The $\phi^4$ model in statistical physics describes the
continous phase transition in the liquid-vapour system, transition to
the superfluid phase in helium, etc. Experimentally measured values in
this model are critical exponents and universal amplitude ratios.
These values can also be calculated in the framework of the
renormalization group approach. It turns out that the obtained series
are divergent asymptotic series, but it is possible to perform Borel
resummation of such a series. To make this procedure more accurate we
need as much terms of the expansion as possible.
The results of the recent six loop analitical calculations of the
anomalous dimensions, beta function and critical exponents of the
$O(N)$ symmetric $\phi^4$ model will be presented. Different technical
aspects of these calculations (IBP method, R* operation and parametric
integration in Feynman representation) will be discussed. The

numerical estimations of critical exponents obtained with Borel
resummation procedure are compared with experimental values and
results of Monte-Carlo simulations.

The big phase space is an infinite dimensional manifold which is the arena
for topological quantum field theories and quantum cohomology (or
equivalently, dispersive integrable systems). tt*-geometry was introduced by
Cecotti and Vafa and is a way to introduce an Hermitian structure on what
would be naturally complex objects, and the theory has many links with
singularity theory, variation of Hodge structures, Higgs bundles, integrable
systems etc.. In this talk the two ideas will be combined to give a
tt*-geometry on the big phase space.

(joint work with Liana David)

Consider the problem of identifying an unknown subspace S from data with erasures and with limited memory available. To estimate S, suppose we group the measurements into blocks and iteratively update our estimate of S with each new block.

In the first part of this talk, we will discuss why estimating S by computing the "running average" of span of these blocks fails in general. Based on the lessons learned, we then propose SNIPE for memory-limited PCA with incomplete data, useful also for streaming data applications. SNIPE provably converges (linearly) to the true subspace, in the absence of noise and given sufficient measurements, and shows excellent performance in simulations. This is joint work with Laura Balzano and Mike Wakin.
 

A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy Nguyen. 

Gauss was the first to give a formula for the number of monic irreducible polynomials of degree n over a finite field. A natural problem is to determine the number of such polynomials for which certain coefficients are prescribed. While some asymptotic and existence results have been obtained, very few exact results are known. In this talk I shall present an algorithm which for any finite field GF(q) of characteristic p expresses the number of monic irreducibles of degree n for which the first l < p coefficients are prescribed, for n >= l and coprime to p, in terms of the number of GF(q^n)-rational points of certain affine varieties defined over GF(q). 
The GF(2) base field case is related to the distribution of binary Kloosterman sums, which have numerous applications in coding theory and cryptography, for example via the construction of bent functions. Using a variant of the algorithm, we present varieties (which are all curves) for l <= 7 and compute explicit formulae for l <= 5; before this work such formulae were only known for l <= 3. While this connection motivates the problem, the talk shall focus mainly on computational algebraic geometry, with the algorithm, theoretical questions and computational challenges taking centre stage.