The concept of an acylindrically hyperbolic group, introduced by D. Osin, generalizes hyperbolic and relatively hyperbolic groups, and includes many other groups of interest: Out(F_n), n>1, most mapping class groups, directly indecomposable non-cyclic right angled Artin groups, most graph products, groups of deficiency at least 2, etc. Roughly speaking, a group G is acylindrically hyperbolic if there is a (possibly infinite) generating set X of G such that the Cayley graph \Gamma(G,X) is hyperbolic and the action of G on it is "sufficiently nice". Many global properties of hyperbolic/relatively hyperbolic groups have been also proved for acylindrically hyperbolic groups. 
In the talk I will discuss a method which allows to construct a common acylindrically hyperbolic quotient for any countable family of countable acylindrically hyperbolic groups. This allows us to produce acylindrically hyperbolic groups with many unexpected properties.(The talk will be based on joint work with Denis Osin.)
 

This talk is concerned with the mathematical and numerical analysis of a steady phase change problem for non-isothermal incompressible viscous flow. The system is formulated in terms of pseudostress, strain rate and velocity for the Navier-Stokes-Brinkman equation, whereas temperature, normal heat flux on the boundary, and an auxiliary unknown are introduced for the energy conservation equation. In addition, and as one of the novelties of our approach, the symmetry of the pseudostress is imposed in an ultra-weak sense, thanks to which the usual introduction of the vorticity as an additional unknown is no longer needed. Then, for the mathematical analysis two variational formulations are proposed, namely mixed-primal and fully-mixed approaches, and the solvability of the resulting coupled formulations is established by combining fixed-point arguments, Sobolev embedding theorems and certain regularity assumptions. We then construct corresponding Galerkin discretizations based on adequate finite element spaces, and derive optimal a priori error estimates. Finally, numerical experiments in 2D and 3D illustrate the interest of this scheme and validate the theory.

Knot theory investigates the many ways of embedding a circle into the three-dimensional sphere. The study of these embeddings is not only important for understanding three-dimensional manifolds, but is also intimately related to many new and surprising phenomena appearing in dimension four. I will discuss how four-dimensional interpretations of some invariants can help us understand surfaces that bound a given link (embedding of several disjoint circles).

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of k-colorings of a graph G on n vertices with maximum degree \Delta is rapidly mixing for k \ge \Delta+2. In 1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters on the set of proper k-colorings for k > (11/6)\Delta, implying rapid mixing for Glauber dynamics. In this paper, we obtain the first improvement beyond the (11/6)\Delta barrier for general graphs by showing rapid mixing for k > (11/6 - \eta)\Delta for some positive constant \eta. The key to our proof is combining path coupling with a new kind of metric that incorporates a count of the extremal configurations of the chain. Additionally, our results extend to list coloring, a widely studied generalization of coloring. Combined, these results answer two open questions from Frieze and Vigoda’s 2007 survey paper on Glauber dynamics for colorings. 

This is joint work with Michelle Delcourt and Luke Postle.

There was major progress on perfect graphs in the early 2000's: Chudnovsky, Robertson, Thomas and I proved the ``strong perfect graph theorem'' that a graph is perfect if and only if it has no odd hole or odd antihole; and Chudnovsky, Cornuejols, Liu, Vuscovic and I found a polynomial-time algorithm to test whether a graph has an odd hole or odd antihole, and thereby test if it is perfect. (A ``hole'' is an induced cycle of length at least four, and an ``antihole'' is a hole in the complement graph.)

What we couldn't do then was test whether a graph has an odd hole, and this has stayed open for the last fifteen years, despite some intensive effort. I am happy to report that in fact it can be done in poly-time (in time O(|G|^{12}) at the last count), and in this talk we explain how.

Joint work with Maria Chudnovsky, Alex Scott, and Sophie Spirkl.

The Erdos-Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs.

1. A theorem of R\"odl asserts that if an n-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to 0 or 1) of size \Omega(n). We prove that if a 3-uniform hypergraph is non-universal then it contains an almost homogeneous set of size \Omega(log n). An example of R\"odl from 1986 shows that this bound is tight.

2. Let R_r(t) denote the size of the largest non-universal r-graph G so that neither G nor its complement contain a complete r-partite subgraph with parts of size t. We prove an Erd\H{o}s--Hajnal-type stepping-up lemma, showing how to transform a lower bound for R_r(t) into a lower bound for R_{r+1}(t). As an application of this lemma, we improve a bound of Conlon-Fox-Sudakov by showing that R_3(t) \geq t^{ct).

Joint work with M. Amir and A. Shapira

Get to know the Mathematrix events of this term!

We were a bit too late with ordering food, so the usual sandwich lunch will only start from week 2. However, there may be some small snacks.

In this talk, we show that a range of non-trivial SU(3) structures can be constructed on large classes of Calabi-Yau three-folds. Among the possible SU(3) structures we find Strominger-Hull systems, suitable for heterotic or type II string compactifications. These SU(3) structures of Strominger-Hull type have a non-vanishing and non-closed three-form flux which needs to be supported by source terms in the associated Bianchi identity. We discuss the possibility of finding such source terms and present first steps towards their explicit construction. Provided suitable sources exist, our methods lead to Calabi-Yau compactifications of string theory with a non Ricci-flat, physical metric which can be written down explicitly and in analytic form. The talk is based on the paper 1805.08499.

The zeta-function of a manifold varies with the parameters and may be evaluated in terms of the periods. For a one parameter family of CY manifolds, the periods satisfy a single 4th order differential equation. Thus there is a straight and, it turns out, readily computable path that leads from a differential operator to a zeta-function. Especially interesting are the specialisations to singular manifolds, for which the zeta-function manifests modular behaviour. We are also able to find, from the zeta function, attractor points. These correspond to special values of the parameter for which there exists a 10D spacetime for which the 6D corresponds to a CY manifold and the 4D spacetime corresponds to an extremal supersymmetric black hole. These attractor CY manifolds are believed to have special number theoretic properties. This is joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.