The rational solutions on an elliptic curve form a finitely generated abelian group, but the maximum number of generators needed is not known. Goldfeld conjectured that if one also fixes the j-invariant (i.e. the complex structure), then 50% of such curves should require 1 generator and 50% should have only the trivial solution. Smith has recently made substantial progress towards this conjecture in the special case of elliptic curves in Legendre form. I'll discuss recent work with Lemke Oliver, which bounds the average number of generators for general j-invariants.

It has been known for almost a hundred years that most polynomials with integral coefficients are irreducible and have a big Galois group. For a few dozen years, people have been interested in whether the same holds when one considers sparse families of polynomials—notably, polynomials with plus-minus 1 coefficients. In particular, “some guy on the street” conjectures that the probability for a random plus-minus 1 coefficient polynomial to be irreducible tends to 1 as the degree tends to infinity (a much earlier conjecture of Odlyzko-Poonen is about the 0-1 coefficients model).

It is a major challenge in Combinatorial Geometry to understand the intersection structure of the edges in a geometric or topological graph, in the Euclidean plane. One of the few "tight" results in this direction is the the Crossing Lemma (due to Ajtai, Chvatal, Newborn, and Szemeredi 1982, and independently Leighton 1983). It provides a relation between the number of edges in the graph and the number of crossings amongst these edges. This line of work led to several Ramsey-type questions of geometric nature. We will focus on two recent advances.

Abstract: 

In 1925, Tarski asked whether a disk in R^2 can be partitioned into finitely many pieces which can be rearranged by isometries to form a square of the same area. The restriction of having a disk and a square with the same area is necessary. In 1990, Laczkovich gave a positive answer to the problem using the axiom of choice. We give a completely explicit (Borel) way to break the circle and the square into congruent pieces. This answers a question of Wagon. Our proof has three main components. The first is work of Laczkovich in Diophantine approximation.

Abstract: The Ising model, and its generalisation, the Potts model, are two classical graph-colouring models for magnetism and antiferromagnetism. Albeit their simple formulation, these models were instrumental in explaining many real-world magnetic phenomena and have found various applications in physics, biology and computer science. While our understanding of these models as modeling magnets has been constantly improving since the early twentieth century, little progress was made in treatment of Potts antiferromagnets.

Abstract: A central problem in complex analysis is how to describe zero sets of power series in terms of their coefficients. In general, it is difficult to obtain precise results for a given function. However, when the function is defined by a power series, whose coefficients are independent random variables, such results can be obtained. Moreover, if the coefficients are complex Gaussians, the results are especially elegant. In particular, in this talk I will discuss some different notions of "rigidity" of the zero sets.

Given a convex polytope P, what is the number of integer points in P? This problem is of great interest in combinatorics and discrete geometry, with many important applications ranging from integer programming to statistics. From a computational point of view it is hopeless in any dimensions, as the knapsack problem is a special case. Perhaps surprisingly, in bounded dimension the problem becomes tractable. How far can one go? Can one count points in projections of P, finite intersections of such projections, etc?

Abstract: Many classical problems on the distribution of prime numbers (and related objects with multiplicative origin) admit function field analogues which can be proved in the large finite field limit. The first results of this type were obtained in the 70's by Swinnerton-Dyer and S. D. Cohen and in recent years there has been a resurgence of activity in this field.

From Betti cohomology to crystalline cohomology

Read more about Landau Lecture 1: From Betti cohomology to crystalline cohomology (colloquium)

Representation theory of non-compact real groups, such as SL(2,R), is a fundamental discipline with uses in harmonic analysis, number theory, physics, and more. This theory is analytical in nature, but in the course of the 20th century it was algebraized and geometrized (the key contributions are by Harish-Chandra for the former and by Beilinson-Bernstein for the latter). Roughly and generally speaking, algebraization strips layers from the objects of study until we are left with a bare skeleton, amenable to symbolic manipulation.

Abstract: If X is an object such that the notion of an automorphism of X is defined (e.g., an algebraic structure, a graph, a topological space, etc.), then one can define an equivalence relation ∼ on X via x ∼ y if and only if α(x) = y for some automorphism α of X. The equivalence classes of ∼ are called the automorphism orbits of X. Say that X is highly symmetric if and only if all elements of X lie in the same automorphism orbit. Finite highly symmetric objects are studied across various mathematical disciplines, e.g. in combinatorics, graph theory and geometry. When

Abstract: Mirror symmetry is a correspondence between symplectic geometry on a manifold M and complex geometry on a mirror manifold W. The question of why one sort of geometry on M should be reflected in another sort of geometry on the topologically distinct manifold W, and the question of how to find W given M, are a priori highly mysterious. One attempt to explain the mysteries of mirror symmetry is the SYZ conjecture, which asserts that the mirror manifold W can be realized as the moduli space of certain objects of a category associated to M.

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an 'enlarged' copy H^+ of a fixed hypergraph H. These include well-known problems such as the Erdos-Sos 'forbidding one intersection' problem and the Frankl-Furedi 'special simplex' problem.