| September 3, 2009 | |
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| October 8, 2009 | |
| Tuesday, October 20, 2009 |
Abstract: A C*-algebra can be thought as a noncommutative topological space or as a collection of infinite matrices of complex numbers endowed with an interesting algebraic and topological structure. The C*-algebras have significant applications in different areas of mathematics (geometry, topology, ergodic theory), parts of physics (quantum mechanics and statistical mechanics) and other sciences. Understanding the structure and classification of C*-algebras was and continues to be one of the most important researh directions in Operator Algebras (Elliott and Kirchberg, I.C.M. 1994, Rordam, I.C.M. 2006). In this talk I will present, in a natural context and giving basic definitions and examples, a joint work with Mikael Rordam (J. Reine Angew. Math. 2007) in which we characterize, in the separable case, for a large and important class of C*-algebras that are "infinite" in some specific sense (introduced 10 years ago by Kirchberg and Rordam) a certain condition of noncommutative zero dimensionality (introduced by Brown and Pedersen) that proved to be very successful in Elliott's well known Classification Program for C*-algebras (I.C.M. 1994). (It is perhaps worth to mention also that many C*-algebras of interest happen-sometimes surprisingly-to satisfy this condition). Some interesting consequences of this result that concern the structure of C*-algebras will be also discussed. Our theorem strongly generalizes a result of Perera and Rordam (J. Funct. Anal. 2004) and, in the separable case, a result of Zhang. Contact person: Guoliang Yu |
| October 22, 2009 | |
| October 29, 2009 | Abstract: A substantial part of extremal combinatorics studies relations existing between densities with which given (fixed size) combinatorial structures may appear in unknown (and presumably very large) structures of the same type. Using basic tools and concepts from algebra, analysis and measure theory, we develop a general framework that allows to treat all problems of this sort in an uniform way and reveal mathematical structure that is common for most known arguments in the area. The backbone of this structure is made by commutative algebras defined in terms of finite models of the associated first-order theory. In this talk I will give a general impression of how things work in this framework, and we will pay a special attention to concrete applications of our methods. Contact person: Mark Sapir |
| November 5, 2009 | Abstract: One distinguishing feature of Hamiltonian dynamical systems is that such systems, with very few exceptions, tend to have numerous fixed and periodic points. In 1984 Conley conjectured that a Hamiltonian diffeomorphism (i.e., the time-one map of a Hamiltonian flow) of a torus has infinitely many periodic points or, more precisely, such a diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period. This fact was proved by Hingston some twenty years later, in 2004. Similar results for Hamiltonian diffeomorphisms of surfaces of positive genus were also established by Franks and Handel. Of course, one can expect the Conley conjecture to to hold for a much broader class of closed manifolds and this is indeed the case. For instance, by now, the conjecture has been proved for the so-called closed, symplectically aspherical manifolds (including tori and surfaces of positive genus) and the Calabi-Yau manifolds using symplectic topological techniques. In this talk we will discuss the underlying reasons for the existence of periodic orbits for Hamiltonian flows and maps and outline a proof of the Conley conjecture. Contact person: Başak Gürel |
| November 12, 2009 | Abstract: Universality limits for random matrices describe the spacings between successive eigenvalues of random matrices, and their distribution. In the unitary case, one way to establish such universality limits is via the the theory of entire functions of exponential type, and de Branges spaces. We shall discuss the method and some recent results. No background on de Branges spaces, or universality is assumed. Contact person: Ed Saff |
| November 19, 2009 | Abstract: Combinatorial independence originated from the work of Rosenthal on characterization of Banach spaces containing l_1 isomorphically. Based on joint work with Wen Huang, David Kerr, and Xiangdong Ye, I will discuss how it leads to unified combinatorial and functional-analytic approaches to the study of various mixing properties in dynamics. Contact person: Guoliang Yu |
| November 26, 2009 | |
| December 3, 2009 | |
| December 10, 2009 | Abstract: The constraint satisfaction problem CSP(G) for a directed graph G is the problem of deciding of an input directed graph H whether there exists a homomorphism from H to G. For the two-element complete directed graph G with no loops this problem is solvable in polynomial time (corresponds to the class of bipartite directed graphs), while for the three-element complete directed graph G with no loops it is NP-complete (corresponds to the class of 3-colorable directed graphs). The dichotomy conjecture formulated by Feder and Vardi in 1993 asserts that for any directed graph (or finite relational structure) G the problem CSP(G) is in P or NP-complete, therefore the intermediate complexity classes, which exist by Ladner's result if P does not equal NP, cannot be realized with constraint satisfaction problems. This conjecture has been verified in numerous special cases but it is still open in general. The latest results have been achieved with the help of universal algebra. First, we can fully characterize in algebraic terms those relational structures G for which the so called "local consistency" algorithm works. Second, we can fully characterize in algebraic terms those relational structures G for which the set of all homomorphisms from H to G can be represented in polynomial space (this is a generalization of solving a system of linear equations over a finite field). We review these results, and show how these two algorithms can be combined to solve an even larger class of problems. Contact person: Ralph McKenzie |
| January 14, 2010 | |
| January 21, 2010 | Abstract: It is a classical result that if C is a compact Riemann surface of genus g at least two then C has at most 84(g-1) automorphisms. On the other hand, suppose that f(z_1,z_2,...,z_n) is a holomorphic function. If f(0)=0 then 1/|f|^2 is not an integrable function. The largest value of t for which 1/|f|^{2t} is integrable is a real number between zero and one which is a measure of how much f vanishes at the origin. The set of all such t, as the holomorphic function f varies, exhibits some unusual boundedness properties. We show that these two apparently unrelated phenomena are in fact closely related. Contact person: Dietmar Bisch |
| January 28, 2010 | |
| February 4, 2010 | Abstract: We are given categories A, B, and S together with functors F:A-->S and G:B-->S such that for each object a of A, there exists an object b of B such that F(a) is isomorphic to G(b). We are asking whether the assignment from a to b can be made functorial. CLL is a theorem of pure category theory, whose statement is complex but that can be summarized as follows: Theorem (P. Gillibert and F. Wehrung, 2008). If A, B, S, F, G form a "larder", then such a thing can be done. This result can be used either to deduce properties of "large" objects (how large being determined by infinite combinatorics) from properties of "small" objects, but also conversely, for example by establishing a lifting property of a finite diagram of finite Boolean semilattices by way of a representation result on uncountable algebras. We illustrate this result by applications to various open problems in universal algebra, lattice theory, and ring theory. Contact person: Ralph McKenzie |
| February 11, 2010 | Abstract: We will explain how planar pictures give rise to a host of associative algebras and subalgebras including some that allow reconstruction of the origingal planar structure. Connections with random matrices, algebraic geometry and low dimensional topology will be explored. Contact person: Dietmar Bisch |
| February 18, 2010 | Abstract: Studying possible relations between a mathematical structure and its description in a formal language Model Theory developed a hierarchy of 'logical perfection'. On the very top of this hierarchy we discovered a new class of structures called Zariski geometries. A joint theorem by Hrushovski and the speaker (1993) indicated that the general Zariski geometry looks very much like an algebraic variety over an algebraically closed field, but in general is not reducible to an algebro-geometric object. Later the present speaker established that a typical Zariski geometry can be explained in terms of a possibly noncommutative 'co-ordinate' algebra. Moreover, conversely, many quantum algebras give rise to Zariski geometries and the correspondence 'Co-ordinate algebra - Zariski geometry' for a wide class of algebras is of the same type as that between commutative affine algebras and affine varieties. General quantum Zariski geometries can be approximated (in a certain model-theoretic sense) by quantum Zariski geometries at roots of unity. The latter are of a finitary type, where Dirac calculus has a well-defined meaning. We use this to give a mathematically rigorous calculation of a Feynman propagator in a few simple cases. Contact person: Mark Sapir |
| February 25, 2010 | Abstract: The analysis of high dimensional data sets is useful in a large variety of applications, from machine learning to dynamical systems: data sets are often modeled as low-dimensional, noisy data sets embedded in high-dimensional spaces; dynamical systems often have very high-dimensional state spaces but sometimes interesting dynamics occurs on low-dimensional sets. We discuss several problems associated with the analysis of the geometry of such sets, and with the approximation of functions on such sets, together with some solutions: in particular we discuss how to construct random walks on such data sets and perform multiscale analysis of them and their applications (especially to machine learning); how to construct robust coordinate systems for data sets; how to estimate reliably the intrinsic dimensionality of the data when only few noisy samples are available. Contact person: Akram Aldroubi |
| March 4, 2010 | Abstract: From habitat degradation and climate change to spatial spread of invasive species, dispersals play a central role in determining how organisms cope with a changing environment. The dispersals of many organisms depend upon local biotic and abiotic factors, and as such are often non-random. In this talk we will discuss some recent progress on the effects of non-random dispersal on two competing species in heterogeneous environments via reaction-diffusion-advection models. Contact persons: Phil Crooke, Zhian Wang, and Glen Webb |
| March 11, 2010 | |
| March 18, 2010 | Abstract: Everybody learns in the introductory group theory course that every group of prime order is cyclic. In fact, there are other numbers with this property. Namely, every group of order n is cyclic if and only if n and φ(n) are relatively prime, where φ is Euler's totient function. This was observed by George A. Miller in 1899. In 1948 Paul Erdos proved that the number of positive integers n < x with this property is asymptotically e-γx/logloglog x, where γ denotes Euler's constant. In this survey talk I will discuss similar results characterizing those numbers n for which every group of order n is abelian, nilpotent, solvable, etc. and giving an asymptotic for the number of positive integers n < x having these properties. In some cases the relevant question is whether a group with some property (for example, a simple group) of a given order exists. Paul Erdos has a forgotten paper on this question. I will speculate why this paper was omitted from his list of publications. Now the classification of finite simple groups yields that there are asymptotically 3.21/3 x1/3/log x simple group orders less than x. I will also tell how I got Erdos number 1. Contact person: Ralph McKenzie |
| Friday, March 26, 2010 3:10 pm |
Abstract: Just like in complex analysis, Liouville theorems in partial differential equations assert that if u is an entire solution of a specific equation and it is contained in an admissible class of functions, then it is the trivial solution. We will start the talk with an overview of Liouville theorems for nonlinear elliptic and parabolic equations. Then we will show some typical applications of Liouville theorems in qualitative studies of solutions. Contact person: Gieri Simonett |
| March 30, 2010 | Abstract: Lie groups have been studied from almost all possible points of view. However, most of the attention has been focused on semi-simple Lie groups. The large scale geometry of a general connected Lie group is a challenging field, where many natural questions are still wide open. In this talk, we will consider a continuous analogue of the "word problem" in Lie groups. We will see that as a corollary of our results, we obtain an algebraic characterization of Gromov-hyperbolic homogeneous Riemannian manifolds. Similar results are obtained in the p-adic setting. An award ceremony will be held before the colloquium at 3:00 p.m. in SC 1425. |
| April 1, 2010 | Abstract: We consider the dichotomy in the title in the context of properties of large sparse finite structures. This in turn leads to the nowhere dense vs somewhere dense dichotomy which can be defined in a surprising variety of different ways. But basically most of these characterizations are static - using properties of the homomorphism order. Recently the nowhere dense classes were characterized by properties of counting function of subobjects. This then relates to Lovasz and al. (for dense graphs) and Benjamini-Schramm (for ultrasparse graphs) statistics. Contact person: Ralph McKenzie |
| April 8, 2010 | Abstract: While univariate polynomial interpolation has been a basic tool of scientific computing for hundreds of years, multivariate polynomial interpolation is much less understood. Already the question from which polynomial space to choose an interpolant to given data has no obvious answer. The talk presents, in some detail, one answer to this basic question, namely the "least interpolant" of Amos Ron and the speaker which, among other nice properties, is degree-reducing, then seeks some remedy for the resulting discontinuity of the interpolant as a function of the interpolation sites, then addresses the problem of a suitable representation of the interpolation error and the nature of possible limits of interpolants as some of the interpolation sites coalesce. The last part of the talk is devoted to a more traditional setting, the complementary problem of finding correct interpolation sites for a given polynomial space, chiefly the space of polynomials of degree <= k for some k, and ends with a particular recipe for good interpolation sites in the square, the Padua points. References: http://pages.cs.wisc.edu/~deboor/multiint/ Contact person: Larry Schumaker |