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Richard Montgomery
Schedule:Logarithmic residues and zero sums of idempotents in Banach algebras abstract: A basic result from complex function theory states that the contour integral of the logarithmic derivative of a scalar analytic function can only vanish when the function has no zeros inside the contour. Question: does this result generalize to the vector-valued case? We suppose the function takes values in a Banach algebra. The answer depends on which Banach algebra. Affirmative answers have been obtained for a class of algebras including polynomial identity Banach algebras by employing non-commutative Gelfand theory, which involves families of matrix representations. Negative answers have been achieved via the construction of non-trivial zero sums of a finite number of idempotents. It is intriguing that we need only five idempotents in all known examples. The idempotent constructions relate to deep problems concerning the geometry of Banach spaces and general topology. The talk, which is accessible to non-specialists, reports on work done jointly with Torsten Ehrhardt (Santa Cruz) and Bernd Silbermann (Chemnitz). The ABC Conjecture abstract: The ABC Conjecture, formulated in the mid-1980's by Oesterle and Masser, is one of the most important conjectures in number theory. It has many deep consequences, but its basic formulation can be given in entirely elementary terms. Earlier this month, a 500-page solution was announced by Shinichi Mochizuki (building on several thousand pages of work he has carried out over the last 20 years). In this talk, I will explain what the conjecture asserts, some evidence that supports it, and a few of its corollaries. Geometry and physics of Arnold diffusion abstract: Many physical systems, such as the Solar system, for instance, can be considered as small perturbations of completely integrable Hamiltonian system. In the case of the Solar system the interaction between planets can be viewed as a small perturbation of the otherwise decoupled system of Kepler problems. Kolmogorov-Arnold-Moser theory, arguably the main development in Hamiltonian dynamics of the past century and a half, shows that small perturbations do not affect qualitative behavior of the majority of solutions of a completely integrable Hamiltonian system (under some general assumptions). This majority forms a Cantor set in the phase space. Arnold diffusion is the phenomenon that is believed to happen in the complement to this ``surviving" set of solutions, for a generic Hamiltonian system. In this talk I will give an intuitive geometrical illustration of this phenomenon for geodesics on the torus, or, equivalently, for the motion of a particle in a periodic potential. With this interpretation, the concepts such as resonances, the ``whiskered tori" and the heteroclinic orbits acquire a simple geometrical meaning. At the end, I will show how Arnold diffusion can manifest itself in a mechanical system. In a chain of coupled pendula any pendulum can pass most of its energy to any other pendulum, no matter how weak the coupling. This leakage can happen in a ``deterministically random" fashion, i.e. along any prescribed itinerary. The geometry of the space of measures and its applications abstract: In 1781, Monge asked about the optimal way to transport a dirtpile from one place to another. Performing such an optimal transport gives a geodesic path in the space of probability measures. The optimal transport problem has had recent applications to Riemannian geometry (collapse with Ricci curvature bounded below). I will describe these developments. On the Rigidity of the Maslov Index for Coisotropic Submanifolds abstract: A rigidity result on the coisotropic Maslov index states that there exists a non-trivial loop (tangent to the characteristic foliation of a stable coisotropic submanifold) with certain bounds on its symplectic area and its Maslov index. This was proved by Ginzburg for symplectically aspherical ambient manifolds. The result also holds for some symplectic manifolds not necessarily aspherical. We shall state the theorem for the “rational case” and discuss some tools used to prove this result. Obama or Romney? abstract: We summarize the results of very recent joint work with the Supreme Court and a multitude of superPACS. Using Mathematics to Optimally Balance Between Cash and Stock in a Portfolio abstract: Consider a portfolio of stock and cash. We want to find the best balance between the stock and cash so as to optimize the risk-adjusted return (called the utility) at some given later date. Under the market assumptions of Black and Scholes, which include no costs for buying or selling stock, we know from Merton (1970) how to achieve this best balance at any time. If we allow the portfolio to also contain derivatives based on the stock, it will have no effect on this optimal expected utility. But what if we include the costs for buying or selling stock? We will show, to leading order, how to optimally trade the portfolio when these costs are small. We will discuss the very different nature of this optimal trading strategy when we have one stock vs. when we have multiple stocks. And finally, we will show how to optimally use derivatives if they are allowed in the portfolio, which can have a significant effect on the optimal expected utility when costs are present. Mirror symmetry in complex dimension 1 abstract: This talk will try to introduce non-specialists to some key concepts in mirror symmetry, by considering some of the simplest examples (such as cylinders or pairs of pants). These examples will be a pretext to discuss the Strominger-Yau-Zaslow picture of mirror symmetry (according to which mirror pairs carry dual torus fibrations), as well as Kontsevich's homological mirror symmetry conjecture (according to which mirror symmetry is an equivalence between two derived categories). I will also try to motivate the appearance of "wrapped" Fukaya categories in the case of open manifolds. The less introductory parts of the talk are based on joint work with Mohammed Abouzaid, Alexander Efimov, Dmitri Orlov, and Ludmil Katzarkov. Quiver mutation and quantum dilogarithm identities abstract: Quiver mutation is an elementary operation on quivers which appeared in physics in Seiberg duality in the 1990s and in mathematics in Fomin-Zelevinsky's definition of cluster algebras in 2002. In this talk, I will show how, by comparing sequences of quiver mutations, one can construct identities between products of quantum dilogarithm series. These identities generalize Faddeev-Kashaev-Volkov's classical pentagon identity and the identities obtained by Reineke. Morally, the new identities follow from Kontsevich-Soibelman's theory of Donaldson-Thomas invariants. They can be proved rigorously using the theory linking cluster algebras to quiver representations. Geometric problems in Biology abstract: Biological studies give rise to many geometrical problems. This is partly because many biological properties are determined or reflected in shape. I'll talk about how geometric issues play a central role in some of these problems, including: 1. Classifying proteins 2. Protein docking 3. The evolutionary tree of old world monkies 4. Diagnosing Alzheimer's disease
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