| September 3, 2009 |
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| October 8, 2009 |
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| 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 |
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| 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 |
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| December 3, 2009 |
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| 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 |
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| January 21, 2010 |
Contact person: Dietmar Bisch |
| January 28, 2010 |
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| February 4, 2010 |
Contact person: Ralph McKenzie |
| February 11, 2010 |
Contact person: Dietmar Bisch |
| February 18, 2010 |
Contact person: Mark Sapir |
| February 25, 2010 |
Contact person: Akram Aldroubi |
| March 11, 2010 |
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| March 25, 2010 |
Contact person: Larry Schumaker |
| April 1, 2010 |
Contact person: Ralph McKenzie |
| April 8, 2010 |
Contact person: Gieri Simonett |
| April 22, 2010 |
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