Erik Bollt : Transport and Global Control of Deterministic and Stochastic Dynamical Systems
- Other Meetings and Events ( 14 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 ( 15 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 ( 16 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 ( 14 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 ( 16 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 ( 17 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 ( 14 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 ( 12 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.
Claude LeBrun : Einstein Metrics, Variational Problems, and Seiberg-Witten Theory
- Other Meetings and Events ( 15 Views )Abstract: One of the major themes of modern differential geometry is the relationship between the curvature and topology of Riemannian manifolds. In this lecture, I will describe some links between curvature and SMOOTH topological invariants --- i.e. invariants which can distinguish between different smooth structures on a given topological manifold. The specific invariants I will discuss are the Seiberg-Witten invariants of 4-manifolds, and I will describe the impact these have on the existence problem for Einstein metrics and some related Riemannian variational problems.
Sheldon Katz : Gopakumar-Vafa Invariants of Moduli Spaces of Holomorphic Curves and Integrality
- String Theory ( 11 Views )In a remarkable paper, Gopakumar and Vafa have used M-theory to assign BPS invariants to families of holomorphic curves which are believed to yield the Gromov-Witten invariants. In this talk, techniques of algebraic geometry are developed for calculating these invariants which agree with mirror symmetry results in examples. The proper mathematical context for complete generalization of these ideas appears to require geometric invariant theory.
Richard Paul Horja : Derived category automorphims from mirrorsymmetry
- Algebraic Geometry ( 10 Views )Inspired by Kontsevich's homological mirror symmetry conjecture, I will show how to construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi-projective Calabi-Yau variety. Examples will be presented. I will also explain the 'local' character of the picture.
Christian Hasse : Perles at Bings House -- Facet Subgraphs of Simple Polytopes
- Other Meetings and Events ( 14 Views )The combinatorial structure of a d-dimensional simple convex polytope -- as given, for example, by the set of the (d-1)-regular subgraphs of facets -- can be reconstructed from its abstract graph [Blind & Mani 1988, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction exists [Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgraphs'' [Perles 1970,1984]. We give examples for the validity of Perles conjecture: In particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we observe that for any counterexample, the boundary of the (simplicial) dual polytope P^* contains a 2-complex without a free edge, and without 2-dimensional homology. Examples of such complexes are known; we use a simple modification of ``Bing's house'' (two walls removed) to construct explicit 4-dimensional counterexamples to Perles' conjecture.
Phil Hanlon : The Combinatorial Laplacian
- Other Meetings and Events ( 10 Views )The combinatorial laplacian is a method for computing the rational homology of an algebraic or simplicial complex. Although the method is quite old, it has recently been applied with success to a number of problems in algebraic combinatorics. We will survey the method and describe some of these recent applications.
Matthew Cushman : The Motivic Fundamental Group
- Algebraic Geometry ( 11 Views )The fundamental group of a topological space is usually defined in terms of homotopy classes of based loops. The group structure is given by composition of loops. If X is a complex algebraic variety, one has an underlying topological space, and hence a fundamental group. Hain showed that the nilpotent completion of the group ring of this topological fundamental group carries a mixed Hodge structure. Grothendieck defined a fundamental group for schemes defined over any field. Applying this to a complex algebraic variety, one obtains the profinite completion of the topological fundamental group. This group comes with a natural action of the absolute Galois group of the field of definition. The above indicates that varieties over fields of characteristic zero have two notions of fundamental group, armed with either a Galois action or a mixed Hodge structure. This is similar to the situation with homology and cohomology groups, where one has both an etale and Betti version carrying Galois actions and Hodge structures. An important guiding principle is that both of these versions of homology and cohomology should come from an underlying ``motivic'' theory. This is a homology and cohomology theory for algebraic varieties over a field k taking values the abelian tensor category of mixed motives over k, denoted M_k. There should be functors from M_k to both the category of Galois modules and mixed Hodge structures. When applied to the motivic homology of a variety X, these functors should yield the etale homology or Betti homology of X. In this way, motives glue these two different theories together more strongly than just the comparison isomorphisms. Nori has recently given a definition of the category of mixed motives. In this talk, we will show how this category relates to the fundamental group. In fact, more generally there is a motivic version of paths between two different points x and y of X which is important for applications. We also show that the multiplication and comultiplication maps are motivic, and compare with Hain's theory.