Adam Jacob : The Yang-Mills flow and the Atiyah-Bott formula on compact Kahler manifolds
- Geometry and Topology ( 117 Views )In this talk I will describe the limiting properties Yang-Mills flow on a holomorphic vector bundle E, in the case where the flow does not converge. In particular I will describe how to determine the L^2 limit of the curvature endomorphism along the flow. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. I will then explain how to use this result to identify the limiting bundle along the flow, which turns out to be independent of metric and uniquely determined by the isomorphism class of E.
Iva Stavrov : On a point-particle gluing construction
- Geometry and Topology ( 117 Views )Analyzing the motion of a small body is often done by making a point-particle approximation. This simplification is not entirely appropriate in general relativity since, roughly speaking, too much mass in too little space creates black holes. In place of point-particles one considers one-parameter families of space-time metrics $\gamma_\varepsilon$ in which $\varepsilon\to 0$ corresponds to a body shrinking to zero size. In addition, certain point-particle limit properties are imposed on $\gamma_\varepsilon$. While there are some examples of such metrics $\gamma_\varepsilon$ (e.g. Schwarzschild-de Sitter space-time), there is no general existence theorem for such space-times. This talk will discuss a gluing construction which produces initial data with desirable point-particle limit properties.
Clifford Johnson : Large N Phases, Holography and Quantum Gravity
- String Theory ( 10 Views )The Holographic principle---in the form of the AdS/CFT correspondence---suggests a relation between processes in quantum gravity and phenomena in ordinary quantum field theory. The old `sum over histories' semi-classical approach to quantum gravity can be revisited in this light, by studying various gravitational instantons in AdS. We show that the AdS/CFT correspondence may be extended to spaces which are only locally asymptotically AdS, by examining the properties of the AdS-Taub-NUT and AdS-Taub-Bolt spacetimes. We also use holography to show that spacetime topology change in quantum gravity is a unitary process, in contrast to suggestions in the old `sum over histories' literature.
Jake Bouvrie : Learning and Synchronization in Stochastic Neural Ensembles
- Graduate/Faculty Seminar ( 96 Views )We consider a biological learning model composed of coupled stochastic neural ensembles obeying a nonlinear gradient dynamics. The dynamics optimize a simple error criterion involving noisy observations provided by the environment, leading to a function that can be used to make decisions in the future. The uncertainty of the resulting decision function is characterized, and shown to be controlled in large part by trading off coupling strength (and/or network topology) against the ambient neuronal noise. Further connections with classical regularization notions in statistical learning theory will also be explored.
Lingjiong Zhu : Self-Exciting Point Processes
- Probability ( 108 Views )Self-exciting point processes are simple point processes that have been widely used in neuroscience, sociology, finance and many other fields. In many contexts, self-exciting point processes can model the complex systems in the real world better than the standard Poisson processes. We will discuss the Hawkes process, the most studied self-exciting point process in the literature. We will talk about the limit theorems and asymptotics in different regimes. Extensions to Hawkes processes and other self-exciting point processes will also be discussed.
Ding Ma : Multiple Zeta Values and Modular Forms in Low Levels
- Number Theory ( 119 Views )In this talk, I will introduce the famous result by Gangl-Kaneko-Zagier about a family of period polynomial relations among double zeta value of even weight. Then I will generalize their result in various ways, from which we can see the appearance of modular forms in low levels. At the end, I will give a generalization of the Eichler-Shimura-Manin correspondence to the case of the space of newforms of level 2 and 3 and a certain period polynomial space.
Christine Berkesch Zamaere : Torus actions and holonomic D-modules
- Algebraic Geometry ( 88 Views )Just as algebraic varieties with group actions admit quotients, we provide a quotient construction for D-modules with torus actions that is with several important properties in algebraic analysis. As an application, we apply tools from toric geometry to obtain new information about hypergeometric systems of PDEs studied by Gauss, Appell, and Lauricella, among others. In particular, we determine when such "Horn systems" are regular holonomic. This is joint work with Laura Felicia Matusevich and Uli Walther.
Gordana Matic : Contact invariant in sutured Floer homology and fillability
- Geometry and Topology ( 127 Views )In the 70's Thurston and Winkelnkemper showed how an open book decomposition of a 3-manifold can be used to construct a contact structure. In 2000 Giroux showed that every contact structure on a 3-manifold can be obtained from that process. Ozsvath and Szabo used this fact to define an invariant of contact structures in their Heegaard Floer homology, providing an important new tool to study contact 3-manifolds. In joint work with Ko Honda and Will Kazez we describe a simple way to visualize this contact invariant and provide a generalization and some applications. When the contact manifold has boundary, we define an invariant of contact structure living in sutured Floer homology, a variant of Heegaard Floer homology for a manifold with boundary due to Andras Juhasz. We describe a natural gluing map on sutured Floer homology and show how it produces a (1+1)-dimensional TQFT leading to new obstructions to fillability.
Hangjun Xu : Constant Mean Curvature Surfaces in Asymptotically Flat Manifolds
- Graduate/Faculty Seminar ( 106 Views )The study of surfaces with constant mean curvature (CMC) goes back to 1841 when Delaunay classified all CMC surfaces of revolution. There has been consistent work on finding CMC hypersurfaces in various ambient manifolds. In this talk, we will discuss some nice properties of CMC surfaces, and then the existence of CMC surfaces in the Schwarzschild, and in general, asymptotically flat manifolds.
Alice Guionnet : The spectrum of non-normal matrices, II: the Brown measure.
- Gergen Lectures ( 250 Views )In this talk, which is a continuation of Wednesday's lecture, we shall describe the natural candidate for the limit of the empirical measure of the eigenvalues of non-normal matrices, the so-called Brown measure. We will give some details about how to prove convergence towards such a limit, but also discuss the instability of such convergence.
Spencer Leslie : Intro to crystal graphs and their connections with number theory
- Graduate/Faculty Seminar ( 183 Views )I will review some basics of crystal bases for highest-weight representations for a semisimple Lie algebra. I will also point to some connections with number theory through Fourier coefficients of Eisenstein series, mostly in type A.
Yifeng Yu : Random Homogenization of Non-Convex Hamilton-Jacobi Equations in 1d
- Applied Math and Analysis ( 99 Views )I will present the proof of the random homogenization of general coercive Hamiltonian in 1d with the form as H(p,x,\omega)=H(p)+V(x, \omega). Some interesting and complex phenomena associated with non-convex Hamiltonian will also be discussed. This is a joint work with Scott Armstrong and Hung Tran.
Ravi Srinivasan : Kinetic theory for shock clustering and Burgers turbulence
- Applied Math and Analysis ( 94 Views )A remarkable model of stochastic coalescence arises from considering shock statistics in scalar conservation laws with random initial data. While originally rooted in the study of Burgers turbulence, the model has deep connections to statistics, kinetic theory, random matrices, and completely integrable systems. The evolution takes the form of a Lax pair which, in addition to yielding interesting conserved quantities, admits some rather intriguing exact solutions. We will describe several distinct derivations for the evolution equation and, time-permitting, discuss properties of the corresponding kinetic system. This talk consists of joint work with Govind Menon (Brown).
Jan Rychtar : The evolution of cooperation kin selection and greenbeard genes
- Probability ( 150 Views )Abstract:One way to convince ourselves that no cooperation can evolve among defectors is via a simple yet one of the most famous games in all of game theory - the Prisoners dilemma (PD) game. The players of this game adopt one of the two strategies: a) a cooperator who pays a cost so that another individual can receive a benefit, or b) a defector who can receive benefits, but it has no cost as it does not deal out any benefits at all. As seen from this formulation, no rational individual would opt to be a cooperator. Yet, we can see cooperation everywhere around us and thus (assuming defectors were here first) there must exist at least one mechanism for its evolution. Nowak (2006, 2012) discusses several of such mechanisms, including the kin selection by which cooperation can spread if the benefits go primarily to genetic relatives. In this talk we will introduce a simple PD-like asymmetric matrix game and show how Hamiltons rule can easily be recovered. We will also introduce a simple PD-like symmetric matrix game to model the evolution of cooperation via greenbeard mechanism, which can be seen as a special case of kin selection.
Shamit Kachru : Fun with Supersymmetric Three-Cycles
- String Theory ( 11 Views )We study the physics of D6 branes wrapped on supersymmetric three-cycles in type IIA string theory. In particular, we argue that although the superpotential in the resulting field theories vanishes to all orders in \alpha', nonperturbative contributions (from disk instantons) should generically be present. We discuss how mirror symmetry can relate these to tree-level sigma model computations in some circumstances. We also present an illustration of the role that background closed string moduli play in the brane worldvolume theory.
Giulia Sacca : Compact Hyperkahler manifolds in algebraic geometry
- Algebraic Geometry ( 116 Views )Hyperkahler (HK) manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. Compact HK manifolds are one of the building blocks for algebraic varieties with trivial first Chern class and their role in algebraic geometry has grown immensely over the last 20 year. In this talk I will give an overview of the theory of compact HK manifolds and then focus on some of my work, including a recent joint work with R. Laza and C. Voisin.
Paul Norbury : Magnetic monopoles on manifolds with boundary
- Geometry and Topology ( 139 Views )Kapustin and Witten introduced interesting boundary value problems for magnetic monopoles on a Riemann surface times an interval. They described the moduli space of such solutions in terms of Hecke modifications of holomorphic bundles over the Riemann surface. I will explain this and prove existence and uniqueness for such monopoles.
Lin Lin : Elliptic preconditioner for accelerating the self consistent field iteration of Kohn-Sham density functional theory
- Applied Math and Analysis ( 126 Views )Kohn-Sham density functional theory (KSDFT) is the most widely used electronic structure theory for molecules and condensed matter systems. Although KSDFT is often stated as a nonlinear eigenvalue problem, an alternative formulation of the problem, which is more convenient for understanding the convergence of numerical algorithms for solving this type of problem, is based on a nonlinear map known as the Kohn-Sham map. The solution to the KSDFT problem is a fixed point of this nonlinear map. The simplest way to solve the KSDFT problem is to apply a fixed point iteration to the nonlinear equation defined by the Kohn-Sham map. This is commonly known as the self-consistent field (SCF) iteration in the condensed matter physics and chemistry communities. However, this simple approach often fails to converge. The difficulty of reaching convergence can be seen from the analysis of the Jacobian matrix of the Kohn-Sham map, which we will present in this talk. The Jacobian matrix is directly related to the dielectric matrix or the linear response operator in the condense matter community. We will show the different behaviors of insulating and metallic systems in terms of the spectral property of the Jacobian matrix. A particularly difficult case for SCF iteration is systems with mixed insulating and metallic nature, such as metal padded with vacuum, or metallic slabs. We discuss how to use these properties to approximate the Jacobian matrix and to develop effective preconditioners to accelerate the convergence of the SCF iteration. In particular, we introduce a new technique called elliptic preconditioner, which unifies the treatment of large scale metallic and insulating systems at low temperature. Numerical results show that the elliptic preconditioner can effectively accelerate the SCF convergence of metallic systems, insulating systems, and systems of mixed metallic and insulating nature. (Joint work with Chao Yang)
Heinrich M. Jaeger : Granular Fluids: Liquids with Vanishing Surface Tension?
- Nonlinear and Complex Systems ( 138 Views )Qualitatively new behavior often emerges when large numbers of similar entities are interacting at high densities, no matter how simple the individual components. One prototypical example is granular matter such as fine dry sand, where individual grains are solids. In this talk I will discuss several striking phenomena, including the formation of jets and their break-up into droplets, where large ensembles of grains behave very much like a liquid, except that they do so without apparent surface tension.
Robin PEMANTLE : Analytic Combinatorics in Several Variables Subtitle: estimating coefficients of multivariate rational power series
- Probability ( 92 Views )The analytic framework for estimating coefficients of a generating function is the same in many variables as in one variable: evaluate Cauchy's integral by manipulating the contour into a "standard" position. That being said, the geometry when dealing with several complex variables can be much more complicated. This talk, drawing on the recent book (with Mark Wilson) of the same title, surveys analytic methods for extracting asymptotics from multivariate generating functions. I will try to give an idea of the main pieces of the puzzle. In particular, I will try to explain in pictures the roles of Morse theory, complex algebraic geometry and hyperbolicity in the asymptotic evaluation of integrals.
Dmitry Vagner : Higher Dimensional Algebra in Topology
- Graduate/Faculty Seminar ( 211 Views )In his letter, "Pursuing Stacks," Grothendieck advocated to Quillen for the use of "higher" categories to encode the higher homotopy of spaces. In particular, Grothendieck dreamt of realizing homotopy n-types as n-groupoids. This powerful idea both opened the field of higher dimensional algebra but also informed a paradigm in which the distinction between topology and algebra is blurred. Since then, work by Baez and Dolan among others further surveyed the landscape of higher categories and their relationship to topology. In this talk, we will explore this story, beginning with some definitions and examples of higher categories. We will then proceed to explain "the periodic table of higher categories" and the four central hypotheses of higher category theory. In particular, these give purely algebraic characterizations of homotopy types, manifolds, and generalized knots; and account for the general phenomena of stabilization in topology. No prerequisites beyond basic ideas in algebraic topology will be expected.
Anton Kapustin : Mirror Symmetry and Noncommutative Geometry
- String Theory ( 9 Views )I review the homological mirror symmetry conjecture of Kontsevich and its relation with topological D-branes. This conjecture is nontrivial even for tori, which allows one to perform some interesting checks. It turns out that in the presence of a general B-field Kontsevich's conjecture must be modified. In particular, on the B-side, coherent sheaves on a Calabi-Yau must be replaced by coherent sheaves on a certain `noncommutative deformation' of the Calabi-Yau. By contrast, an algebraic description of the A-model branes, if it exists, should involve a noncommutative algebra even for a vanishing B-field.
Simon Brendle : Singularity formation in geometric flows
- Geometry and Topology ( 289 Views )Geometric evolution equations like the Ricci flow and the mean curvature flow play a central role in differential geometry. The main problem is to understand singularity formation. In this talk, I will discuss recent results which give a complete picture of all the possible limit flows in 2D mean curvature flow with positive mean curvature, and in 3D Ricci flow.
Richard Schoen : Ricci flow and 1/4-pinching
- Gergen Lectures ( 239 Views )In this series of three lectures we will describe positivity conditions on Riemannian metrics including the classical conditions of positive sectional, Ricci, and scalar curvature. We will discuss open problems and recent progress including our recent proof of the differentiable sphere theorem (joint with Simon Brendle). That proof employs the Ricci flow, so we will spend some time explaining that technique. Finally we will discuss problems related to positive scalar curvature including some high dimensional issues which occur in that theory. If time allows we will describe recent progress on black hole topologies. These lectures, especially the first two, are intended for a general audience.
Xiuqing Chen : Global weak solution for kinetic models of active swimming and passive suspensions
- Applied Math and Analysis ( 95 Views )We investigate two kinetic models for active suspensions of rod-like and ellipsoidal particles, and passive suspensions of dumbbell beads dimmers, which couple a Fokker-Planck equation to the incompressible Navier-Stokes or Stokes equation. By applying cut-off techniques in the approximate problems and using compactness argument, we prove the existence of the global weak solutions with finite (relative) entropy for the two and three dimensional models. For the second model, we establish a new compact embedding theorem of weighted spaces which is the key in the compactness argument. (Joint work with Jian-Guo Liu)
Ronnie Sircar : Games with Exhaustible Resources
- Probability ( 149 Views )We study N-player repeated Cournot competitions that model the determination of price in an oligopoly where firms choose quantities. These are nonzero-sum (ordinary and stochastic) differential games, whose value functions may be characterized by systems of nonlinear Hamilton-Jacobi-Bellman partial differential equations. When the quantity being produced is in finite supply, such as oil, exhaustibility enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative, but expensive, resource (for example solar technology for energy production), and give an asymptotic approximation in the limit of small exhaustibility. We illustrate the two-player problem by numerical solutions, and discuss the impact of limited oil reserves on production and oil prices in the dupoly case. Joint work with Chris Harris (Cambridge University) and Sam Howison (Oxford University).
John Cain : Stabilization of Periodic Wavetrains in Excitable Media
- Nonlinear and Complex Systems ( 105 Views )Cardiac cells, like toilets, are excitable: Giving a sufficiently strong push to the handle of a quiescent toilet elicits a dramatic response (flush) followed by a gradual return to the resting state. Likewise, supplying a sufficiently strong electrical stimulus to a quiescent cardiac cell elicits a prolonged elevation of the membrane potential (an action potential).
Suppose that one end of an excitable fiber of cardiac cells is paced periodically. If the period is large, the generic response is a stable periodic wave train of the sort associated with normal, coordinated contraction of heart muscle tissue. Reducing the period (think "speeding up the heart rate") can cause the onset of an instability which can have devastating physiological consequences. Echebarria and Karma (Chaos, 2002) argued that if one attempts to stabilize the periodic wave train by using feedback control to perturb the pacing period, success can be achieved only within some small radius of the stimulus site. Those authors used a special case of the ETDAS control method that Dan Gauthier and Josh Socolar devised.
Here, I will offer an explanation as to WHY algorithms like ETDAS, applied locally, cannot achieve global results in this context. Then, I'll argue that it actually IS possible to stabilize the periodic wave train if the perturbations are chosen more carefully. While these findings may seem encouraging from an experimental or clinical standpoint, I will close by describing some recent work of Flavio Fenton which I believe is even more promising.