Ofer Aharony : `Little String Theories of NS 5-Branes and Holography in Linear Dilaton Backgrounds
- String Theory ( 14 Views )We argue that vacua of string theory which asymptote at weak coupling to linear dilaton backgrounds are holographic (like AdS backgrounds). The full string theory in such vacua is ``dual'' to a theory without gravity in fewer dimensions. The dual theory is generically not a local quantum field theory. Excitations of the string vacuum, which can be studied in the weak coupling region using worldsheet methods, give rise to observables in the dual theory. An interesting example is string theory in the near-horizon background of parallel NS5-branes, the CHS model, which is dual to the decoupled NS5-brane theory (``little string theory''). This duality can be used to study some of the observables in this theory and some of their correlation functions.
Mina Aganagic : D-Branes, Mirror Symmetry and Counting Holomorphic Discs
- String Theory ( 14 Views )A class of special Lagrangian subspaces of Calabi-Yau manifolds is considered and their mirrors, certain holomoprhic varieties of the mirror geometry, are identified. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We discuss flat coordinates on the moduli spaces, and a certain quantum ambiguity which influences disc counting. Highly non-trivial new predictions for disc counting are obtained.
Allan Adams : LISA in the Sky with Dark Matter
- String Theory ( 17 Views )Space-based laser interferometers such as LISA may prove spectacularly effective not only in listening to gravitational radiation but also in the direct detection of galactic halo dark matter streaming through the solar system. This talk will describe the basic strategy and touch on some entertaining open possibilities.
Bobby Acharya : M Theory, G_2 Holonomy spaces and N=1 Theories in Four Dimensions
- String Theory ( 20 Views )After reviewing some reasons why one might be interested in G_2 holonomy compactifications of M theory, I shall review some of the recent results about such vacua, which are mainly due to Atiyah, Maldacena, Vafa, Witten, and myself in various collaborations. Topics covered will mainly center around how various aspects of super Yang-Mills theory can be related to properties of M theory in such spacetime backgrounds. These will include instantons, domain walls and confinement.
Wendy Zhang : Drop breakup: asymmetric cones in viscous flow
- Other Meetings and Events ( 20 Views )Dynamic singularities are ubiquitous. They arise in mathematical models of phenomena as grand as star formation or as familiar as the breakup of a thread of honey as it is being added to tea. Drop breakup allows one to study dynamics close to a singularity in a simple context which is also accessible to experiments. Recent works have revealed that a viscous liquid drop close to breakup looks self-similar---the drop profile looks the same if the length scales are rescaled appropriately. A new numerical strategy is developed to capture the drop breakup dynamics and show good agreement with experimental measurements. Surprisingly, the presence of even small amounts of viscous dissipation in the surrounding can dramatically alter the self-similar profile. In particular, when no exterior viscous dissipation is present, the thread profile is symmetric about the point of pinch-off. When small amounts of exterior viscous dissipation are present, the thread profile becomes severely asymmetric. An understanding of the final breakup process is crucial in elucidating the mechanisms underlying the formation of satellite drops, an issue relevant to the development of ink-jet printing technologies and emulsification processes.
Michael Thaddeus : Mirror Symmetry and Langlands Duality
- Other Meetings and Events ( 17 Views )Strominger, Yau and Zaslow have proposed that Calabi-Yau mirror partners should be families of special Lagrangian tori over the same base which are dual to each other. I will exhibit some striking evidence for this proposal: pairs of Calabi-Yau orbifolds satisfying the requirements of SYZ for which the expected equality of stringy Hodge numbers can be completely verified. This is in a sense the first enumerative evidence supporting SYZ. The construction I will present is the first of several suggesting a general principle: for any construction using a compact Lie group to construct a Calabi-Yau, mirror partners arise from applying the construction to Langlands dual groups.
Jonathan Mattingly : Ergodicity of Stochastically Forced PDEs
- Other Meetings and Events ( 17 Views )Stochastic PDEs have become important models for many phenomenon. Nonetheless, many fundamental questions about their behavior remain poorly understood. Often such SPDE contain different processes active at different scales. Not only does such structure give rise to beautiful mathematics and phenomenon, but I submit that it also contains the key to answering many seemingly unrelated questions. Questions such as ergodicity and Mixing. Given a stochastically forced dissipative PDE such as the 2D Navier Stokes equations, the Ginzburg-Landau equations, or a reaction diffusion equation; is the system Ergodic ? If so, at what rate does the system equilibrate ? Is the convergence qualitatively different at different physical scales ? Answers to these an similar questions are basic assumptions of many physical theories such as theories of turbulence. I will try both to convince you why these questions are interesting and explain how to address them. The analysis will suggest strategies to explore other properties of these SPDEs as well as numerical methods. In particular, I will show that the stochastically forced 2D Navier Stokes equations converges exponentially to a unique invariant measure. I will discuss under what minimal conditions one should expect ergodic behavior. The central ideas will be illustrated with a simple model systems. Along the way I will explain how to exploit the different scales in the problem and how to overcome the fact that the problem is an extremely degenerate diffusion on an infinite dimensional function space. The analysis points to a class of operators in between STRICTLY ELLIPTIC and HYPOELLIPTIC operators which I call EFFECTIVELY ELLIPTIC. The techniques use a representation of the process on a finite dimensional space with memory. I will also touch on a novel coupling construction used to prove exponential convergence to equilibrium.
Richard Kenyon : Random maps from Z2 to Z
- Other Meetings and Events ( 21 Views )One of the most basic objects in probability theory is the simple random walk, which one can think of as a random map from Z to Z mapping adjacent points to adjacent points. A similar theory for random maps from Z2 to Z had until recently remained elusive to mathematicians, despite being known (non-rigorously) to physicists. In this talk we discuss some natural families of random maps from Z2 to Z. We can explicitly compute both the local and the large-scale behavior of these maps. In particular we construct a "scaling limit" for these maps, in a similar sense in which Brownian motion is a scaling limit for the simple random walk. The results are in accord with physics.
A. S. Fokas : Differential Forms, Spectral Theory and Boundary Value Problems
- Other Meetings and Events ( 19 Views )A new method will be reviewed for analyzing boundary value problems for linear and integrable nonlinear PDEs. This method involves the following:
- Given a PDE for q(x), x in Rn, construct a closed (n-1)-differential form W(x,k), k in Cn-1.
- Given a convex domain D contained in Rn, perform the spectral analysis of W
- Given appropriate boundary conditions, analyze the global relation
For linear PDEs, the relation of this method with the Ehrenpreis-Palamodov principle, as well as the relations with applied techniques such as the Weiner-Hopf technique will be discussed.
Erik Bollt : Transport and Global Control of Deterministic and Stochastic Dynamical Systems
- Other Meetings and Events ( 17 Views )Associated with a dynamical system, which evolves single initial conditions, the Frobenius-Perron operator evolves ensemble densities of initial conditions. Including a brief tutorial on the topic, we will present our new applications of this global and statistical point of view:
- The inverse Frobenius-Perron problem (IFPP) is a global open-loop strategy to control chaos by constructing a "nearby" dynamical system with desirable invariant density. We reduce the question of stabilizing an arbitrary invariant density to the question of a hyperplane intersecting a unit hyperbox; several controllability theorems follow. Applications will be described.
- Well-known models have been found to exhibit new and interesting dynamics under the addition of stochastic perturbations. Using the Frobenius-Perron operator for stochastic dynamical systems, we develop new tools designed to predict the effects of noise and to pinpoint stochastic transport regions in phase space. As an example, we study a model from population dynamics for which chaos-like behavior can be induced, as the standard deviation of the noise is increased. We identify how stochastic perturbations destabilize two attracting orbits, effectively completing a heteroclinic orbit, to create chaos-like behavior. Other physical applications will also be discussed.
Jim Arthur : The principle of functorialty: an elementary introduction
- Colloquium ( 19 Views )The principle of functoriality is a central question in present day mathematics. It is a far reaching, but quite precise, conjecture of Langlands that relates fundamental arithmetic information with equally fundamental analytic information. The arithmetic information arises from the solutions of algebraic equations. It includes data that classify algebraic number fields, and more general algebraic varieties. The analytic information arises from spectra of differential equations and group representations. It includes data that classify irreducible representations of reductive groups. The lecture will be a general introduction to these things. If time permits, we shall also describe recent progress that is being made on the problem.
Gang Tian : Geometry and Analysis of low-dimensional manifolds
- Gergen Lectures ( 19 Views )In this series of talks, I will focus on geometry and analysis of manifolds of dimension 2, 3 or 4. The first talk is a general introduction of this series. I will start the talk by reviewing some classical theories on Riemann surfaces and their recent variations in geometric analysis. Then we survey some recent progress on 3- and 4-manifolds. I hope that this talk will show some clues how geometric equations can be applied to studying geometry of underlying spaces. In the second talk, I will discuss recent works on the Ricci flow and its application to the geometrization of 3-manifolds, in particular, I will briefly discuss Perelman's work towards the Poincare conjecture. In last talk, I will discuss geometric equations in dimension 4 and how they can be applied to studying geoemtry of underlying 4-spaces. Some recent results will be discussed and some open problems will be given.
Gang Tian : Geometry and Analysis of low-dimensional manifolds
- Gergen Lectures ( 17 Views )In this series of talks, I will focus on geometry and analysis of manifolds of dimension 2, 3 or 4. The first talk is a general introduction of this series. I will start the talk by reviewing some classical theories on Riemann surfaces and their recent variations in geometric analysis. Then we survey some recent progress on 3- and 4-manifolds. I hope that this talk will show some clues how geometric equations can be applied to studying geometry of underlying spaces. In the second talk, I will discuss recent works on the Ricci flow and its application to the geometrization of 3-manifolds, in particular, I will briefly discuss Perelman's work towards the Poincare conjecture. In last talk, I will discuss geometric equations in dimension 4 and how they can be applied to studying geoemtry of underlying 4-spaces. Some recent results will be discussed and some open problems will be given.
Leo P. Kadanoff : Drips and Jets: Singularities, Topology Changes, and Scaling for Fluid Interfaces
- Gergen Lectures ( 19 Views )We investigate the behavior of the interface between two fluids. We are interested in the singularities which develop when the bridge connecting two pieces of fluid goes to zero thickness. One physical situation is Hele-Shaw flow: two fluids are trapped between parallel glass plates and feel frictional forces. Another situation is a cylindrically symmetrical stream, for example, in a dripping faucet. In a third case, a fluid in an electric or magnetic field which pulls off a piece of fluid ending in a sharp point. The last case has an interface between a light and a heavier fluid being sucked up as in a drinking straw. At a critical value of the sucking, a very thin bridge of the heavy fluid is formed. Analytical, numerical, and experimental methods are used to describe what happens right around the pinchoff of these bridges. Much of the resulting behavior can be understood via scaling arguments.
Gerhard Huisken : Parabolic Evolution Equations for the Deformation of Hypersurfaces
- Gergen Lectures ( 20 Views )A smooth one-parameter family F0 : Mnx [0,T) ---> (Nn+1,g) of hypersurfaces in a Riemannian manifold (N(n+1),g) is said to move by its curvature if it satisfies an evolution equation of the form
(d/dt) F(p,t) = f(p,t) p Mn, t [0,T),
such that at each point of the surface its speed in normal direction is a function $f$ of the extrinsic curvature of the hypersurface. Examples such as the flow by mean curvature, flow by Gauss curvature or flow by inverse mean curvature arise naturally both in Differential Geometry, where they exhibit fascinating interactions between the extrinsic curvature of the surfaces and intrinsic geometric properties of the ambient manifold, and in Mathematical Physics, where they serve as models for the evolution of interfaces in phase transitions. The first lecture gives a general introduction to the main examples and phenomena and highlights some recent results. The second lecture shows how parabolic rescaling techniques can be combined with a priori estimates to study and in some cases classify possible singularities of the mean curvature flow. The series concludes with applications of hypersurface families in General relativity, including a recent proof of an optimal lower bound for the total energy of an isolated gravitating system by Huisken and Ilmanen.
Tom Kepler : Microevolution in the Immune System: A Computational Systems Approach
- Colloquium ( 18 Views )Vaccines protect their recipients by inducing long-term structural changes in populations of immune cells. Part of that restructuring is exactly analogous to Darwinian Selection. New antibody molecules are created by somatic mutation of existing antibody genes. Subsequently, the immune cell populations that possess these mutated receptors overtake the "wild-type" immune cells due to the selective advantage they have acquired. Thus the immune system is vastly better prepared to recognize and eliminate the eliciting pathogen the next time around.
New sequencing and biosynthesis technologies, together with mathematical and computational tools, now allow us to investigate this fascinating and important phenomenon more deeply than ever before. I will illustrate this development with examples from the immune response to HIV infection.
Jianqiang Zhao : Arithmetic and Geometry of Multiple Polylogarithms
- Algebraic Geometry ( 17 Views )In this talk I will describe a proof of a conjecture made by Beilinson et al concerning the motivic complex in the weight three case. Then I will explain a new way to define the analytic continuation of the multiple polylogarithms which provides an appproach to defining the good variations of mixed Hodge-Tate structures associated to the multiple polylogarithms explicitly. As an application I will define the single-valued real analytic version of these multi-valued functions and which should be connected to the special values of multiple Dedekind zeta functions over general number fields.