Mathematical Sciences Institute (MSI)
MSI Colloquia
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Mathematical Sciences Institute (MSI)
MSI Colloquia
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MSI Colloquia AbstractsUnless otherwise stated, MSI Colloquia are held in the Bernhard Neumann Seminar Room (G35) on the ground floor of the John Dedman Mathematical Sciences Building, Bldg 27 (Map). To have a MSI Colloquium listed in this page, email the details to Paul Leopardi or Adam Rennie. View MSI weekly bulletin (includes MSI Colloquium details for the week). 2007 MSI Colloquia 2008 MSI Colloquia
Date: January 15, 2009 Abstract: Date: January 22, 2009 Abstract: Date: January 29, 2008 Abstract: I will talk about classification in algebraic geometry, aiming at the 3-dimensional case of Mori theory, reviewing the familiar motivation coming from complex curves. Gorenstein rings appear throughout the theory in many different roles, often closely associated to the canonical class, the determinant of the cotangent bundle. I will explain, with plenty of examples, how we can write down the equations of many threefolds and how this fits in to the broader picture of classification. Date: February 5, 2009 Abstract: Date: February 12, 2009 Abstract: Date: February 19, 2009 Abstract: We consider the portfolio optimization problem of maximizing the asymptotic growth rate under a combination of fixed and proportional costs. Expressing the asymptotic growth rate in terms of the risky fraction process, the problem can be transformed to controlling a diffusion. In the case of one risky asset and one bond we arrive at a one-dimensional diffusion and can use the corresponding quasi-variational inequalities to obtain the explicit shape together with the existence of an optimal impulse control strategy. This optimal strategy is given by only four parameters, two for the stopping boundaries and two for the new risky fractions the investor chooses at these times. In the case of two or more risky assets we propose a flexible class of strategies which extends the class of optimal strategies in the one risky asset case. For such strategies we show a certain kind of stationarity and obtain the asymptotic return rate by renewal theoretic methods. Date: February 26, 2009 Abstract: Photonic crystals are dielectric structures with periodicity on the scale of the wavelength of light. A glance at any optics journal shows their rapidly expanding applications to signal processing, sensing, bandgap and negative-index materials, and the exciting possibility of fast integrated optical computers. Calculating their `band structure' (ie the traveling waves they support) is a PDE eigenvalue problem similar to that of the Laplacian, ie the first few vibration modes of a drum. The major new features are i) (quasi-)periodic boundary conditions (the drum is a torus), ii) the wavenumber varies in space, being piecewise constant, and iii) the problem is now parameter-dependent (the so-called Bloch phases). We introduce a new approach: imposing the boundary conditions on the walls of the lattice cell using layer potentials (sources of waves distributed according to some density function), which turns the problem into a integral equation on the boundary, which may be solved numerically. Using a finite number of `ghosts' or images of the walls, we preserve the desirable second-kind nature of this integral equation. This has many advantages over many standard methods, and results in `spectral' (ie geometric) reduction of errors with numerical effort, allowing high accuracy to be achieved quickly. I will review the necessary background ideas from numerical analysis. Joint work with Leslie Greengard (NYU) Date: March 5, 2009 Abstract: The integrable chiral Potts model began as a special case of the $3$-state chiral clock model and was found to be parametrized by a genus 10 curve. I shall talk about some further developments, especially in relation with mathematics, including its relation with quantum groups, loop algebra and basic hypergeometric series at roots of unity. Date: March 12, 2009 Abstract: In a quite vague way, harmonic analysis can be described as a set of techniques to estimate sums using non-trivial cancellations. L2 estimates usually involve some orthogonality, and can, in appropriate situations, be extended to Lp for 1< p <\infty. However, for objects like singular integrals, L1 estimates usually do not hold. In the early 70's, with the work of Fefferman and Stein in particular, it appeared that a way around this problem could be to replace L1 by the (real variable) Hardy space H1. This idea turned out to be quite right, and became the classical approach to p=1 estimates. Since then harmonic analytic techniques have been developed in a variety of contexts: from non-commutative analysis to stochastics, via PDE and geometric analysis. And, in recent years, adequate generalizations of H1 have appeared in these various fields. In this talk, I'll give an overview of these developments, focusing on what they have in common: appropriate notions of averaging, and the idea that behind each problem there is a substitute to the euclidean Laplacian. Date: March 19, 2009 Abstract: Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order problems and to models involving lower dimension density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and the calculus of variations. In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others the higher order plays a fundamental role. Applications to phase transitions, to the equilibrium of foams under the action of surfactants, imaging, micromagnetics, thin films, and quantum dots will be addressed. Date: March 26, 2009 Abstract: Poisson geometry is the mid-step between classical differential geometry and non-commutative geometry. From the viewpoint of geometer, a Poisson manifold, namely a manifold with a Poisson bracket on the function space, is foliated by symplectic leaves. A main task in Poisson geometry is to quantize (to connect to non-commutative geometry). This is much studied in the 90's by the schools of Weinstein, Kontsevich, Cattaneo-Felder, etc..� Here I will talk on a related topic (or a mid-step) on how to integrate Poisson manifolds and how it is a special case of a more general integration problem. Date: April 2, 2009 Abstract: In their celebrated paper of 1939 Myers and Steenrod showed that the group of isometries of a Riemannian manifold acts properly on the manifold. This fact has many important consequences. In particular, it implies that the group of isometries is a Lie group in the compact-open topology. This result triggered extensive studies of closed subgroups of the isometry groups of Riemannian manifolds. The peak of activities in this area occurred in the 1950s-70s, with many outstanding mathematicians involved: Kobayashi, Nagano, Yano, H.-C. Wang, Egorov, to name a few. In particular, Riemannian manifolds whose isometry groups possess subgroups of sufficiently high dimensions were explicitly determined. I will speak about proper actions in the complex-geometric setting. In this setting (real) Lie groups act properly by holomorphic transformations on complex manifolds. My general aim is to build a theory parallel to the theory that exists in the Riemannian case. In my lecture I will survey recent classification results for complex manifolds that admit proper actions of high-dimensional groups. Date: April 9, 2009 Abstract: For real-analytic CR-manifolds we discuss necessary and sufficient conditions that these can be locally realized as tube manifolds and clarify the question of the 'right' equivalence between two local tube realizations. This can be reduced to a purely algebraic description in terms of Lie theory. Date: April 16, 2009 Abstract: Date: April 23, 2009 Abstract: In this talk, I will explain how to generalize Rieffel's deformation formula for a class of symplectic exponential solvable Lie group actions. This is based on quantization technics for rank one symplectic hermitian symmetric spaces of non-compact type, in the case where the associated homogeneous space is of group-type. After a short explanation of the geometric setup (that forces us to consider such groups only), I will explain how wavelette-analysis and provides powerful tools to establish suitable estimates in this situation. In particular, an interesting generalization of the Calderon-Vaillancourt theorem will be proven. Date: April 30, 2009 Abstract: The Hilbert transform for a domain in the complex plane is the operator which maps the boundary values of a harmonic function in the domain to the boundary values of its harmonic conjugate function. This operator, as well as the Cauchy integral operator on the boundary, are important examples of singular integral operators and have been much studied in harmonic analysis, in particular in the case of a non smooth boundary.� A way to calculate the Hilbert transform, whose kernel depends in an implicit way on the domain, is to use that it factors into a product of the Cauchy integral and the inverse of a double layer potential operator. In this talk I will discuss extensions of this method for calculating Hilbert transforms / harmonic conjugate functions by solving a double layer potential equation, to Lipschitz domains in higher dimensional euclidean spaces. This makes use of the Cauchy reproducing integral formula for the Hodge--Dirac system $(d+\delta)f=0$ (which generalizes the Cauchy--Riemann system in the plane). Date: May 7, 2009 Abstract: The loop space LM of a Riemannian manifold M is itself an interesting infinite dimensional manifold.� LM has a family of Riemannian metrics indexed by a Sobolev parameter.� We can construct characteristic classes for LM by using the Wodzicki residue instead of the usual matrix trace.� The Pontrjagin classes of LM vanish, but the secondary or Chern-Simons classes may be nonzero. Date: May 14, 2009 Abstract: Random fractals are used to model irregular and disordered media and processes which exhibit statistical self similarity over a range of scales. Examples occur in stochastic processes such as brownian motion, various physical media and statistical mechanics. I will discuss some recent work involving new classes of random fractals and the analysis of their properties. The talk will be of an expository nature for non experts. Date: May 21, 2009 Abstract: Postponed until June 4 Date: May 28, 2009 Abstract: The usual mathematical "wheel" is the circle acted upon by the rotation group SO(2). There are plenty of natural differential operators on this usual circle, manufactured from the basic d/d\theta. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of natural differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres under various symmetry groups. Some familiar differential operators emerge, such as "div," "grad," and "curl" but also some less familiar ones such as the mapping "strain -> stress" in linearised elasticity. These constructions are part of a general theory (of the "Bernstein-Gelfand-Gelfand resolution" derived from the representation theory of semisimple Lie algebras) but they have numerous unexpected applications, for example in suggesting a new stable finite-element scheme in linearised elasticity (due to Arnold, Falk, and Winther). Date: June 4, 2009 Abstract: Integrable (exactly solved) quantum many-body problems have a long history, dating back to Bethe's solution of the Heisenberg spin chain in 1931. For a long time many of the key models, like the one-dimensional model of interacting bosons solved by Lieb in 1963, were thought of as toy models of interest only to mathematical physicists. Indeed, the field of exactly solved models has inspired numerous developments in mathematics. However, physicists are now paying close attention to these models due to the spectacular advances in trapping and cooling atomic matter in quantum atom optics. These developments include the experimental realization of the one-dimensional model of interacting bosons. In this talk I will review these developments and discuss our recent mathematical work on the exactly solved model of interacting fermions of relevance to current experiments.
Date: July 16, 2009 Note: This Colloquium will be held in the Haydon-Allen Tank, and is part of the Workshop on Spectral Theory and Harmonic Analysis. Abstract: To extend the determinant of the Laplacian to non-compact surfaces one has to deal with the presence of continuous spectrum and the fact that the heat kernel is not trace-class. I will explain how to use renormalized integrals to extend the definition of the determinant and find a formula for its variation among asymptotically regular metrics. I will also report on the behaviour of normalized Ricci flow on these metrics and use this to show that the maximum value of the determinant of the Laplacian occurs at constant curvature metrics. This is joint work with Clara Aldana and Frederic Rochon. Date: July 23, 2009 Abstract: This talk is about a book in progress with Richard Brent (ANU, Canberra, Australia). "Modern Computer Arithmetic" collects in the same document all state-of-the-art algorithms in multiple precision arithmetic (integers, integers modulo n, floating-point numbers). The best current reference on that topic is volume 2 from Knuth's The art of computer programming, which misses some new important algorithms (divide and conquer division, other variants of FFT multiplication, floating-point algorithms, ...) Our aim is to give detailed algorithms: (i) for all operations (not just multiplication as many text books), (ii) for all size ranges (not just schoolbook methods or FFT-based methods), (iii) and including all details (for example how to properly deal with carries for integer algorithms, or a rigorous analysis of roundoff errors for floating-point algorithms). The talk will say a few words about the history of this book, and then will focus on a few algorithms from the book, some of which are not well known. URL of the book: http://www.loria.fr/~zimmerma/mca/pub226.html Date: July 30, 2009 Abstract: A standard tool in mathematical analysis is the refining sequence of partitions of the Euclidean space into cubes with side-lenghts equal to powers of two, the so-called dyadic cubes. Associated to the system of dyadic cubes, there is a natural basis of square-integrable functions, the Haar system, which has proven useful e.g. in the analysis of singular integral operators. In their work on singular integrals related to quite general measures, Nazarov, Treil and Volberg observed that, while certain matrix coefficients of such an operator with respect to the Haar basis do not admit any good control, such bad situations only occur "rarely", which can be made precise in the sense of probability by working with a randomly chosen dyadic system, instead of a fixed one. I will discuss how these random cubes are chosen in the original Euclidean framework, and also what they could be in a more general metric space. Date: August 6, 2009 Abstract: We consider the one-parameter family of iterated function systems (IFS) on the unit interval: $([0,1],g_0,g_1)$, where $g_0(x)=tx$, $g_1(x)=tx+1-t$ are contractions on $[0,1]$ and $t\in(0,1)$ is a parameter. Let us denote by $A_t$ the attractor of this system. An element $\omega\in \prod_0^\infty\{0,1\}$ is called address of $x\in A_t$ if $x=\lim_{j\rightarrow\infty}g_{\omega_0}\circ g_{\omega_1}\circ \cdots \circ g_{\omega_j}(0)$. It is easily seen that $x=(1-t)\sum_{j=0}^\infty\omega_jt^j$. For $t\in(1/2,1)$ almost all points $x\in A_t$ have more than one address. The set of addresses of a point $x\in A_t$ possesses a unique lexicographically largest element. Following Barnsley we call this element the top at $t$ and denote the set of all tops at $t$ by $\Omega_t$. Given address $\tau$ we find a set of $t\in(0,1)$ such that $\tau \in \Omega_t$. Then we describe $\Omega_t$ and give the explicit construction of this set. Date: August 10, 2009 Abstract: Accurate estimates of volatility parameters are needed in option pricing. Generalized Autoregressive Conditional Heteroscedastic (GARCH) models and Random Coefficient Autoregressive (RCA) models have been used for volatility modelling. Following Thompson and Thavaneswaran (1999) combined estimating functions are used to estimate the parameters in the volatility models. It turns out that the combined estimating function is more informative for autoregressive processes with GARCH errors and for RCA models. The combination of the least squares (LS) estimating function and the least absolute deviation (LAD) estimating function with application to GARCH model error identification is discussed as an application. An improved option pricing formula for Black-Scholes model with GARCH volatility is also discussed in some detail. Date: August 13, 2009 Abstract: First, I will give several examples of Lie groupoids to try motivate their use in mathematics. In particular I will introduce the tangent groupoid of a manifold and other deformation groupoids (geometric strict deformation quantization). In the second part of the talk I will explain the role of these deformation groupoids in index theory (a la Atiyah-Singer): construction of push-forward maps, analytic index morphisms, and I will finish by sketching a (conceptually) simple proof of the Atiyah-Singer index theorem for compact manifolds using ideas of Connes, and if time allows it, explain how this could be adapted to other geometrical situations. Date: August 20, 2009 Abstract: Complexity theory attempts to sort out which classes of problems are "easy" to solve (e.g., computing determinants, counting perfect matchings of planar graphs etc...) and which are "hard" - computing permanents and the traveling salesman problem are conjectured to be hard - this is essentially the famous P v.s. NP conjecture. Recently tools from geometry have been applied to this problem. I will survey recent work, including Valiant's theory of holographic algorithms and the Mulmuley-Sohoni approach to P v.s. NP and their relations with notions from geometry and representation theory such as spinors and saturation. This talk should be accessible to mathematicians and computer scientists. Date: August 27, 2009 Abstract: We outline some algorithms for testing irreducibility and (if they are not irreducible) finding factors of polynomials over finite fields. In the talk we consider the case of characteristic 2, but the algorithms generalise to other small positive characteristics. We describe a very efficient algorithm for finding the smallest factor of a sparse polynomial of high degree. We also compare several algorithms for testing irreducibility of sparse polynomials. The motivation for the development of these algorithms was a search for primitive trinomials whose degree is the exponent of a Mersenne prime. Computational results include new primitive trinomials of degree 42643801 and 43112609 (currently the two largest known Mersenne exponents). This is joint work with Paul Zimmermann (INRIA, Nancy). Date: September 3, 2009 Abstract: The infinite divergent sum 1+1+...+1+.... can be assigned a finite value which corresponds to the Riemann zeta function evaluated at zero. It can then be viewed as the "number" of integer points on the one dimensional cone corresponding to the positive real line. Generalising this to higher dimensions raises the question as how to "count" integer points in a cone. For a specific class of cones, this amounts to evaluating multiple zeta functions at zero, which requires a renormalisation procedure. We shall describe different renormalisation procedures one can use to "count" integer points on a cone, which are inspired by the methods implemented by physicists when evaluating Feynman integrals. Date: September 10, 2009 Abstract: In 1981 John Hutchinson showed that a hyperbolic iterated function system (IFS) has an attractor. More recently, R. Atkins, M. Barnsley, A. Vince, and D. C. Wilson have shown that there are more geometric conditions which also guarantee the existence of an attractor. In particular, they showed that if an affine IFS is either topologically contractive or non-antipodal, then it has an attractor. In addition, this attractor has a coding map. (Such an attractor will be referred to as self-replicating.) Interest in this result was stimulated by M. Barnsley’s recently developed technique for morphing a digital photograph into a work of art. A key idea in his method is that these geometric conditions guarantee the existence of an attractor. The research presented in this talk is driven by our efforts to generalize the theorem for an affine IFS to projective and bilinear iterated function systems. The following questions drive the discussion. What conditions on an IFS guarantee the existence of an attractor? What conditions ensure a projective IFS defined on real projective n-space Pn has a self-replicating attractor? What conditions ensure a projective IFS defined on P2 has a self-replicating attractor contained in the set K = [0,1] x [0,1]? What conditions ensure a bilinear IFS defined on the space R2 has a self-replicating attractor contained in the set K = [0,1] x [0,1]? Can general conditions be found which guarantee an IFS will not have a self-replicating attractor? While a topologically contractive projective IFS will always have a self-replicating attractor, bilinear transformations are far less tractable. In particular, examples will be presented, where a topologically contractive bilinear IFS with a single function may have two fixed points or a point of period two. This talk should be accessible to non-specialists. The ideas and results presented in this talk resulted from collaborations with R. Atkins, M. Barnsley, and A. Vince. Date: September 17, 2009 Abstract: Date: 3.00pm Monday September 21, 2009 (G35) Abstract: Recent progress on the Kakeya conjecture. This is an updated and reformatted version of Terry Tao's Fefferman conference lecture. Date: 3.30pm Tuesday September 22, 2009 (G35) Abstract: It often happens that a solution of an extremal problem in geometry has more regularity and nicer features than one has an a priori right to expect. I will show how a simple topological problem - when does an immersed curve on a surface bound an immersed subsurface? - is unexpectedly related to linear programming in nonseparable Banach spaces, and gives rise to geometric and dynamical rigidity and discreteness of symplectic representations. Date: 5.30pm Tuesday September 22, 2009 (Theatre 1, Manning Clarke) Abstract: The prime numbers are a fascinating blend of both structure and randomness. It is widely believed that beyond the `obvious' structures in the primes, they otherwise behave as if they were distributed randomly; this `pseudorandomness' then underlies our belief in many unsolved conjectures about the primes, from the twin prime conjecture to the Riemann hypothesis. This pseudorandomness has been frustratingly elusive to actually prove rigorously, but recently there has been progress to establish new results about the primes, such as that they contain arbitrarily long arithmetic progressions. Some of these developments will be discussed in this lecture. This lecture is presented by the Mathematical Sciences Institute. It is free and open to the public. Refreshments provided after the lecture. Date: 2.00pm Wednesday September 23, 2009 (G35)
Abstract: Given a polynomial in two (or more variables), one may study the zero locus from the point of view of different mathematical subjects (number theory, algebraic geometry, ...). I will explain how tropical geometry allows to encode all topological aspects by elementary combinatorial objects called "tropical varieties." Date: 3.30pm Wednesday September 23, 2009
(Baume AGR, Peter Baume Building) Abstract: Recent progress in additive prime number theory. This is an updated and reformatted version of Terry Tao's AMS lecture on this topic. Date: 2.30pm Thursday September 24, 2009 (G35)
Abstract: SCL answers the question: "what is the simplest surface in a given space with prescribed boundary?" where "simplest" is interpreted in topological terms. This topological definition is complemented by several equivalent definitions - in group theory, as a measure of non-commutativity of a group; and in linear programming, as the solution of a certain linear optimization problem. On the topological side, scl is concerned with questions such as computing the genus of a knot, or finding the simplest 4-manifold that bounds a given 3-manifold. On the linear programming side, scl is measured in terms of certain functions called quasimorphisms, which arise from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures). In these talks we will discuss how scl in free and surface groups is connected to such diverse phenomena as the existence of closed surface subgroups in graphs of groups, rigidity and discreteness of symplectic representations, bounding immersed curves on a surface by immersed subsurfaces, and the theory of multi-dimensional continued fractions and Klein polyhedra. Date: September 24, 2009 Abstract: The Strominger-Yau-Zaslow conjecture gives an intrinsic explanation for Homological Mirror Symmetry in the case of Calabi Yau manifolds. I will explain that by extending the SYZ conjecture beyond the Calabi-Yau case, one may associate a Landau-Ginzburg mirror to generic hypersurfaces in toric varieties. The key idea is to use tropical geometry to reduce the problem to understanding the mirror of hyperplanes.
Date: October 1, 2009
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Date: October 8, 2009
Abstract: After a brief introduction to the notions of strongly continuous semigroups and related quadratic forms we will recall how they are applied to classical nonlinear parabolic partial differential equations. In particular, we indicate the representation of fractional Sobolev spaces in R^n and on smooth domains in terms of the corresponding heat semigroup. We further explain the notion of mild solution of related stochastic partial differential equations, where the noise is interpreted as an element of an appropriate dual space. Such equations have been studied in the literature with several tools. The semigroup approach enables us to consider similar problems in certain nonlinear metric measure spaces. In particular, parabolic equations on self-similar fractals admitting a Laplace operator can be treated. This will be explained at the end of the talk. Date: October 15, 2009 Abstract: Date: October 22, 2009 Abstract: Tiling in nature is as old as the honey bee. The mathematical theory goes back at least to Johannes Kepler. After a brief history of tiling in Euclidean space, some results on digit tiling are presented. This is a type of self-replicating tiling based on the notion of an iterated function system as developed by Hutchinson, Barnsley, and others. The tile boundaries in such a tiling are usually fractal.
Date: October 29, 2009
Abstract: Inverse problems are aimed, for example, at finding material laws, identifying not directly observable functions or making a diagnosis based on measuring external physical quantities. A characteristic property of such problems is their ill-posedness. Small perturbations in the data may lead to arbitrarily large errors in the solution. In this context, the nature of ill-posedness for different inverse problems is strongly varying and the stable approximate solution requires sophisticated and well-adapted methods, so-called regularization methods. For initial illustration, we present some effects and results for the mathematical and numerical treatment of an inverse heat transfer problem occurring in the engineering practice of simulating electric fault arc tests. In a next part the talk is concerned with the convergence theory of general linear regularization methods with appropriate qualifications for linear ill-posed operator equations Ax=y (*) in a Hilbert space setting. We discuss the interplay of smoothing properties of the forward operator A and smoothness assumptions on the solution x' of (*) expressed by general source conditions and their approximation using distance functions. The main goal of the concluding part of the presentation, devoted to ill-posed nonlinear operator equations F(x)=y in Banach spaces, is to show that new mathematical concepts of nonlinear regularization theory can immediately be applied to inverse option pricing and used for the stable approximate solution of calibration problems that arise in financial markets. The serious treatment of such inverse problems in finance requires certainly stochasticians, but even more specialists in functional analysis.
Date: November 12, 2009
Abstract: Synchronization of dynamical networks has attracted extensive attention recently. But most existing results are about networks with identical nodes. This talk addresses the issue of synchronization for dynamical networks with non-identical nodes. Several criteria for bounded synchronization and asymptotic synchronization are presented. The problem of how to achieve synchronization via controller design is also studied.
Date: November 19, 2009
Abstract: We describe a general method for enumerating the distinct self-similar fractals that arise as attractors of certain families of iterated function systems, using a little group theory to analyse the symmetries of the attractors. The talk will be illustrated by a range of examples. |
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