We give an overview of joint work with Lewis Topley on modular W-algebras. In particular, we outline the classification 1-dimensional modules for modular W-algebras for gl_n, which in turn this leads to a classification of minimal dimensional modules for reduced enveloping algebras for gl_n.
Cycles are fundamental objects in graph theory. A spanning cycle in a graph is also called a Hamiltonian cycle. The celebrated Dirac's Theorem in 1952 shows that every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ contains a Hamiltonian cycle. In recent years, there has been a strong focus on extending Dirac’s Theorem to hypergraphs. We survey the results along the line and mention some recent progress on this problem. Joint work with Yi Zhao.
A system of linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic solution. The finite partition regular systems were completely characterised by Rado in terms of a simple property of their matrix of coefficients. As a result, finite partition regular systems are very well understood.
Much less is known about infinite systems. In fact, only a very few families of infinite partition regular systems are known. I'll explain a relatively new method of constructing infinite partition regular systems, and describe how it has been applied to settle some basic questions in the area.
A fundamental theorem of Wilson states that, for every graph $F$, every sufficiently large $F$-divisible clique has an $F$-decomposition. Here $G$ has an $F$-decomposition if the edges of $G$ can be covered by edge-disjoint copies of $F$ (and $F$-divisibility is a trivial necessary condition for this). We extend Wilson's theorem to graphs which are allowed to be far from complete (joint work with B. Barber, D. Kuhn, A. Lo).
I will also discuss some results and open problems on decompositions of dense graphs and hypergraphs into Hamilton cycles and perfect matchings.
The maximal subgroups of the exceptional groups of Lie type
have been studied for many years, and have many applications, for
example in permutation group theory and in generation of finite
groups. In this talk I will survey what is currently known about the
maximal subgroups of exceptional groups, and our recent work on this
topic. We explore the connection with extending morphisms from finite
groups to algebraic groups.
Joint work with: Sina Greenwood, Brian Raines and Casey Sherman
Abstract: We say a space $X$ with property $\C P$ is \emph{universal} for orbit spectra of homeomorphisms with property $\C P$ provided that if $Y$ is any space with property $\C P$ and the same cardinality as $X$ and $h:Y\to Y$ is any (auto)homeomorphism then there is a homeomorphism$g:X\to X$ such that the orbit equivalence classes for $h$ and $g$ are isomorphic. We construct a compact metric space $X$ that is universal for homeomorphisms of compact metric spaces of cardinality the continuum. There is no universal space for countable compact metric spaces. In the presence of some set theoretic assumptions we also give a separable metric space of size continuum that is universal for homeomorphisms on separable metric spaces.
I will give an overview of some of the most interesting algebraic-lattice theoretical results on bilattices. I will focus in particular on the product construction that is used to represent a subclass of bilattices, the so-called 'interlaced bilattices', mentioning some alternative strategies to prove such a result. If time allows, I will discuss other algebras of logic related to bilattices (e.g., Nelson lattices) and their product representation.
In topos-valid point-free topology there is a good analogue of regular measures and associated measure theoretic concepts including integration. It is expressed in terms of valuations, essentially measures restricted to the opens. A valuation $m$ is $0$ on the empty set and Scott continuous, as well as satisfying the modular law $$ m(U \cup V) + m(U \cap V) = m(U) + m(V). $$
\\Of course, that begs the question of why one would want to work with topos-valid point-free topology, but I'll give some general justification regarding fibrewise topology of bundles and a more specific example from recent topos work on quantum foundations.
\\The focus of the talk is the valuation locale, an analogue of hyperspaces: if $X$ is a point-free space (locale) then its valuation locale $VX$ is a point-free space whose points are the valuations on $X$. It was developed by Heckmann, by Coquand and Spitters, and by myself out of the probabilistic powerdomain of Jones and Plotkin.
\\I shall discuss the following results, proved in a draft paper "A monad of valuation locales" available at http://www.cs.bham.ac.uk/~sjv/Riesz.pdf:
The technical core is an analysis of simple maps to the reals. They can be used to approximate more general maps, and provide a means to reducing the calculations to finitary algebra. In particular the free commutative monoid $M(L)$ over a distributive lattice $L$, subject to certain relations including ones deriving from the modular law, can be got as a tensor product in a semilattice sense of $L$ with the natural numbers. It also satisfies the Principle of Inclusion and Exclusion (in a form presented without subtraction).
Several formulations of structural optimization problems based on linear and nonlinear semidefinite programming will be presented. SDP allows us to formulate and solve problems with difficult constraints that could hardly be solved before. We will show that sometimes it is advantageous to prefer a nonlinear formulation to a linear one. All the presented formulations result in large-scale sparse (nonlinear) SDPs. In the second part of the talk we will show how these problems can be solved by our augmented Lagrangian code PENNON. Numerical examples will illustrate the talk.
Joint work with Michael Stingl.