Volumes of hyperbolic 3-manifolds
Details
Speaker: David Gabai (Princeton)
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Speaker: David Gabai (Princeton)
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Speaker: Simon Donaldson (Imperial)
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Speaker: Stephen Smale (CUHK)
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Speaker: William Thurston (Cornell)
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Speaker: Curtis T. McMullen (Harvard)
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Speaker: John Morgan (Stony Brook)
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Speaker: Michael Atiyah (Edinburgh)
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Abstract: A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions
of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic manifolds arising as a quotient of the upper half-plane by a discrete “arithmetic” subgroup of SL2(R) (for example, SL2(Z), and in this case the corresponding eigenfunctions are called Maass cusp forms). In this case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions become equi-distributed. I will discuss some recent progress which has led to a resolution of this conjecture, and also on a holomorphic analog for classical modular forms.
Speaker: Kannan Soundararajan (Stanford)
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Abstract: We will discuss ergodic theory over the moduli space of compact Riemann surfaces, and its connections with algebraic geometry, Teichmüller theory and billiard tables with optimal dynamics.
Speaker: Curtis T. McMullen (Harvard)
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Abstract: Let G be a connected reductive group over a number field and let H be an endoscopic group of G. A conjecture of Langlands predicts that there exists a correspondence between automorphic representations of H(A) and automorphic representations of G(A), where A is the ring of adeles of the ground field. Langlands’ idea of proof is to compare the Arthur-Selberg’s trace for- mulas of H and G. It’s necessary to solve many problems, in particular two problems of harmonic analysis over a local field: the transfer conjecture and the fundamental lemma. These two questions have remained open until the decisive result of Ngô Bao Châu, two years ago. In my talk, I’ll try to explain what is the endoscopic transfer and what is the fundamental lemma. I will give several statements of that lemma, more or less sophisticated. I’ll try to explain the situation at the present time. In fact, all the useful problems are resolved even if certain related questions of harmonic analysis remain open.
Speaker: Jean-Loup Waldspurger (Jussieu)
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Abstract: Questions in automorphic forms and number theory often get tied up with the magnificent, largely conjectural, edifice of functoriality, a simple instance being the desire to know if certain four-dimensional Galois representations occurring inside the cohomology of Siegel modular threefolds are symplectic. Of particular importance, besides base change, is the transfer of automorphic forms from orthogonal and symplectic groups to the general linear group, which sheds light on many prob- lems. Crucial progress has been made of late in the work of Arthur via the twisted trace formula, extending the earlier results known for generic cusp forms, which had relied on the elegant converse theorem insight of Piatetski-Shapiro. Part of what makes Arthur’s approach work is the incredible recent progress on the (different guises of) fundamental lemma due to Ngô, Waldspurger and others. This talk will try to introduce the basic global statements, a few ideas, and applications.
Speaker: Dinakar Ramakrishnan (Caltech)
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Abstract: The classification of non-compact hyperbolic 3-manifolds with finitely-generated fundamental groups depends on an understanding of the topology and asymptotic geometry of their ends. A number of advances in recent years have made this classification possible, and more. I will discuss the background and features of this theory, and its applications to a fuller understanding of the ways in which these manifolds (compact and non-compact) cover and approximate each other.
Speaker: Yair Minsky (Yale)