Abstract: The notion of normality was intended to capture random behaviour in the digits of numbers, but some clearly patterned numbers passs the normality test; we propose a stronger test of normality that is passed by almost all numbers but failed by Champernowne's number. We also propose a modular definition of normality to generalize the original definition and make a wild conjecture.

Abstract: We will deal with a gap in a result of Bjorn Poonen. He found explicit examples of hypersurfaces of degree d≥3 and dimension n≥1 over any field, such that the group of automorphisms over the algebraic closure is trivial, except for the pairs (n,d)=(1,3) or (2,4). Examples of the former pair, cubic curves, do not exist. We deal with the remaining case, quartic surfaces. For any field k of characteristic at most 19 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher characteristics. Over the rationals we also construct an example on which the set of rational points is Zariski dense.

Abstract: Dirichlet's theorem on primes in arithmetic progressions characterizes those linear polynomials which take on prime values infinitely often. However, this is where the current state of knowledge ends. For the case of polynomials with higher degrees, heuristic arguments lead us to believe that for an irreducible polynomial with integer coefficients, if the leading coefficient is positive and the polynomial has no fixed prime divisor, then the polynomial represents primes infinitely often. I will discuss the case for quadratic polynomials with an emphasis on the work of Iwaniec.

Abstract: I'll give an expository talk, following the recent article of Soundararajan, on the theorem of Goldston, Yildirim, and Pintz that there are infinitely many primes p such that the next prime q satisfies q – p = o(log p).

Abstract: Recurrence sequences appear in many areas of mathematics and are widely studied. A recurrence sequence of order n is defined by a polynomial of degree n and n initial values. Given a sequence, there are several important questions that one can ask about the sequence, for example: "Which primes divide at least one number in the sequence? What is their (relative) density in the set of all primes?" and "Which primes divide several consecutive numbers in the sequence? What is their density in the set of all primes?" This last question is the one I will address. Though density is at heart an analytic problem, we will explore ways of rephrasing density questions in an algebraic way, using the Chebotarev Density Theorem.

Abstract: For many questions in explicit arithmetic geometry of curves, one needs detailed information on the rational points of the Jacobian of the curve. A first step is to bound the free rank of the finitely generated group that they form. For hyperelliptic curves [curves admitting a model of the form y² = f(x)], we have fairly good methods for producing bounds, and curves of genus 2 are always hyperelliptic. Curves of genus 3 (for instance smooth plane quartics) are generally not hyperelliptic. A straightforward generalization of the standard methods to these curves would lead to infeasible computational tasks involving number fields up to degree 756. We propose a modification, which requires number fields up to degree 28 and is sometimes just about feasible.

Abstract: Zeta functions and L-functions contain arithmetic information when evaluated at special values, such as s = 0. In the 1970s, Stark conjectured that the derivatives of L-functions at s = 0 can be evaluated by certain algebraic units. Under certain circumstances, these “Stark units” should also produce abelian extensions of number fields. After introducing the First Order Stark Conjecture, we explore an extended version in which no prime splits completely.