L3

This is the second of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. (Still work in progress.)

Behrend showed that conventional Donaldson-Thomas invariants can be written as the Euler characteristic of the moduli space of semistable sheaves weighted by a "microlocal obstruction function" \mu.

In previous work, the speaker defined Donaldson-Thomas type invariants "counting" coherent sheaves on a Calabi-Yau 3-fold using

Euler characteristics of sheaf moduli spaces, and more generally, of moduli spaces of "configurations" of sheaves. However, these invariants are not deformation-invariant.

We now combine these ideas, and insert Behrend's microlocal obstruction \mu into the speaker's previous definition to get new generalized Donaldson-Thomas invariants. Microlocal functions \mu have a multiplicative property implying that the new invariants transform according to the same multiplicative transformation law as the previous invariants under change of stability condition.

Then we show that the invariants counting pairs in the previous seminar are sums of products of the new generalized Donaldson-Thomas invariants. Since the pair invariants are deformation invariant, we can deduce by induction on rank that the new generalized Donaldson-Thomas invariants are unchanged under deformations of the underlying Calabi-Yau 3-fold.

Refining Julia Robinson's 1949 work on the undecidability of the first order theory of Q, we prove that Z is definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points.

Let E be an elliptic curve over the rationals. To get an asymptotic to the number of primes p

Recently there has been increasing interest in discrete analogues of classical operators in harmonic analysis. Often the difficulties one encounters in the discrete setting require completely new approaches; the most successful current approaches are motivated by ideas from classical analytic number theory. This talk will describe a menagerie of new results for discrete analogues of operators ranging from twisted singular Radon transforms to fractional integral operators both on R^n and on the Heisenberg group H^n. Although these are genuinely analytic results, key aspects of the methods come from number theory, and this talk will highlight the roles played by theta functions, Waring's problem, the Hypothesis K* of Hardy and Littlewood, and the circle method.

We recall that an elliptic curve is a curve of genus one with a rational point on it. Certain algorithms for determining the structure of the group of rational points on an elliptic curve produce a whole set of curves of genus one and then require that we determine which of these curves has a rational point.

Unfortunately no algorithm which has been proved to terminate is known for doing this. Such an algorithm or proof would probably have profound implications for the study of elliptic curves and may shed light on the Birch and Swinnerton-Dyer conjecture.

This talk will be about joint work with Samir Siksek (Warwick) on the development of a new algorithmic criterion for determining that a given curve of genus one has no rational points. Both the theory behind the criterion and recent attempts to make the criterion computationally practical, will be detailed.

It is a wide open question whether the set of rational points on a smooth quartic surface in projective three-space can be nonempty, yet finite. In this talk I will treat the case of diagonal quartics V, which are given by: a x^4 + b y^4 + c z^4 + d w^4 = 0 for some nonzero rational a,b,c,d. I will assume that the product abcd is a square and that V contains at least one rational point P. I will prove that if none of the coordinates of P is zero, and P is not contained in one of the 48 lines on V, then the set of rational points on V is dense. This is based on joint work with Adam Logan and David McKinnon.

We consider finite sequences $h = (h_1, . . . h_s)$ of real polynomials in $X_1, . . . ,X_n$ and assume that

the semi-algebraic subset $S(h)$ of $R^n$ defined by $h1(a1, . . . , an) \leq 0$, . . . , $hs(a1, . . . , an) \leq 0$ is

bounded. We call $h$ (quadratically) archimedean if every real polynomial $f$, strictly positive on

$S(h)$, admits a representation

$f = \sigma_0 + h_1\sigma_1 + \cdots + h_s\sigma_s$

with each $\sigma_i$ being a sum of squares of real polynomials.

If every $h_i$ is linear, the sequence h is archimedean. In general, h need not be archimedean.

There exists an abstract valuation theoretic criterion for h to be archimedean. We are, however,

interested in an effective procedure to decide whether h is archimedean or not.

In dimension n = 2, E. Cabral has given an effective geometric procedure for this decision

problem. Recently, S. Wagner has proved decidability for all dimensions using among others

model theoretic tools like the Ax-Kochen-Ershov Theorem.