L6

In the first part of the talk I will review Bob Fernholz' theory of functionally generated portfolios. In the second part I will discuss questions related to the existence of short-term arbitrage opportunities.
This is joint work with Bob Fernholz and Ioannis Karatzas

  
The Cartan model computes the equivariant cohomology of a smooth manifold X with 
differentiable action of a compact Lie group G, from the invariant functions on 
the Lie algebra with values in differential forms and a deformation of the de Rham 
differential. Before extracting invariants, the Cartan differential does not square 
to zero. Unrecognised was the fact that the full complex is a curved algebra, 
computing the quotient by G of the algebra of differential forms on X. This 
generates, for example, a gauged version of string topology. Another instance of 
the construction, applied to deformation quantisation of symplectic manifolds, 
gives the BRST construction of the symplectic quotient. Finally, the theory for a 
X point with an additional quadratic curving computes the representation category 
of the compact group G.

Ramsey classes may be viewed as the top of the line of Ramsey properties. Classical and not so classical examples of Ramsey classes of finite structures were recently extended by many new examples which make the characterisation of Ramsey classes  realistic (and in many cases known). Particularly I will cover recent  joint work with J. Hubicka.
 

The size Ramsey number r'(H) of a graph H is the smallest number of edges in a graph G which is Ramsey with respect to H, that is, such that any 2-colouring of the edges of G contains a monochromatic copy of H. A famous result of Beck states that the size Ramsey number of the path with n vertices is at most bn for some fixed constant b > 0. An extension of this result to graphs of maximum degree ∆ was recently given by Kohayakawa, Rödl, Schacht and Szemerédi, who showed that there is a constant b > 0 depending only on ∆ such that if H is a graph with n vertices and maximum degree ∆ then r'(H) < bn^{2 - 1/∆} (log n)^{1/∆}. On the other hand, the only known lower-bound on the size Ramsey numbers of bounded-degree graphs is of order n (log n)^c for some constant c > 0, due to Rödl and Szemerédi.

Together with David Conlon, we make a small step towards improving the upper bound. In particular, we show that if H is a ∆-bounded-degree triangle-free graph then r'(H) < s(∆) n^{2 - 1/(∆ - 1/2)} polylog n. In this talk we discuss why 1/∆ is the natural "barrier" in the exponent and how we go around it, why we need the triangle-free condition and what are the limits of our approach.

The chromatic number of the Erdős–Rényi random graph G(n,p) has been an intensely studied subject since at least the 1970s. A celebrated breakthrough by Bollobás in 1987 first established the asymptotic value of the chromatic number of G(n,1/2), and a considerable amount of effort has since been spent on refining Bollobás' approach, resulting in increasingly accurate bounds. Despite this, up until now there has been a gap of size O(1) in the denominator between the best known upper and lower bounds for the chromatic number of dense random graphs G(n,p) where p is constant. In contrast, much more is known in the sparse case.

In this talk, new upper and lower bounds for the chromatic number of G(n,p) where p is constant will be presented which match each other up to a term of size o(1) in the denominator. In particular, they narrow down the optimal colouring rate, defined as the average colour class size in a colouring with the minimum number of colours, to an interval of length o(1). These bounds were obtained through a careful application of the second moment method rather than a variant of Bollobás' method. Somewhat surprisingly, the behaviour of the chromatic number changes around p=1-1/e^2, with a different limiting effect being dominant below and above this value.

There is an obvious product-free subset of the symmetric group of density 1/2, but what about the alternating group? An argument of Gowers shows that a product-free subset of the alternating group can have density at most n^(-1/3), but we only know examples of density n^(-1/2 + o(1)). We'll talk about why in fact n^(-1/2 + o(1)) is the right answer, why
Gowers's argument can't prove that, and how this all fits in with a more general 'product mixing' phenomenon. Our tools include some nonabelian Fourier analysis, a version of entropy subadditivity adapted to the symmetric group, and a concentration-of-measure result for rearrangements of inner products.