Mostrando recursos 1 - 20 de 46

  1. Recent Developments in Mathematical Quantum Chaos

    Zelditch, Steve
    This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigenfunctions with anomalous concentration. We cover the lower bounds on entropies of quantum limit measures due to Anantharaman, Nonnenmacher, and Rivière on compact Riemannian manifolds with Anosov flow. These lower bounds give new constraints on the possible quantum limits. We also cover the non-QUE result of Hassell in the case of the Bunimovich stadium. We include some discussion of Hecke eigenfunctions and recent results...

  2. Phase Transitions, Minimal Surfaces and A Conjecture of De Giorgi

    Savin, O.

  3. The Arf-Kervaire Invariant Problem in Algebraic Topology: Introduction

    Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C.
    This paper gives the history and background of one of the oldest problems in algebraic topology, along with a short summary of our solution to it and a description of some of the tools we use. More details of the proof are provided in our second paper in this volume, The Arf-Kervaire invariant problem in algebraic topology: Sketch of the proof. A rigorous account can be found in our preprint The non-existence of elements of Kervaire invariant one on the arXiv and on the third author’s home page. The latter also has numerous links to related papers and talks we have given on the subject since announcing our...

  4. Survey on the Fundamental Lemma

    Châu, Ngô Bao
    This is a survey on the recent proof of the fundamental lemma. The fundamental lemma and the related transfer conjecture were formulated by R. Langlands in the context of endoscopy theory for automorphic representations in "L-indistinguishability for SL(2)," Canad. J. Math. 31 (1979), no. 4, 726–785. Important arithmetic applications follow from the endoscopy theory, including the transfer of automorphic representations from classical groups to linear groups and the construction of Galois representations attached to automorphic forms via Shimura varieties. Independent of applications, endoscopy theory is instrumental in building a stable trace formula that seems necessary to any decisive progress toward Langlands’ conjecture on functoriality of automorphic representations. ¶ There are already...

  5. Wellposedness of the two- and three-dimensional full water wave problem

    Wu, Sijue

  6. Some recent results on representations of p-adic special orthogonal groups

    Waldspurger, Jean-Loup

  7. Universal formulas for counting Nodal curves on surfaces

    Tzeng, Yu-Jong
    These notes are supplementary to the author’s lectures at the 2010 conference on Current Developments in Mathematics, held on November 19-20 in Cambridge Massachusetts.

  8. On the Friedlander-Milnor conjecture for groups of small rank

    Morel, Fabien

  9. The Arf-Kervaire problem in algebraic topology: Sketch of the proof

    Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C.
    We provide a sketch of the proof of the non-existence of Kervaire Invariant one manifolds using equivariant homotopy theory. A treatment of the statement and history of the problem can be found in “The Arf-Kervaire problem in algebraic topology: Introduction”. Our goal here is to introduce the reader to the techniques used to prove the result and to familiarize them with the kinds of computations needed.

  10. Unearthing the visions of a master: harmonic Maass forms and number theory

    Ono, Ken
    Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These non-holomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outline the general facets of the theory, and we give several applications to number theory: partitions and q-series, modular forms, singular moduli, Borcherds products, extensions of theorems of Kohnen-Zagier and Waldspurger on modular L-functions, and the work of Bruinier and Yang on Gross-Zagier formulae. What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery.

  11. Properly embedded minimal planar domains with infinite topology are Riemann minimal examples

    Meeks, William H.; Peréz, Joaquín

  12. On the classification of topological field theories

    Lurie, Jacob
    Our goal in this article is to give an expository account of some recent work on the classification of topological field theories. More specifically, we will outline the proof of a version of the cobordism hypothesis conjectured by Baez and Dolan in [2].

  13. Very large graphs

    Lovász, László

  14. The evolution problem in general relativity

    Dafermos, Mihalis
    These notes accompany a set of lectures for the Current Developments in Mathematics conference, Harvard, November 22, 2008.

  15. Notes on the Seiberg-witten equations, the Weinstein conjecture and embedded contact homology

    Taubes, Clifford Henry

  16. Techniques for the analytic proof of the finite generation of the canonical ring

    Siu, Yum-Tong

  17. Lectures on stability and constant scalar curvature

    Phong, D.H.; Sturm, Jacob
    An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kähler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson’s infinite-dimensional GIT, and conditions on the closure of orbits of almost-complex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, Tian-Yau-Zelditch approximations, estimates for moment maps, complex Monge-Amp`ere equations and pluripotential theory, and the Kähler-Ricci flow.

  18. Recent progress in GW-invariants of Calabi-Yau threefolds

    Li, Jun

  19. Finite generation of a canonical ring

    Kawamata, Yujiro
    The purpose of this note is to review an algebraic proof of the finite generation theorem due to Birkar-Cascini-Hacon-McKernan [5] whose method is based on the Minimal Model Program (MMP). An analytic proof by Siu [57] will be reviewed by Mihai Paun.

  20. Three topics in additive prime number theory

    Green, Ben
    We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes.

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