Clay Research Fellowship
Plenary Lecture - 1
10.30 - 11.30, July 26 (Tue), 2016 Room: Pecatu Hall 1 and 2
Title: Small (and large) gaps between primes
Many of the most famous and most important questions on the distribution of primes can be cast as solving systems of linear equations with prime variables. The twin prime conjecture, Goldbach’s conjecture, k-term arithmetic progressions of primes and most questions about small gaps between primes can all be seen in this manner, as well as several questions with applications to diophantine geometry or cryptography. We will describe some of the progress on these questions, with a particular emphasis on establishing weak forms of some of these questions which has led to new results on bounded gaps between primes and large gaps between primes, amongst other things.
||Sudhir Ramakant GHORPADE
Indian Institute of Technology, Bombay
Plenary Lecture - 2
13.30 - 14.30, July 26 (Tue), 2016 Room: Pecatu Hall 1 and 2
Title: Number of zeros of multivariate polynomials over finite fields
A univariate polynomial of degree with coefficient in a field has at most zeros in . Likewise, a bivariate homogeneous polynomial of degree over has at most non-proportional zeros in or in other words, at most zeros in the projective space . However, multivariate polynomials will, in general, have infinitely many zeros. But when is the finite field with elements, it makes sense to ask for similar degree-based bounds on the number of zeros of one or more multivariate polynomials of given degrees. We consider in particular, the following question. Let be positive integers and let denote the ring of polynomials in variables with coefficients in and the -dimensional projective space over .
Question: What is the maximum number, say , of common zeros that a system of linearly independent homogeneous polynomials of degree in can have in ?
A remarkable conjecture by Tsfasman and Boguslavsky made about two decades ago predicted an explicit and rather complicated formula for at least when . This was already known to be valid in the case , thanks to the results of Serre (1991) as well as S∅rensen (1991), The conjectured formula for was shown to be true in the case by Boguslavsky (1997). In this talk, we will outline these developments and report on a recent progress in a joint work with Mrinmoy Datta where we show that the Tsfasman-Boguslavsky Conjecture holds in the affirmative if and is false in general if . We will also mention some newer conjectures and results that are partly obtained in collaboration with Peter Beelen and Mrinmoy Datta.
The University of Warwick
Plenary Lecture - 3
10.30 - 11.30, July 27 (Wed), 2016 Room: Pecatu Hall 1 and 2
Title: Taming infinities
Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of ”renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until now.
Plenary Lecture - 4
13.30 - 14.30, July 27 (Wed), 2016 Room: Pecatu Hall 1 and 2
Title: Noncommutative real algebraic geometry of Kazhdan’s property (T)
Kazhdan’s property (T) is a representation-theoretic property of groups, which was introduced by Kazhdan in 1967, but has found numerous applications in an amazingly large variety of subjects from representation theory and ergodic theory to combinatorics (expanders) and the theory of networks. I will start with a gentle introduction to the emerging subject of ”noncommutative real algebraic geometry,” a subject which deals with equations and inequalities in noncommutative algebras over the reals, with the help of analytic tools such as representation theory and operator algebras. I will then present a surprisingly simple proof that a group G has property (T) if and only if a certain inequality in the group algebra R[G] is satisfied. This in- equality is rather amenable to computer-assisted analysis and would be useful in finding new examples of property (T) groups or better bounds of the Kazhdan constants of known property (T) groups.
||Sug Woo Shin
Plenary Lecture - 5
10.30 - 11.30, July 28 (Thu), 2016 Room: Pecatu Hall 1 and 2
Title: Motives with Galois group of type G2-construction of Gross and Savin revisited
Serre asked whether there exists a motive (over Q) with Galois group G2. Put it in another way, the question is to find (a compatible family of) ell-adic Galois representations whose image has Zariski closure G2. This has been answered affirmatively since 2010 by Dettweiler and Reiter, Khare-Larsen-Savin, Yun, and Patrikis (including generalizations to exceptional groups other than G2). In this talk I revisit the construction of Gross-Savin (which was conditional when proposed in 1998) which aims to realize such a motive in the cohomology of a Siegel modular variety of genus 3 via exceptional theta correspondence between G2 and PGSp6. Then I will explain that the construction is now unconditional due to my recent work with Arno Kret on the construction of GSpin(2n + 1)-valued Galois representations in the cohomology of Siegel modular varieties.
Institut Teknologi Bandung
Plenary Lecture - 6
13.30 - 14.30, July 28 (Thu), 2016 Room: Pecatu Hall 1 and 2
Title: Fractional Integrals and Morrey Spaces
The talk will consist of two parts. In the first part, I will give a brief survey on Hardy-Littlewood maximal operator and fractional integral operators, especially about their boundedness properties on (generalized) Morrey spaces. The two operators were intially studied by G. Hardy & J. Littlewood and M. Riesz in the 1920’s and the 1930’s. Their boundedness on Morrey spaces was first proved by D. Adams and also by F. Chiarenza and M. Frasca in the 1970’s and the 1980’s. The results were generalized among others by E. Nakai in 1995, and I will present some recent results in this direction to indicate how ‘hot’ the topic is in the last two decades. In the second part, I will talk about basic but important properties of Morrey spaces, namely their inclusion properties. Here I will present some recent results not only for the ‘strong’ Morrey spaces but also weak type Morrey spaces, including their generalized versions. I shall also present some related results, and mention some future works.
||Wee Teck Gan
National University of Singapore
Plenary Lecture - 7
15.00 - 16.00, July 28 (Thu), 2016 Room: Pecatu Hall 1 and 2
Title: Twisted Bhargava cubes and boxes
In his groundbreaking thesis work from 2001, Manjul Bhargava has extended Gauss’s composition laws for binary quadratic forms to higher degree forms. One crucial ingredient in his work is the parametrisation of the or- bits of lattice points in a prehomogeneous vector space by quantities of arithmetic interest. Using the SL(2,Z)x SL(2,Z)xSL(2,Z)-action on 2x2x2 cubes, he gave a simpler description of Gauss’s law, and using the SL(2,Z) x SL(3,Z) x SL(3,Z)-action on 2x3x3 boxes, he discovered a composition law on binary cubic forms. In this lecture, we revisit these two orbit problems over fields (rather than integers) and consider twisted versions of them. In particular, we shall discuss the question: what are the orbits parametrised by?
||Van H. Vu
Percey F. Smith Professor of Mathematics, Yale
Plenary Lecture - 8
10.30 - 11.30, July 29 (Fri), 2016 Room: Pecatu Hall 1 and 2
Title: Random functions
Studying the zeroes of of a function and its derivates (local maxima and minima, saddle points etc) is among the most basic and important problems in mathematics. Needless to say, there are hundreds of theorems on the topic, many give valuable information in some special cases, while being totally useless in others. There are no universally good estimates. In this talk, I am going to tell the story about what happens in the average case, namely when our function is random, following the footsteps of Littlewood, Offord, Kac, Erdos, Ibragimov and many leading mathematicians of the last century, and concluding with a current development that finished of some of the most basic problems.
Institute of Mathematics, CAS, Beijing
Plenary Lecture - 9
15.30 - 16.30, July 29 (Fri), 2016 Room: Pecatu Hall 1 and 2
Title: Recent progress on multiplier ideal sheaves and optimal L2 extensions
In this talk, we’ll present our recent proof of Demailly’s strong openness conjecture about multiplier ideal sheaves, solution of optimal L2 extension problem, and their applications.