Mathematical Sciences Institute (MSI)
MSI Colloquia
|
|
Mathematical Sciences Institute (MSI)
MSI Colloquia
|
|
|
|
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 Florica Cirstea or Adam Rennie. View MSI weekly bulletin (includes MSI Colloquium details for the week). 2007 MSI Colloquia
Date: January 15, 2008 Abstract: Q-curvature is a notion introduced by Paneitz-Branson in the early 1980th initially defined on manifolds of dimension four; and which is a direct generalization of the Gaussian curvature on compact surface. In recent years, there has been intensive effort to study the subject mainly due to its conformal invariant property and its connection to some geometric invariants in the asymptotic hyperbolic, conformally compact Einstein manifolds. In this talk, I will give a brief survey of the subject with emphasize on applications to problems in conformal geometry. Date: January 16, 2008 Abstract: Pseudo-conformal structure is a CR-structure with a chosen contact form. There is a well defined notion of volume and area, as well as natural differential operators on such manifold. I will talk about the minimal surface equation and the CR-Yamabe equation in 3-D. Date: January 24, 2008 Abstract: In 1986, Hardt, Kinderlehrer and Lin established the existence and partial regularity of minimizers of the liquid crystal energy with the Oseen-Frank density. In this talk, we will discuss a new flow for the liquid crystal systems. We will show that existence of global partial regular solution of the heat flow in two dimensional case and prove the existence of weak solution in dimension three. Date: February 14, 2008 Abstract: The famous question of Kac is whether one can hear the shape of a drum. Or more precisely, whether all eigen frequencies of a drum determine the drum. In general the answer to the latter question is negative. The eigen frequencies are equal if and only if there exists a unitary operator which intertwines the Laplacians on the two drums. In this talk we discuss what happens if the unitary operator is replaced by an order isomophism, i.e., if it maps positive functions to positive functions. Or equivalently, if the diffusion processes on the two drums are equal. Date: February 28, 2008 Abstract: Some leading experts in optimization take the view that convergence theory is necessary for the development of a successful new algorithm. The speaker has found, however, that this view has two severe disadvantages. Firstly, several algorithms have become highly useful without suitable convergence theory. Secondly, available techniques for proving convergence may impose restrictions on new algorithms that exclude huge improvements in practice over existing software. Examples of these situations will be presented and discussed, most of them being from unconstrained optimization. Date: March 6, 2008 Abstract: I will delve a little into the origins, mostly in integro-differential equations, of the Fredholm condition for operators on a Hilbert space. The use of the index in determing solutions of equations was later turned on its head by Atiyah and Singer when they used the index of elliptic operators to find invariants of manifolds. Starting with Breuer in the 60s, the notion of Fredholm has been extended into many new domains. It turns out that there is a pleasing unity amongst these generalisations. Date: March 13, 2008 Abstract: The talk gives a survey of recent results in
an interesting area at the intersection of the theory of partial
differential equations, differential geometry and stochastic analysis.
The main object of study is the weak elliptic equation
Date: March 18, 2008 Abstract: The problem that we are struggling with goes back to Alberto Calderon's celebrated and beautiful work on his complex interpolation spaces [A0;A1]s, and is now about 44 years old: Suppose that T is a linear operator such that T: A0 --> B0 is compact and T: A1 --> B1 is bounded, or even compact. Despite many partial results, including some quite recent ones, we still do not know whether, in general, T :[A0;A1]s --> [B0;B1]s is compact. Unless the audience prefers otherwise, I will briefly recall some of the history and applications of interpolation theory and the definitions and some of the basic relevant facts about Calderon's spaces. I will also indicate various connections with Fourier series. Indeed Fourier series are apparently a significant part of the arsenal we have for attacking this problem. (Svante Janson has characterised Calderon's spaces via sequence spaces of Fourier coefficients, and Fedor Nazarov has used Fourier series to give an "almost counterexample" to a closely related question.) I will also briefly mention a recent application by Bartels, Jensen and Mueller of a partial solution of Calderon's problem to miscible oil recovery, that is, to numerical solution of PDEs by finite element methods. Some background about these things can be found at http://www.math.technion.ac.il/emcwikel/compact Date: March 27, 2008 Abstract: We discuss the proof that in suitable dimensions the branch of positive solutions of the problem -Laplacian u= r exp u in D with u=0 on the boundary of D has infinitely many bifurcation points. We also discuss related equations. /P> Date: April 1, 2008 Abstract: After a discussion of selected classical results on actions of finite symmetry groups on compact Riemann surfaces, an overview of the complex 2-dimensional setting will be presented. Recent developments will be exemplified by an analysis of K3-surfaces with special symplectic symmetry. Date: April 3, 2008 Abstract: The spectral flow of a path of selfadjoint Fredholm operators counts the eigenvalues changing sign along the path (with signs and multiplicities). I will explain its relation to the winding number and I will use it to give a general definition of the spectral flow for paths of unbounded selfadjoint Fredholm operators. I will also explain how integral formulas for the spectral flow can be derived from this definition. Date: April 10, 2008 Abstract: Science has reached the point where, in order to shorten the time required to breed a new plant variety and enhance the efficiency of the breeding, a plant breeder requires molecular information. For wheats, such information is recovered from a study of the rheological properties of wheat-flour doughs made from the flours of different wheat varieties. The interpretation of such results involves, among other things, the modelling of the reptation dynamics of polymers as a function of their molecular weight distributions. Utilizing the fact that, in order to guarantee sensible physics, the relaxation modulus of the Boltzmann causal integral equation of linear viscoelasticity “must” be a completely monotone function, an analytic relationship can be derived and solved that connects the measured relaxation spectrum of a polymer to its molecular weight distribution. The result raises interesting questions, which will be discussed, about how the relationship between the relaxation spectrum of a particular polymer and its molecular weight distribution should be modelled. Date: April 17, 2008 Abstract: In 1995 Bost and Connes constructed a remarkable C*-dynamical system with the Galois group of the maximal abelian extension of Q as the symmetry group and the Riemann zeta-function as the partition function. Since then similar systems have been constructed for arbitrary number fields and for higher dimensions. I will explain how analysis of the phase transition of these systems leads to study of actions of groups on adelic spaces, with the uniqueness of a phase for large temperatures related to ergodicity of certain actions and to equidistribution results in number theory. Date: April 24, 2008 Abstract: This talk will be about the interaction between mathematical statistics and health research policy. Over the last twenty years, first medical research policy, and then health research policy more broadly, have gradually come to be dominated by a set of research methods collectively known as Evidence-Based Medicine. Evidence-Based Medicine includes a detailed statistical methodology but no mechanism for updating this methodology in the light of new developments in mathematical statistics. I will introduce this statistical methodology in terms of the concept of a hierarchy of evidence, and discuss its likely consequences for both statistical and medical innovation. As a brief case study, I will discuss the recent panic over potentially fatal side-effects of the widely used drug rosiglitazone. Date: May 1, 2008 Abstract: The talk is based on joint results with L. Baratchart and F. Nazarov. For $2\le p<\infty$, we consider a problem of approximating in the norm of $L^p$ a matrix function $\Phi$ on the unit circle by matrix functions analytic in the unit disc. It turns out that the space of matrix functions in $L^p$ splits into two massive subsets: the set of respectable functions and the set of weird functions. For respectable $m\times n$ matrix functions $\Phi$ the distance o the set of analytic functions is equal to the norm of the Hankel operator $H_\Phi$ from $H^q(C^n)$ to $H^2_-(C^m)$, where $1/p+1/q=1/2$. For weird functions $\Phi$ the distance is greater than the norm of the Hankel operator. We have found another distance formula that works for all matrix functions in $L^p$. We also consider related factorization formulae, describe all best approximants and characterize badly approximable matrix functions. Date: May 8, 2008 Abstract: Moduli spaces are one of the beauties of algebraic geometry: {\it sets} of isomorphism classes of objects that turn out to carry a natural {\it algebraic} structure. Many general questions are of special interest for such moduli spaces and lead to a beautiful interplay between the geometry of the objects individually and in families.Ê In my talk, I will try to introduce and illustrate these ideas. The moduli spaces I will discuss are those of algebraic curves---widely studied and applied in mathematical physics, symplectic geometry and number theory. The questions I will ask about them are from birational geometry and deal with maps from these spaces to complex projective spaces. Date: May 15, 2008 Abstract: We consider strictly convex surfaces that shrink according to a fully nonlinear geometric evolution equation. Such surfaces convergence to points in finite time and, after appropriate rescaling, to round spheres. We present the result and the algorithm to find the main test function. Date: May 22, 2008 Abstract: Recently there has been considerable interest in the Fluctuation Theorem (FT). The Evans-Searles FT shows how time reversible microscopic dynamics leads to irreversible macroscopic behavior as the system size or observation time increases. We discuss this theorem as well as the Crooks Fluctuation Theorem and the Jarzynski Equality. We show that the argument of the FT, the dissipation function, plays a central role in nonlinear response theory and derive the Dissipation Theorem, giving exact relations for the nonlinear response of classical N-body systems that are more widely applicable than previous expressions. These expressions should be verifiable experimentally. When linearized they reduce to the well known Green-Kubo expressions for linear response. Date: May 29, 2008 Abstract: The study of Complex Systems has received a lot of attention recently motivated by deeper studies of natural phenomena, including the emergence of systems biology as a discipline and the increasing complexity of society, its infrastructure and economy. In this talk some views on what we mean by complex systems in mathematical terms will be presented. The ARC Centre of Excellence MASCOS will be used to illustrate how a framework of relevant mathematical tools can be presented. The talk will then turn to complex networks which form a large class of complex systems where the focus is on complexity arising from extensive interconnectedness. Theories for dynamical behaviour (such as chaos, synchronisation, collapse) of complex networks have been a hot topic in science in the last decade. A key concept is synchronization whereby all the nodes behave similarly in some sense. However, the existing results, for models motivated by oscillations in biology, are limited by assumptions which do not typically hold in many other networks. In this talk, some extensions to the theory including non-identical nodes, more general synchronous manifolds, role of system structure and regions of synchronisation will be described. Then open questions on stabilization and a brief outline of the need for a systems and control theory of such networks will be presented Biography: DAVID J HILL received the BE and BSc degrees from the University of Queensland, Australia, in 1972 and 1974, respectively. He received the PhD degree in Electrical Engineering from the University of Newcastle, Australia, in 1976. He is currently a Professor and Australian Research Council Federation Fellow in the Research School of Information Sciences and Engineering at The Australian National University. He is also Deputy Director of the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems. He has held academic and substantial visiting positions at the universities of Melbourne, California (Berkeley), Newcastle (Australia), Lund (Sweden), Sydney and Hong Kong (City University). His research interests are in network systems science, stability analysis, nonlinear control and applications. He is a Fellow of the Institution of Engineers, Australia, the Institute of Electrical and Electronics Engineers, USA and the Australian Academy of Science; he is also a Foreign Member of the Royal Swedish Academy of Engineering Sciences.Date: June 5, 2008 Abstract: We will describe a very novel, new way to look at dualizing complexes; dualizing complexes traditionally form the backbone of the theory of Grothendieck duality. The talk will begin with a quick review of the classical theory, then it will mention some results that appeared in recent papers by Krause, Jorgensen and Iyengar-Krause and finally we will describe a string of very recent theorems by the speaker and Daniel Murfet. Murfet's results are part of his PhD thesis at the ANU; the first article in which this series of results will appear is by the speaker, and has just been accepted in Inventiones. Date: June 12, 2008 Abstract: Modern dynamical systems come in three basic flavours. Topological (action of a continuous map on a topological state space), Smooth (action of a differentiable map on a differentiable manifold) and Ergodic (action of a measure-preserving map on a measure space). It is a convenient fact that most interesting examples contain features from two or more categories simultaneously. Through a number of concrete examples, we will see different ways ergodic theory can arise in such systems, sometimes naturally and sometimes unexpectedly. At the same time one can get a feel for the kind of results that are amenable to the theory. One of the most interesting situations is when one can prove that a given system has an invariant measure, however there is no closed form expression for it. In such cases it is a nontrivial problem to obtain quantitative information from ergodic theory. This brings us into line with a number of current and active areas of research.
Date: July 10, 2008 Abstract: Starting with the work of Littlewood, Offord and Kac in the 1930s and 1940s, there is now a substantial literature on the number and location of real zeros of polynomials with random real coefficients. Using an idea of Kurt Mahler, I will consider similar questions for random polynomials over other fields such as the p-adics and find simple, non-asymptotic expressions for the number of simultaneous zeros of several natural systems of such polynomials. A crucial ingredient in the proof is a theorem on the distribution of the elementary divisors of random matrices with entries from fields such as the p-adics. This latter result uses some observations of Richard Brent and Brendan McKay and has connections with Jason Fulman's probabilistic proof of the Rogers-Ramanujan identities. Date: July 17, 2008 Abstract: Recent advances in the theory of quasilinear and fully nonlinear elliptic PDE based on nonlinear potential theory and harmonic analysis methods will be discussed. Model problems involve the p-Laplacian and k-Hessian operators which have numerous applications in analysis and geometry. Fundamental integral inequalities, dyadic models and sharp estimates of solutions will be treated. A solution to the existence problem, and a complete characterization of removable singularities will be given for a class of nonlinear PDE of Lane-Emden type. Date: July 24, 2008 Abstract: In this talk we will discuss the local integrability of distributions $q$ satisfying $Dq=div f$ for some matrix-valued map $f=(f^i_j)$ in the Hardy space $H^1$. As a consequence, we will discuss the existence and the local representation of the hydrostatic pressure, and the derivation of Euler-Lagrange equations for volume-preserving, elastic energy-minimizing vector fields in $R^n$. (Joint work with Dr Aram Karakhanyan, a former postdoc at ANU) Date: July 31, 2008 Abstract: By a "billiard" I mean a bounded plane domain D, with smooth (enough) boundary. Quantum billiards is the study of properties of eigenfunctions of the Laplacian on D, i.e. solutions of $\Delta u = Eu$, where $u$ is a function on D vanishing at the boundary and $E$ is a real number, in the limit as $E \to \infty$. This large-E limit is the "classical limit" in which eigenfunctions exhibit behaviour related to the classical billiard system (a billiard ball moving around inside D, bouncing off the boundary). I will talk about Quantum Ergodicity, which is the property that "most of" the eigenfunctions become uniformly distributed in D, asymptotically as $E \to \infty$, i.e. they are the same size, on average, in all parts of the domain D; and the related property of Quantum Unique Ergodicity, which is the same property with the words "most of" deleted. There has been a conjecture open for the last 20 years or so, that certain domains called "stadium domains" are quantum ergodic but not quantum unique ergodic, which I solved very recently. I will motivate and discuss this conjecture and talk a little about the proof, which is surprisingly simple. Date: August 7, 2008 Abstract: Advances in 3D imaging technology are enabling us to peer inside an enormous range of materials and create the need for quantitative measures of the complex structures being unveiled. The most fundamental description of an object is in terms of its topology: how many pieces, and what types of holes does it have? To answer this question we turn to work from the past decade or so on algorithms for extracting topological information from scattered point data. One of the most important concepts to emerge is that of topological persistence - a method for quantifying the significance of features using good old-fashioned homology. This talk will introduce the theory of persistent homology and describe how we are adapting it to the study of 3D digital images. Perhaps unsurprisingly, this is not as simple as it first appears. It turns out the problems stem from topological issues of a different kind - that of adjacency in digital grids. Date: August 14, 2008 Abstract: Abstract: A topological field theory in d dimensions associates to each (d-1)-dimensional closed manifold M an inner-product space V(M), and to each d-dimensional manifold W with boundary M a vector v(W) in V(M), satisfying certain natural axioms; for example, V(-) takes disjoint unions to tensor products, and behaves well under diffeomorphisms. There are many flavours of topological field theories - one may for example assume that all of the manifolds are oriented, or spin, or carry a free action of a finite group G. It turns out that the two-dimensional case is especially simple: two-dimensional topological field theories are equivalent to commutative algebras with inner product (also known as commutative Frobenius algebras). In this talk, we relate this to a result in topology. Harvey has introduced a manifold with boundary containing the (6g-6)-dimensional Teichmueller space of genus g closed Riemann surfaces as its interior, and we define a filtration F(i) of this space such that the inclusion of F(i) into F(i+1) is i-connected. (The proof is an application of a triangulation of Teichmueller space constructed by Harer.) This result and its generalizations explain many pheonomena in topological field theory, including theorems of Moore and Seiberg, Moore and Segal, and Turaev. Date: August 21, 2008 Abstract: An important question in the regularity theory of minimal submanifolds, which remains largely open, is understanding the local behavior of the submanifolds near points with tangent planes of multiplicity $>1$ (``branch points''). This talk will describe some well known as well as recent results concerning this question for various classes of minimal submanifolds, focusing mainly on the class of stable minimal hypersurfaces (i.e. those having non-negative second variation of area). We shall discuss natural geometric conditions which prevent the existence of branch points, and in the absence of these conditions, ways to construct examples with branch points as well as what is known in general about the size of the subset of branch points and the asymptotic behavior of the submanifolds near them. Date: August 28, 2008 Abstract: Finite element methods provide a powerful tool in engineering analysis and numerical solutions of boundary value problems. These methods are based on variational principle and therefore have intrinsic mathematical beauty and can be applied to complicated and nonlinear problems. The other advantage of finite element methods is that when applied to differential equations they naturally fit into the concept of so-called weak solutions in Sobolev spaces. There are many situations in solving differential equations where classical solution does not exist and one has to rely on the weak solutions. In the first part of the talk, we give a brief introduction to the finite element methods starting from the concept of weak solutions. We mainly concentrate on the problem of structural mechanics and show the applications of finite elements in linear and nonlinear problems of structural mechanics. The second part of the talk is concentrated on the modern research in the field of nearly incompressible materials. As it is well-known that the finite element approximations based on low order methods do not converge uniformly when applied to nearly incompressible elasticity, many different methods are proposed to alleviate this problem. We examine the classical Hu-Washizu mixed formulation for plane and three dimensional problems in linear elasticity with the emphasis on behavior in the incompressible limit. We also show the extension to nonlinear materials. The approach based on Hu-Washizu formulation is applied for the meshes with quadrilaterals and hexahedra. However, the Hu-Washizu formulation based on lowest order finite elements does not yield a stable formulation when the meshes consist of simplices. In this case, we propose a novel method based on primal and dual meshes and bases leading us again to the displacement-based formulation. Numerical examples will be presented for all considered approaches. Date: September 4, 2008 Abstract: There are two "integrability" criteria in statistical mechanics. One is the star-triangle equation, also known as the Yang-Baxter equation; the other is a generalization of Gaussian integration to fermionic or bosonic systems. In this talk I plan to describe both criteria in a historical context and from different points of view, omitting the more technical details. Then I will discuss some of our recent results obtained using these techniques, showing some of our latest results for the pair correlation functions in the (planar) Z-invariant Ising model and the quantum Ising chain, ending with a few remarks on the chiral Potts model. This talk will be aimed at a non-specialist audience. Date: September 11, 2008 Abstract: We construct a von Neumann algebra generated by regular representations of the infinite-dimensional nilpotent group B_0^Z. These representations were defined and studied by Alexander Kosyak and depend on a quasi-invariant Gaussian measure on a group in which the group B_0^Z is dense. We prove that under a certain sufficient condition on the measure, these von Neumann algebras are type III_1 factors (according to the classification of Alain Connes).Ê Date: September 15, 2008 (Note: Monday) Abstract: In recent years, evolving hypersurfaces with normal speed equal to a function of its principal curvatures has been applied in several cases to prove new geometric inequalities. In this talk we will describe some of these results and discuss a recent result of the speaker, how a weak level-set formulation of the flow with speed equal to a positve power of the mean curvature can be used to give a new proof of the isoperimtric inequality in R^n up to dimension 8. We also show how this technique can be extended to complete, simply connected 3-manifolds with nonpositive sectional curvature to yield a new proof of the Euclidean isoperimetric inequality on such manifolds. Date: September 18, 2008 Abstract: The classical birth-death processes provide useful models for practitioners of population modeling. Simple variations of these processes can often lead to difficult, if not intractable, equations. Partly due to these mathematical complexities many researcher have employed various approximation and numerical methods in their studies of these processes. One approximation method that has been used with good results replaces the probability of a specific state transition rate by an average transition rate. In this survey talk, the history and relevant mathematics of this approximation technique will be detailed. Some recent applications in population modeling and epidemiology will be presented. Date: September 25, 2008 Abstract: The consistency problem in statistics has to do with the asymptotic properties of estimates as the observed data set size grows without bound. Problems with a similar flavour occur in the analysis of the numerical proprties of the algorithms used in the estimation process. For example, the Gauss-Newton algorithm for minimizing a nonlinear sum of squares has generically a first order convergence rate. However, under appropriate circumstances, this rate asymptotes to second order as the number of observations tends to infinity. This result goes quite a way towards explaining the observed very satisfactory performance of this method. The Bock iteration is a method for estimating parameters in a system of ordinary differential equations from observations made on the state variables. The first distinguishing feature is the explicit treatment of the system of differential equations as constraints on the estimation problem. Typically this constraint is formulated as a set of algebraic constraints by discretizing the differential system. The basic estimation approach is to apply Newton's method to find a stationary point of the Lagrangian of the resulting system. The second distinguishing feature is a simplification, similar in spirit to that used in Gauss-Newton, made by ignoring second order partial derivatives in the Newton iteration. Considerable success in using this modification has been reported. The asymptotics of the Bock iteration are examined in this talk. An unusual feature is the derivation of estimates for the size of the Lagrange multipliers. It is shown that fast convergence rates similar to those associated with the Gauss-Newton iteration can be achieved. However, the conditions under which this proves possible are distinctly more restrictive. Date: October 2, 2008 Abstract: Date: October 9, 2008 Abstract: The Riemann Mapping Theorem, a special case of Hilbert's XXII problem, is equivalent to mapping a planar domain to the unit disk by a quasiconformal (QC) mapping; see Ahlfors and Bers 1960. These QC mappings generalize to higher dimensions (where conformal mappings are trivial). The main problem was to characterize the QC images of the unit ball, i.e., QC balls. In his reviews Ahlfors emphasized the related question of characterizing the images of the unit sphere under QC mappings of space, i.e., quasispheres. In their plenary talks at the ICM, both Ahlfors and Gehring reiterated these questions. Our solution comes from reflection: a sense-reversing idempotent homeomorphism of \hat{R}^3 onto itself. Any reflection F of the sphere \hat{R}^3 has a set T of fixed points forming a topological sphere. Bing showed the existence of a wild reflection, i.e. the complementary domains are not simply connected. He asked for an explicit example which we construct (and show a picture of). On the other hand smooth reflections are tame so Sullivan, Heinonen, Semmes etc asked if there are wild QC reflections. Nevertheless, we prove: {\it The complementary domains of a QC reflection are simply connected. } A uniform sphere is a topological sphere whose blowups only have topolological spheres as limits. So we have { \it T is the fixed set of a QC reflection iff it is a uniform sphere. } We show how this solves Ahflors' problem of characterising quasispheres. The QC Riemann Mapping Theorem is a one-sided version of the result for quasispheres. We discuss the general ideas, as well as illustrating with pictures and applications. Date: October 16, 2008 Abstract: A standard first graduate course in functional analysis will cover Banach and Hilbert spaces, dual spaces, weak topologies, bounded linear operators on Banach spaces, and perhaps something on Banach lattices and on Banach algebras. I have developed a variant of this theory. In this one replaces the norm on a Banach space E by a sequence of norms, one on each of the spaces E^n. This enables us to develop new results on the following topics, among others: (1) the geometry of Banach spaces and absolutely summing operators; (2) multi-continuous linear operators, generalizing the regular operators on a Banach lattice; (3) a more general theory of orthogonality in Banach spaces, and a new duality theory; (4) applications to modules over Banach algebras, especially L^2(G) over the group algebra L^1(G) for a locally compact group G; (5) connections with the theory of amenable groups and algebras. I will try to sketch some of these new theories. Date: October 23, 2008 Abstract: Classical Hodge theory deals with projective and smooth algebraic varieties defined over $\mathbb C$. If $X$ is such a variety defined over the subfield $\mathbb Q$, then associated to $X$ are two cohomology theories:\\ \\ 1) The Betti cohomology of $X$, denoted $H^*_B(X)$. It is defined as the singular cohomology of the underlying space of the complex points of $X$, $X(\mathbb C)$, with coefficients in $\mathbb Q$. \\ \\ 2) the algebraic de Rham cohomology of $X$, denoted by $H^*_{dR}(X)$. This is defined as the hypercohomology of the de Rham complex $\Omega^*_{X/\mathbb Q}$.\\ \\ There is a canonical isomorphism $H^*_{dR}(X)\otimes_{\mathbb Q}\mathbb C$ $\simeq$ $H^*_B(X)\otimes_{\mathbb Q}\mathbb C$. If we choose bases for these two $\mathbb Q$-vector spaces, this isomorphism is given by an element in $GL_N(\mathbb C)$, called the period matrix. In general little is known about the entries in this matrix, although the degree of transcendence of the field generated over $\mathbb Q$ by its entries, was conjectured by Grothendieck to have dimension equal to that of a reductive algebraic group defined over $\mathbb Q$, the Mumford-Tate group associated to $X$.\\ \\ We will discuss the $p$-adic analogues of this classical situation. Beginning with the work of Barsotti, Tate, Grothendieck during the 1950's and 1960's, the theory of $p$-divisible groups developed. In his Nice ICM talk, Grothendieck raised the problem of the "mysterious functor". In 1978 Fontaine introduced a ring, $B_{crys}$, and a field, $B_{dR}$, containing it, each equipped with additional structure, in order to obtain $p$-adic analogues of the period isomorphism described above. In Fontaine's 1982 Annals paper, a number of conjectures relating $p$-adic etale cohomology to crystalline and de Rham cohomology were made. These conjectures have all been established due to the work, over the last thirty years of various mathematicians including Fontaine, Bloch, Breuil, Colmez, Faltings, Gabber, Hyodo, Illusie, Kato, Kisin, Messing, Niziol, Olsson, Tsuji, .... Just as with classical Hodge theory, $p$-adic Hodge theory has had important applications in number theory and arithmetic geometry, including, in particular, the proof, by Khare and Wintenberger , of Serre's Modularity Conjecture. Date: October 30, 2008 Abstract: I plan to explain how the theory of compact Lie groups motivated the study in algebraic topology of finite loop spaces and other objects such as "p-compact groups". Here p is a prime and I will present two reasons why one should work one prime at a time (and how one does this). The last part of the talk will be concerned with joint work with N. Castellana and J.A. Crespo about larger, infinite, loop spaces. Date: November 6, 2008 Abstract: Date: November 13, 2008 Abstract: Date: November 20, 2008 Abstract: In recent joint work with Martha Yip we gave a combinatorial formula for Macdonald polynomials. The formula is a weighted sum of paths and the construction of the paths is completely elementary. The mystery is that these paths are describing subtle information about fancier objects: loop groups, integrable hierarchies of differential equations, representation theory and cohomology theories. I will try to formulate some of my speculations about how these objects are related. The underlying symmetry is certainly touching many parts of modern mathematics and it is all the more amazing that the elementary combinatorics of paths has something deep to say about it all. Date: November 27, 2008 Abstract: Date: December 4, 2008 Abstract: |
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
|
Page last updated: 7 January, 2008 Please direct all enquiries to: MSI webmaster Page authorised by: Director, MSI |
| The Australian National University - CRICOS Provider Number 00120C |
for
measures
on
or
a manifold
,
where
is
a second order elliptic operator and the above equation is understood
in the following weak sense:
for
all compactly supported smooth functions
.
Similarly, parabolic equations for measures are defined. A typical
example of
where
is
a vector field. However,
.
An important motivation to study such equations is that the Fokker-Planck-Kolmogorov
equations for transition probabilities and stationary distributions of diffusion processes are
of this type. Major problems concern existence and uniqueness of solutions, existence of densities,
and regularity and bounds for densities. In the past decade considerable progress has been achieved in this area.
However, there are challenging open problems with very simple formulations. All necessary concepts
will be explained in the talk; no special knowledge is assumed.