L3

We highlight a technique for studying edge colourings of multigraphs, due to Tashkinov. This method is a sophisticated generalisation of the method of alternating paths, and builds upon earlier work by Kierstead and Goldberg. In particular we show how to apply it to a number of edge colouring problems, including the question of whether the class of multigraphs that attain equality in Vizing's classical bound can be characterised.

This talk represents joint work with Jessica McDonald.

In 1961 Hajos conjectured that if a graph contains no subdivsion of a clique of order t then its chromatic number is less than t. In 1981, Erdos and Fajtlowicz showed that the conjecture is almost always false. We show it is almost always true. This is joint work with Keevash, Mohar, and McDiarmid.

In joint work with Tim Perutz, we give a complete characterization of the Fukaya category of the punctured torus, denoted by $Fuk(T_0)$. This, in particular, means that one can write down an explicit minimal model for $Fuk(T_0)$ in the form of an A-infinity algebra, denoted by A, and classify A-infinity structures on the relevant algebra. A result that we will discuss is that no associative algebra is quasi-equivalent to the model A of the Fukaya category of the punctured torus, i.e., A is non-formal. $Fuk(T_0)$ will be connected to many topics of interest: 1) It is the boundary category that we associate to a 3-manifold with torus boundary in our extension of Heegaard Floer theory to manifolds with boundary, 2) It is quasi-equivalent to the category of perfect complexes on an irreducible rational curve with a double point, an instance of homological mirror symmetry.

I will discuss a conjectural Bogomolov-Gieseker type inequality for "tilt-stable" objects in the derived category of coherent sheaves on smooth projective threefolds. The conjecture implies the existence of Bridgeland stability conditions on threefolds, and also has implications to birational geometry: it implies a slightly weaker version of Fujita's conjecture on very ampleness of adjoint line bundles.

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.

If two L-functions are added together, the Euler product is destroyed.

Thus the linear combination is not an L-function, and hence we should

not expect a Riemann Hypothesis for it. This is indeed the case: Not

all the zeros of linear combinations of L-functions lie on the

critical line.

However, if the two L-functions have the same functional equation then

almost all the zeros do lie on the critical line. This is not seen

when they have different functional equations.

We will discuss these results (which are due to Bombieri and Hejhal)

during the talk, and demonstrate them using characteristic polynomials

of random unitary matrices, where similar phenomena are observed. If

the two matrices have the same determinant, almost all the zeros of

linear combinations of characteristic polynomials lie on the unit

circle, whereas if they have different determinants all the zeros lie

off the unit circle.

It is known for long that the set of possible compactifications of a topological space (up to homeomorphism) is in order-preserving bijection to "strong inclusion" relations on the lattice of open sets. Since these relations do not refer to points explicitly, this bijection has been generalised to point-free topology (a.k.a. locales). The strong inclusion relations involved are typically "witnessed" relations. For example, the Stone-Cech compactification has a strong inclusion witnessed by real-valued functions. This makes it natural to think of compactification in terms of d-frames, a category invented by Jung and Moshier for bitopological Stone duality. Here, a witnessed strong inclusion is inherent to every object and plays a central role.

We present natural analogues of the topological concepts regularity, normality, complete regularity and compactness in d-frames. Compactification is then a coreflection into the category of d-frames dually equivalent to compact Hausdorff spaces. The category of d-frames has a few surprising features. Among them are:

• The real line with the bitopology of upper and lower semicontinuity admits precisely one compactification, the extended reals.
• Unlike in the category of topological spaces (or locales), there is a coreflection into the subcategory of normal d-frames, and every compactification can be factored as "normalisation" followed by Stone-Cech compactification.