- Damien Roy: droy at uottawa.ca
- Daniel Fiorilli: Daniel.Fiorilli@uottawa.ca

**Speaker:**Damien Roy (U. of Ottawa)**Title:**Spectra of exponents of approximation.**Abstract:**A basic question in Diophantine approximation is to determine how well a fixed point in real n-space can be approximated by rational points or more generally by linear subspaces defined over Q of some fixed dimension d. This is measured, for each dimension d, by two real numbers: an exponent of best approximation and an exponent of uniform approximation. The extreme cases where d=1 or d=n-1 are especially important.- For best approximation, Khintchine established, in 1926, a pair of inequalities relating the exponents in dimension 1 and n-1. These where shown to be best possible by Jarnik in 1935-36. In 2009, Michel Laurent refined these estimates through a series of inequalities relating the whole set of n-1 exponents of best approximation. It was shown by the speaker that these inequalities in fact describe all possible (n-1)-tuples of such exponents, the so-called spectrum of these exponents.
- For uniform approximation, Jarnik gave, in 1938, several systems of inequalities relating the exponents in dimension 1 and n-1. These were improved by Oleg German in 2012 and, last week, Antoine Marnat proved that the elegant inequalities of German are also best possible.

**Speaker:**Jennifer Park (McGill University and CRM)**Title:**Effective Chabauty for symmetric powers of curves.**Abstract:**Faltings' theorem states that curves of genus g> 1 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry and tropical geometry to show that we can also give an effective bound on the number of rational points outside of the special set of the d-th symmetric power of X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.

**Speaker:**Daniel Fiorilli (U. of Ottawa)**Title:**Moments for primes in arithmetic progressions**Abstract:**In this talk I would like to discuss recent progress on the first and second moment of primes in arithmetic progressions. Time-permitting, I will discuss :- My thesis results on the average discrepancy (on average over q, there tends to be less primes congruent to 1 modulo q than to 6 modulo q).
- Vaughan's approximation which has similar discrepancies.
- A probabilistic model for the variance, which makes predictions in ranges where all other techniques fail to even yield a conjecture (including assuming the Hardy-Littlewood conjecture). This comes from a rigorous computation of the large deviations of an associated random variable.

**Speaker:**Marc Munsch (U. de Montréal, CRM)**Title:**Character sums and congruences equations**Abstract:**I will discuss some recent joint work with Igor Shparlinski on character sums and its applications to some congruence equations. These have a lot of applications, for instance they give us some information about the distribution of quadratic residues. We will focus on the solvability of au=x modulo a prime p, where x lies in an interval I and u in some approximate subgroup U of F_{p}. It has been recently shown by Cilleruelo and Garaev that this equation is solvable for almost all a, provided that |I|> p^{5/8+ε}and |U| > p^{3/8}(where U is a subgroup or a set of consecutive powers of a primitive root).We will prove, for almost all primes, some bounds on character sums which allow us to obtain similar results for a wider range of sizes for U and I. The proof combines classical analytic number theory with combinatorics techniques.

**Speaker:**Hugo Chapdelaine (U. Laval)**Title:**Introduction to the Archimedean rank one Stark conjecture**Abstract:**One may think of the Stark conjectures as providing some refinements taking place on the outskirts of the fashionable Langlands program which aims at relating automorphic representations of algebraic groups to Galois representations of the Galois group through their L-functions. I'll start by giving a gentle introduction to the rank one Archimedean Stark conjecture which relates the first derivative of a partial L-function at s=0 associated to a number field K to the logarithm of (a conjectural) S-unit living in an abelian extension of K. I'll present a simple (and new) proof of this conjecture in the simplest case, namely when the base field K is equal to Q in order to give the flavor of what one has to do in order to prove this conjecture. Secondly, I'll introduce a family of real analytic Eisenstein series (GL_{2}automorphic forms) and explain how they can be used to prove that classical circular numbers (Stark units when the base field is Q) can be interpreted as an arithmetic intersection (in the sense of Arakelov theory) of two torsion points on a generalised elliptic curve.

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