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Current Developments in Mathematics
Current Developments in Mathematics
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...
Savin, O.
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...
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...
Wu, Sijue
Waldspurger, Jean-Loup
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.
Morel, Fabien
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.
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.
Meeks, William H.; Peréz, Joaquín
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].
Lovász, László
Dafermos, Mihalis
These notes accompany a set of lectures for the Current Developments in
Mathematics conference, Harvard, November 22, 2008.
Taubes, Clifford Henry
Siu, Yum-Tong
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.
Li, Jun
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.
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.