Summer Term 2008

Friday 25 April 2008 Tom Fisher (Cambridge) The arithmetic of plane cubics

In this talk I will describe the process of 3-descent on elliptic curves over the rationals, as has recently been made more explicit in joint work with Cremona, O'Neil, Simon and Stoll. I will begin by reviewing some of the classical geometry related to the Hesse pencil of plane cubics. I will then define the group of rational points (or Mordell-Weil group) of an elliptic curve, and explain how computing its rank is related to searching for rational points on plane cubics. The aim of a 3-descent calculation is then, starting from an elliptic curve, to find the relevant plane cubics.

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Friday 2 May 2008 Ezra Getzler (Northwestern) Lie theory for differential graded Lie algebras

I show how to associate to a nilpotent differential graded Lie algebra which vanishes in degree -n and below an n-groupoid. This construction generalizes the case of a Lie algebra (it gives the associated Lie group) and when the dg Lie algebra is abelian (i.e. a chain complex), it becomes the Eilenberg-MacLane space of the complex.

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Friday 9 May 2008 Spiro Karigiannis (Oxford) What are G2 manifolds?

For any fixed algebraic structure on a real vector space, one can consider a smoothly varying family of such structures on the tangent spaces of a manifold. In particular the exceptional algebraic structure of the octonions naturally leads one to define $G_2$ manifolds and their distinguished minimal submanifolds and Yang-Mills connections. The subject of $G_2$ manifolds involves a beautiful interplay of non-associative algebra, differential geometry, and non-linear global analysis.

I will present an introduction to these topics for a general audience, paying particular attention to the similarities and differences of $G_2$ geometry with respect to the geometries of Kahler manifolds and of
3-manifolds. I plan to end with a survey of some recent developments in the field.

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Friday 16 May 2008 Mihai Ciucu (Georgia Tech) The field of average tile orientations in random tilings with holes

The study of random tilings of planar lattice regions goes back to the solution of the dimer model in the 1960's by Kasteleyn, Temperley and Fisher, but received new impetus in the early 1990's, and has since
branched out in several directions in the work of Cohn, Kenyon, Okounkov, Sheffield, and others.

In this talk, we focus on the interaction of holes in random tilings, a subject inspired by Fisher and Stephenson's 1963 conjecture on the rotational invariance of the monomer-monomer correlation on the square
lattice. In earlier work, we showed that the correlation of a finite number of holes on the triangular lattice is given asymptotically by a superposition principle closely paralleling the superposition principle for electrostatic energy.

We now take this analogy one step further, by showing that the discrete field determined by considering at each unit triangle the average orientation of the lozenge covering it converges, in the scaling limit, to the electrostatic field.

Our proof involves a variety of ingredients, including Laplace's method for the asymptotics of integrals, Newton's divided difference operator, and hypergeometric function identities.

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Friday 23 May 2008 TBA

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Friday 30 May 2008 Bernd Sturmfels (Berkeley) Powers of Linear Forms

What is the dimension of the space of polynomials of a certain degree that are annihilated by certain powers of fixed vector fields ? We derive formulas by degenerating Cox-Nagata rings to toric algebras by
means of sagbi bases induced by families of such vector fields. For del Pezzo surfaces, this degeneration implies the Batyrev-Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n-space at n+3 points, sagbi bases of Cox-Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D'Cruz-Iarobbino
and Buczynska-Wisniewski. This study, which is joint with Zhiqiang Xu, emphasizes explicit computations and offers a new approach to fat points.

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Friday 6 June 2008 Artur Czumaj (Warwick - DIMAP) Testing expansion in bounded degree graphs really fast

We consider the problem of testing expansion in bounded degree graphs. We focus on the notion of vertex-expansion: an a-expander is an undirected graph G = (V,E) in which every subset U of V of at most |V|/2
vertices has a neighbourhood of size at least a|U|. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time approximately O(n^{1/2}).

We will prove our result in the framework of property testing, which will be described in the talk. We prove that the property testing algorithm proposed by Goldreich and Ron (2000) with appropriately set parameters accepts every a-expander with probability at least 2/3 and rejects every graph that is epsilon-far from an a*-expander with probability at least 2/3, where a* = O(a^2/(d^2 log(n/epsilon))), d is the maximum degree of the graphs, and a graph is called epsilon-far from an a*-expander if one has to modify (add or delete) at least epsilon d n of its edges to obtain an a*-expander. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is O(d^2 n^{1/2} log(n/epsilon)/(a^2 epsilon^3)). We will also briefly discuss the recent improvements due to Kale and Seshadhri, Nachmias and Shapira, who reduced the dependency in the expansion of the rejected graphs from O(a^2/(d^2 log(n/epsilon))) to O(a^2/d^2).

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Friday 13 June 2008 Bjorn Poonen (Berkeley) Undecidability in number theory

Hilbert's Tenth Problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers. Around 1970, Matiyasevich, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists. But the answer to the analogous question with Z replaced by Q is still unknown, and there is not even agreement among experts as to what the answer should be. I will discuss this and Hilbert's Tenth Problem over other rings of arithmetic interest.

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Friday 20 June 2008 TBA

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