
Thursday, February 1, 2007 UBC Campus, Room WMAX 110 (PIMS)

3:003:50 
Adrian Belshaw (SFU)
Strong normality and modular normality
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.

3:504:10 
tea break 
4:105:00 
Ronald van Luijk (PIMS/SFU/UBC)
Quartic K3 surfaces without nontrivial automorphisms
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.



Thursday, March 1, 2007 UBC Campus, Room WMAX 110 (PIMS)

3:003:50 
Vishaal Kapoor (UBC)
Almostprimes represented by quadratic polynomials
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.

3:504:10 
tea break 
4:105:00 
Greg Martin (UBC)
GoldtonYildirimPintz and small gaps between primes
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).



Thursday, March 15, 2007 SFU Campus, Room ASB 10900 (IRMACS)

3:003:50 
Amy Goldlist (UBC)
Prime divisors of certain quartic linear recurrences
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.

3:504:10 
tea break 
4:105:00 
Peter Borwein (SFU)
The Liouville function and friends



Thursday, March 29, 2007 UBC Campus, Room WMAX 110 (PIMS)

3:003:50 
Nils Bruin (SFU)
Fake 2descent on the Jacobian of a genus3 curve
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 y^{2} = 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.

3:504:10 
tea break 
4:105:00 
Stefan Erickson (Colorado College)
Variations on a theme of Stark
Abstract:
Zeta functions and Lfunctions contain arithmetic information when
evaluated at special values, such as s = 0. In the 1970s, Stark conjectured
that the derivatives of Lfunctions 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.

