L6

Manin’s conjecture predicts the asymptotic behavior of the number of rational points of bounded height on Fano varieties over number fields. We prove this conjecture for a family of nonsplit singular quartic del Pezzo surfaces over arbitrary number fields. For the proof, we parameterize the rational points on such a del Pezzo surface by integral points on a nonuniversal torsor (which is determined explicitly using a Cox ring of a certain type), and we count them using a result of Barroero-Widmer on lattice points in o-minimal structures. This is joint work in progress with Marta Pieropan.

There are many interesting problems surrounding the unit group U(RG) of the ring RG, where R is a commutative ring and G is a finite group. Of particular interest are the finite subgroups of U(RG). In the seventies, Zassenhaus conjectured that any u in U(ZG) is conjugate, in the group U(QG), to an element of the form +/-g, where g is an element of the group G. This came to be known as the "(first) Zassenhaus conjecture". I will talk about the recent construction of a counterexample to this conjecture (this is joint work with L. Margolis), and recent work on related questions in the modular representation theory of finite groups.

Signed permutation modules of symmetric groups and Iwahori-hecke algebras
 

Markoff triples are integer solutions  of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond.  After reviewing some of these, we will discuss  joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular,  that for almost all primes the induced graph is connected.  Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.
Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces  $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.
 

We will discuss a basic framework to deal with coherent sheaves on schemes over $\mathbb{Z}$, involving infinite-dimensional results on the geometry of numbers. As an application, we will discuss basic results, old and new, on arithmetic ampleness, such as Serre vanishing, Nakai-Moishezon, and Bertini. This is joint work with Jean-Benoît Bost.

We present a state-sum construction of TFTs on r-spin surfaces which
uses a combinatorial model of r-spin structures. We give an example of
such a TFT which computes the Arf invariant for r even. We use the
combinatorial model and this TFT to calculate diffeomorphism classes of
r-spin surfaces with parametrized boundary.

We will briefly revisit Voronoi summation in its classical form and mention some of its many applications in number theory. We will then show how to use the global Whittaker model to create Voronoi type formulae. This new approach allows for a wide range of weights and twists. In the end we give some applications to the subconvexity problem of degree two $L$-functions. 

Exponents of Diophantine approximation are defined to study specific sets of real numbers for which Dirichlet's pigeonhole principle can be improved. Khintchine stated a transference principle between the two exponents in the cases  of simultaneous approximation and approximation by linear forms. This shows that exponents of Diophantine approximation are related, and these relations can be studied via so called spectra. In this talk, we provide an optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation for both simultaneous approximation and approximation by linear forms. This is joint work with Nikolay Moshchevitin.

It is an old problem in number theory to count number fields of a fixed degree and having a fixed Galois group for its Galois closure, ordered by their absolute discriminant, say. In this talk, I shall discuss some background of this problem, and then report a recent work with Stanley Xiao. In our paper, we considered quartic $D_4$-fields whose ring of integers has a certain nice algebraic property, and we counted such fields by their conductor.

Let $P$ be a random polynomial of degree $d$ such that the leading and constant coefficients are 1 and the rest of the coefficients are independent random variables taking the value 0 or 1 with equal probability. Odlyzko and Poonen conjectured that $P$ is irreducible with probability tending to 1 as $d$ grows.  I will talk about an on-going joint work with Emmanuel Breuillard, in which we prove that GRH implies this conjecture. The proof is based on estimates for the mixing time of random walks on $\mathbb{F}_p$, where the steps are given by the maps $x \rightarrow ax$ and $x \rightarrow ax+1$ with equal probability.