Imperial

The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and G2 manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.

SEMINAR CANCELLED

Let G be a finitely generated group generated by g_1,..., g_n. Consider the alphabet A(G) consisting of the symbols g_1,..., g_n and the symbols '+' and '-'. The words in this alphabet represent elements of the integral group ring Z[G]. In the talk we will investigate the computational problem of deciding whether a word in the alphabet A(G) determines a zero-divisor in Z[G]. We will see that a version of the Atiyah conjecture (together with some natural assumptions) imply decidability of the zero-divisor problem; however, we'll also see that in the group (Z/2 \wr Z)^4 the zero-divisor problem is not decidable. The technique which allows one to see the last statement involves "embedding" a Turing machine into a group ring.

Counting the number of curves of degree $d$ with $n$ nodes (and no further singularities) going through $(d^2+3d)/2 - n$ points in general position in the projective plane is a problem which was already considered more than 150 years ago. More recently, people conjectured that for sufficiently large $d$ this number should be given by a polynomial of degree $2n$ in $d$. More generally, the Göttsche conjecture states that the number of $n$-nodal curves in a general $n$-dimensional linear subsystem of a sufficiently ample line bundle $L$ on a nonsingular projective surface $S$ is given by a universal polynomial of degree $n$ in the 4 topological numbers $L^2, L.K_S, (K_S)^2$ and $c_2(S)$. In a joint work with Vivek Shende and Richard Thomas, we give a short (compared to existing) proof of this conjecture.