Issue 1, Volume 39, March 2014

2014

1. Preface Preface Ngô Bao Châu PDF

   1
2. Congruent numbers John Coates PDF

   ▸ Abstract

   This paper is an expository account of the recent important progress made by Ye Tian on the ancient congruent number problem.

   3-10
3. Recent progress on the Gross–Prasad conjecture Wee Teck Gan PDF

   ▸ Abstract

   We report on recent progress on both the local and global Gross–Prasad conjectures for unitary groups.

   11-33
4. Parabolic flows in complex geometry Duong H. Phong PDF

   ▸ Abstract

   An informal introduction is provided to some current problems on canonical metrics in complex geometry, from the point of view of non-linear partial differential equations. The emphasis is on the parabolic flow approach and open problems.

   35-54
5. Perfectoid spaces and the weight-monodromy conjecture, after Peter Scholze Takeshi Saito PDF

   ▸ Abstract

   The monodromy weight conjecture is one of the main remaining open problems on Galois representations. It implies that the local Galois action on the $\ell$-adic cohomology of a proper smooth variety is almost completely determined by the traces. Peter Scholze proved the conjecture in many cases including smooth complete intersections in a projective space, using a new powerful tool in rigid geometry called perfectoid spaces. The main arguments of the proof as well as basic ingredients in the theory of perfectoid spaces are presented.

   55-68
6. The Tate conjecture for K3 surfaces—a survey of some recent progress Vasudevan Srinivas PDF

   ▸ Abstract

   The most important open questions in the theory of algebraic cycles are the Hodge Conjecture, and its companion problem, the Tate Conjecture. Both these questions attempt to give a description of those cohomology classes on a nonsingular proper variety which are represented by algebraic cycles, in terms of intrinsic structure which is present on the cohomology of such a variety (namely, a Hodge decomposition, or a Galois representation). For the Hodge conjecture, the case of divisors (algebraic cycles of codimension 1) was settled long ago by Lefschetz and Hodge, and is popularly known as the Lefschetz (1,1) theorem, though there is little general progress beyond that case. However, even this case of divisors is an open question for the Tate Conjecture, in general, even for divisors on algebraic surfaces. After giving an introduction to these problems, I will discuss the recent progress on the Tate Conjecture for K3 surfaces, around works of M. Lieblich, D. Maulik, F. Charles and K. Pera.

   69-85
7. Representation of finite groups: conjectures, reductions, and applications Pham Huu Tiep PDF

   ▸ Abstract

   In this survey, we discuss some basic problems in representation theory of finite groups, including some long-standing conjectures of Alperin, Brauer, and others. A possible approach to some of these problems is to use the classification of finite simple groups to reduce the problem in consideration to some, more specific, questions about simple groups. We will describe recent progress on reduction theorems in this direction. We will also outline applications of these results to various problems in group theory, number theory, and algebraic geometry.

   87-109