conservation laws and shock waves will be given. Finally I would like to present two of our current research projects. One is on the mathematical analysis of shock diffraction by convex cornered wedges, and the other one is on the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws.
Fri, Jan 17, 2:25PM, VV901 Adrianna Gillman (Dartmouth) Fast direct solvers for linear partial differential equations
Fri, Jan 17: Adrianna Gillman (Dartmouth)
Fast direct solvers for linear partial differential equations
The cost of solving a large linear system often determines what can and cannot be modeled computationally in many areas of science and engineering. Unlike Gaussian elimination which scales cubically with the respect to the number of unknowns, fast direct solvers construct an inverse of a linear in system with a cost that scales linearly or nearly linearly. The fast direct solvers presented in this talk are designed for the linear systems arising from the discretization of linear partial differential equations. These methods are more robust, versatile and stable than iterative schemes. Since an inverse is computed, additional right-hand sides can be processed rapidly. The talk will give the audience a brief introduction to the core ideas, an overview of recent advancements, and it will conclude with a sampling of challenging application examples including the scattering of waves.
Thur, Jan 23: Mykhaylo Shkolnikov (Berkeley)
Intertwinings, wave equations and growth models
We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal.
Jan 24: Yaniv Plan (Michigan)
Low-dimensionality in mathematical signal processing
Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. This conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from MRI to quantum state tomography. It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? We explore this question under modern models of low-dimensionality and measurement acquisition.
Thur, Jan 30: Urbashi Mitra (USC)
Underwater Networks: A Convergence of Communications, Control and Sensing
The oceans cover 71% of the earth’s surface and represent one of the least explored frontiers, yet the oceans are integral to climate regulation, nutrient production, oil retrieval and transportation. Future scientific and technological efforts to achieve better understanding of oceans and water-related applications will rely heavily on our ability to communicate reliably between instruments, vehicles (manned and unmanned), human operators, platforms and sensors of all types. Underwater acoustic communication techniques have not reached the same maturity as those for terrestrial radio communications and present some unique opportunities for new developments in information and communication theories. Key features of underwater acoustic communication channels are examined: slow speed of propagation, significant delay spreads, sparse multi-path, time-variation and range-dependent available bandwidth. Another unique feature of underwater networks is that the cost of communication, sensing and control are often comparable resulting in new tradeoffs between these activities. We examine some new results (with implications wider than underwater systems) in channel identifiability, communicating over channels with state and cooperative game theory motivated by the underwater network application.
Feb 7: David Treumann (Boston College)
Functoriality, Smith theory, and the Brauer homomorphism
Smith theory is a technique for relating the mod p homologies of X and of the fixed points of X by an automorphism of order p. I will discuss how, in the setting of locally symmetric spaces, it provides an easy method (no trace formula) for lifting mod p automorphic forms from G^{sigma} to G, where G is an arithmetic group and sigma is an automorphism of G of order p. This lift is compatible with Hecke actions via an analog of the Brauer homomorphism from modular representation theory, and is often compatible with a homomorphism of L-groups on the Galois side. The talk is based on joint work with Akshay Venkatesh. I hope understanding the talk will require less number theory background than understanding the abstract.
Feb 14: Alexander Karp (Columbia Teacher's College)
History of Mathematics Education as a Research Field and as Magistra Vitae
The presentation will be based on the experience of putting together and editing the Handbook on the History of Mathematics Education, which will be published by Springer in the near future. This volume, which was prepared by a large group of researchers from different countries, contains the first systematic account of the history of the development of mathematics education in the whole world (and not just in some particular country or region). The editing of such a book gave rise to thoughts about the methodology of research in this field, and also about what constitutes an object of such research. These are the thoughts that the presenter intends to share with his audience. From them, it is natural to pass to an analysis of the current situation and how it might develop.
Feb 21: Svetlana Jitomirskaya (UC-Irvine)
Analytic quasiperiodic cocycles
Analytic quasiperiodic matrix cocycles is a simple dynamical system, where analytic and dynamical properties are related in an unexpected and remarkable way. We will focus on this relation, leading to a new approach to the proof of joint continuity of Lyapunov exponents in frequency and cocycle, at irrational frequency, first proved for SL(2,C) cocycles in Bourgain-Jitom., 2002. The approach is powerful enough to handle singular and multidimensional cocycles, thus establishing the above continuity in full generality. This has important consequences including a dense open version of Bochi-Viana theorem in this setting, with a completely different underlying mechanism of the proof. A large part of the talk is a report on a joint work with A. Avila and C. Sadel.
Feb 28: Michael Shelley (Courant)
Mathematical models of soft active materials
Soft materials that have an "active" microstructure are important examples of so-called active matter. Examples include suspensions of motile microorganisms or particles, "active gels" made up of actin and myosin, and suspensions of microtubules cross-linked by motile motor-proteins. These nonequilibrium materials can have unique mechanical properties and organization, show spontaneous activity-driven flows, and are part of self-assembled structures such as the cellular cortex and mitotic spindle. I will discuss the nature and modeling of these materials, focusing on fluids driven by "active stresses" generated by swimming, motor-protein activity, and surface tension gradients. Amusingly, the latter reveals a new class of fluid flow singularities and an unexpected connection to the Keller-Segel equation.
March 28: Michael Lacey (GA Tech)
The Two Weight Inequality for the Hilbert Transform
The individual two weight inequality for the Hilbert transform asks for a real variable characterization of those pairs of weights (u,v) for which the Hilbert transform H maps L^2(u) to L^2(v). This question arises naturally in different settings, most famously in work of Sarason. Answering in the positive a deep conjecture of Nazarov-Treil-Volberg, the mapping property of the Hilbert transform is characterized by a triple of conditions, the first being a two-weight Poisson A2 on the pair of weights, with a pair of so-called testing inequalities, uniform over all intervals. This is the first result of this type for a singular integral operator. (Joint work with Sawyer, C.-Y. Shen and Uriate-Tuero)