Saturday, July 20, 2013
Morning
- 7h45 – 8h30: Registration
- 8h30-8h45: Tea time
- 8h45-10h15: Duong H. Phong (Columbia University, US), Non-linear heat flows in complex geometry.
- 10h15-10h30: Tea break
- 10h30-12h00: Vansudevan Srinivas (Tata Institute, India), The Tate Conjecture for K3 surfaces.
Afternoon
- 14h00-14h15: Tea time
- 14h15-15h45: Takeshi Saito (University of Tokyo, Japan), The weight-monodromy conjecture and perfectoid spaces.
- 15h45-16h00: Tea break
- 16h00: Round table discussion on Annual Meeting 2014.
Sunday, July 21, 2013
Morning
- 8h30-8h45: Tea time
- 8h45-10h15: Gan Wee Teck (National University of Singapore), Recent progress on the Gross-Prasad Conjecture.
- 10h15-10h30: Tea break
- 10h30-12h00: John Coates (Cambridge University, UK), Congruent Numbers.
Contents
1. Prof. John Coates – Cambridge University, UK
Title: Congruent Numbers
Abstract:
“The congruent number problem is the oldest unsolved major problem in number theory, and, in retrospect, the simplest and most down to earth example of the conjecture of Birch and Swinnerton-Dyer. After a brief description of the history of the problem, I shall discuss Y. Tian’s beautiful recent extension to composite numbers, with arbitrarily many prime factors, of Heegner’s original proof of the existence of infinitely many congruent numbers. I hope also to say a little at the end about possible generalizations of Tian’s work to other elliptic curves.”
2. Prof. Gan Wee Teck – National University of Singapore
Title: Recent Progress in the Gross-Prasad Conjecture
Abstract:
“I will discuss the local and global Gross-Prasad conjecture concerning the restriction of a representation or an automorphic form of a classical group to a smaller such group. In particular, I will discuss the resolution of the local conjecture by Waldspurger, Beuzart-Plessis and progress for the global conjecture due to Jacquet-Rallis, Wei Zhang, Yifeng Liu and Hang Xue. I will also mention a refinement of the global conjecture due to Ichino and Ikeda.”
3. Prof. Dương Hồng Phong – Columbia University, USA
Title: Non-linear heat flows in complex geometry
Abstract:
“A fundamental result in complex geometry is the Uniformization Theorem, which asserts the existence of a metric of constant scalar curvature on complex curves. The analogue in higher dimensions would be the existence of a canonical metric, with or without singularities, as required by the underlying geometry. A powerful approach to the construction of such metrics is as fixed points of a non-linear heat flow. We discuss several such flows, including the Yang-Mills flow and the K\”ahler-Ricci flow, with emphasis on issues of long-time existence, convergence, or formation of singularities.”
4. Prof. Takeshi Saito – University of Tokyo, Japan
Title: The monodromy weight conjecture and perfectoid spaces
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 l-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. In the talk, I plan to sketch the main arguments of the proof after briefly introducing basic ingredients in the theory of perfectoid spaces.”
5. Prof. Vasudevan Srinivas – Tata Institute, India
Title: The Tate conjecture for K3 surfaces
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 codimen-sion 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 and D. Maulik, F. Charles and K. Pera.”