- Damien Roy: droy at uottawa.ca

**Speaker:**Francesco Pappalardi (U. of Roma 3)**Title:**Properties of reductions of groups of rational numbers**Abstract:**Let Γ be a multiplicative subgroup ofQ ^{*}and let p be a prime for which the valuation v_{p}(x)=0 for every x in Γ. Then the group Γ_{p}={x (mod p) : x in Γ} is a well defined subgroup ofF _{p}^{*}. We will consider various properties of Γ_{p}as p varies and propose various new results in analogy with the old Artin Conjecture for Primitive roots.

**Speaker:**Mohammad Bardestani (U. of Ottawa)**Title:**On a theorem of Dedekind and its application in statistical Galois theory**Abstract:**Using an important theorem of Dedekind and the Chebotarev density theorem, we will discuss some statistical behavior of polynomials modulo prime numbers.

**Speaker:**Abdellah Sebbar (U. of Ottawa)**Title:**Equivariant functions and vector-valued modular forms**Abstract:**For any discrete group Γ and any 2-dimensional complex representation ρ of Γ, we introduce the notion of ρ-equivariant functions, and we show that they are parameterized by vector-valued modular forms. We also provide examples arising from the monodromy of differential equations.

**Speaker:**Kenneth S. Williams (Carleton U.)**Title:**A Product-to-Sum Formula with Applications to Sums of Squares**Abstract:**A product-to-sum formula is proved for a certain class of infinite products and applied to the evaluation of the number of representations of a positive integer as a sum of an even number of squares.

**Speaker:**Daniel Fiorilli (U. of Michigan)**Title:**Nuclear physics and number theory**Abstract:**While the two fields named in the title seem unrelated, there is a strong link between them. Indeed, random matrix theory makes predictions in both fields, and some of these predictions can be verified rigorously on the number theory side. This amazing connection came to life during a meeting between Freeman Dyson and Hugh Montgomery at the Institute for Advanced Study. Random matrices are now known to predict many statistics about zeta functions, such as moments, low-lying zeros and correlations between zeros. The goal of this talk is to discuss this connection, focusing on number theory. We will cover both basic facts about the zeta functions and recent developments in this active area of research.

**Speaker:**Piotr Maciak (École Polytechnique de Lausanne)**Title:**Bounds for the Euclidean minima of number fields and function fields**Abstract:**The Euclidean division is a basic tool when dealing with the ordinary integers. It does not extend to rings of integers of algebraic number fields in general. It is natural to ask how to measure the "deviation" from the Euclidean property, and this leads to the notion of Euclidean minimum. The case of totally real number fields is of special interest, in particular because of a conjectured upper bound (conjecture attributed to Minkowski). The talk will present some recent results concerning abelian fields of prime power conductor. We will also define Euclidean minima for function fields and give some bounds for this invariant. We furthermore show that the results are analogous to those obtained in the number field case.

**Speaker:**Christelle Vincent (Stanford U.)**Title:**Weierstrass points on Drinfeld modular curves**Abstract:**We will start by introducing Weierstrass points, a finite set of points of geometric interest on curves of genus greater than or equal to 2, and state some results about Weierstrass points on classical modular curves. We will then introduce and give some motivation for the study of the so-called Drinfeld setting, a function field analogue of some aspects of the theory of modular forms and elliptic curves. In this setting, Drinfeld constructed families of modular curves defined over a complete, algebraically closed field of positive characteristic. Finally, we will present some tools from the theory of Drinfeld modular forms that we developed to study the Weierstrass points of these curves and the results that we have obtained.

**Speaker:**Christopher Daw (University College London)**Title:**The André-Oort conjecture for a product of modular curves**Abstract:**In this talk, we will explain the André-Oort conjecture for a product of two modular curves. We will give the unconditional proof of Pila, relying on the Pila-Zannier strategy and, in particular, the Pila-Wilkie counting theorem from o-minimality.

**Speaker:**Greg Doyle (Carleton U.)**Title:**Quadratic Form Gauss Sums**Abstract:**Let n and r be positive integers, p a prime and s any integer coprime to p.

For an r-dimensional quadratic form Q, we call the exponential sum in r variables with summands given by exp(2πiQs/p^{n}) to be a quadratic form gauss sum, and denote this sum by G(Q;s;p^{n}).

When r=1, G(Q;s;p^{n}) is the well known quadratic Gauss sum, first examined by Gauss in the early 19th century.

Using elementary methods, we show that if the coefficients of Q satisfy certain divisibility properties, we may express G(Q;s;p^{n}) as a product of r quadratic Gauss sums. We then discuss where we may apply this result.

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