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RTG SeminarThursday, February 22, 2018LCB 222 - 3:30PM Beatrice Pozzetti, Heidelberg University Maximal representations Abstract: An important application of bounded cohomology is the theory of maximal representations: a class of exceptionally well behaved homomorphisms of fundamental groups of Kaehler manifolds (most notably fundamental groups of surfaces and finite volume ball quotients) in Hermitian Lie groups (as Sp(2n,R) or SU(p,q)). I will discuss recent rigidity results for maximal representations of fundamental groups of ball quotients on infinite dimensional symmetric spaces (joint with Duchesne and Lecureux) as well as surprising geometric properties of the more flexible maximal representations of fundamental groups of surfaces (joint with Burger). Wednesday, August 31, 2016LCB 219 - 3:30PM Jenny Wilson, Stanford University Representation theory and higher-order stability in the configuration spaces of a manifold Abstract: Let F_k(M) denote the ordered k-point configuration space of a connected open manifold M. Work of Church and others shows that for a given manifold, as k increases, this family of spaces exhibits a phenomenon called homological "representation stability" with respect to the natural symmetric group actions. In this talk I will explain what this means, and describe a higher-order "secondary stability" phenomenon among the unstable homology classes. The project is work in progress, joint with Jeremy Miller. Wednesday, February 3 , 2016LCB 219 - 3:15PM Burt Totaro , UCLA Group cohomology and algebraic cycles Abstract: The classifying space BG of a compact Lie group G (such as a finite group) can be viewed as a direct limit of complex algebraic varieties. As a result, it makes sense to consider the "Chow ring" of algebraic cycles on BG. This ring maps to the cohomology ring of BG, but they are usually not the same. The Chow ring seems to have a close relation to the cobordism of BG. We survey what is known. Friday, October 2, 2015LCB 219 - 3:10PM Kathryn Mann, UC Berkeley Automatic continuity for homeomorphism groups Abstract: To what extent does the algebraic structure of a topological group determine its topology? Many (but not all) examples of real Lie groups G have a unique Lie group structure, meaning that every abstract isomorphism G -> G is necessarily continuous. In this talk, I'll discuss a recent much stronger result for groups of homeomorphisms of manifolds: every homomorphism from Homeo(M) to any other separable topological group is necessarily continuous. This is part of a broader program to show that the topology and algebraic structure of the group of homeomorphisms (or diffeomorphisms) of a manifold M are intimately linked, and also deeply connected to the topology of M itself. Time permitting, we'll discuss applications in geometric topology, groups acting on manifolds, and connections with a new program to study the quasi-isometry type of homeomorphism groups. Monday April 20, 2015LCB 222 - 3:00PM On volumes of representations Abstract: In many instances one can define the notion of volume of a representation of the fundamental group of a closed manifold M into a simple (non-compact) Lie group G. This is so for instance if M is a surface and the symmetric space associated to G is hermitian, that is carries an invariant 2-form, or if M is a 3-manifold and G is a complex group, equivalently the associated symmetric space carries an invariant 3-form. When M is not compact the definition of volume of a representation presents interesting difficulties; in this talk we will show how bounded cohomology can be used to define an invariant generalizing the volume of a representation and we will see how this invariant is connected with the deformation theory of such representations. This is joint work with Michelle Bucher and Alessandra Iozzi. Tueday February 18, 2014LCB 215 - 3:30PM The Yang-Mills flow on Kaehler manifolds Abstract: The fundamental work of Donaldson and Uhlenbeck-Yau proves the the smooth convergence of the Yang-Mills flow of stable integrable unitary connections on hermitian vector bundles over Kaehler manifolds. This was generalized by Bando and Siu to incorporate certain (singular) hermitian structures on reflexive sheaves. Bando-Siu also conjectured what happens when the initial sheaf is unstable; namely, that the limiting behavior should be controlled by the Harder-Narasimhan filtration of the sheaf. In this talk I will describe the solution to this question, which draws on the work of several authors. Thursday November 21, 2013LCB 219 - 4:00PM Dynamical degrees of birational transformations of projective surfaces Abstract: The dynamical degree λ(f) of a birational transformation f measures the exponential growth rate of the degree of the formulae that define the n-th iterate of f. I will describe the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of λ(f) and the structure of the conjugacy class of f. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed, well ordered set of algebraic numbers. |
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