The prime spectrum of a quantum algebra has a finite stratification in terms
of a set of distinguished primes called H-primes, and we can study these
strata by passing to certain nice localizations of the algebra.  H-primes
are now starting to show up in some surprising new areas, including
combinatorics (totally nonnegative matrices) and physics, and we can borrow
techniques from these areas to answer questions about quantum algebras and
their localizations.    In particular, we can use Grassmann necklaces -- a
purely combinatorial construction -- to study the topological structure of
the prime spectrum of quantum matrices.

A subgroup Gamma of a semisimple algebraic group G is called strongly dense if every subgroup of Gamma is either cyclic or Zariski-dense. I will describe a method for building strongly dense free subgroups inside a given Zariski-dense subgroup  Gamma of G, thus providing a refinement of the Tits alternative. The method works for a large class of G's and Gamma's. I will also discuss connections with word maps and expander graphs. This is joint work with Bob Guralnick and Michael Larsen.

Given an Artin stack $X$, there is growing evidence that there should be an associated `category of B-branes', which is some subcategory of the derived category of coherent sheaves on $X$. The simplest case is when $X$ is just a vector space modulo a linear action of a reductive group, or `gauged linear sigma model' in physicists' terminology. In this case we know some examples of what the category B-branes should be. Hori has conjectured a physical duality between certain families of GLSMs, which would imply that their B-brane categories are equivalent. We prove this equivalence of categories. As an application, we construct Homological Projective Duality for (non-commutative resolutions of) Pfaffian varieties.

The InFoMM CDT Group Meetings will follow the format of the OCIAM group meetings. We hope they will facilitate good communication between the Academic and Student community so that the research activities remain closely connected, opportunities for additional interaction are easily identified, and cross-fertilisation of ideas can be catalysed. 

"In a paper from 2001, Diestel and Leader characterised uncountable graphs with normal spanning trees through a class of forbidden minors. In this talk we investigate under which circumstances this class of forbidden minors can be made nice. In particular, we will see that there is a nice solution to this problem under Martin’s Axiom. Also, some connections to the Stone-Chech remainder of the integers, and almost disjoint families are uncovered.”

Point-free topology can often seem like an algebraic almost-topology, 
> not quite the same but still interesting to those with an interest in 
> it. There is also a tradition of it in computer science, traceable back 
> to Scott's topological model of the untyped lambda-calculus, and 
> developing through Abramsky's 1987 thesis. There the point-free approach 
> can be seen as giving new insights (from a logic of observations), 
> albeit in a context where it is equivalent to point-set topology. It was 
> in that tradition that I wrote my own book "Topology via Logic".
> 
> Absent from my book, however, was a rather deeper connection with logic 
> that was already known from topos theory: if one respects certain 
> logical constraints (of geometric logic), then the maps one constructs 
> are automatically continuous. Given a generic point x of X, if one 
> geometrically constructs a point of Y, then one has constructed a 
> continuous map from X to Y. This is in fact a point-free result, even 
> though it unashamedly uses points.
> 
> This "continuity via logic", continuity as geometricity, takes one 
> rather further than simple continuity of maps. Sheaves and bundles can 
> be understood as continuous set-valued or space-valued maps, and topos 
> theory makes this meaningful - with the proviso that, to make it run 
> cleanly, all spaces have to be point-free. In the resulting fibrewise 
> topology via logic, every geometric construction of spaces (example: 
> point-free hyperspaces, or powerlocales) leads automatically to a 
> fibrewise construction on bundles.
> 
> I shall present an overview of this framework, as well as touching on 
> recent work using Joyal's Arithmetic Universes. This bears on the logic 
> itself, and aims to replace the geometric logic, with its infinitary 
> disjunctions, by a finitary "arithmetic type theory" that still has the 
> intrinsic continuity, and is strong enough to encompass significant 
> amounts of real analysis.

Last year, G. Kings and D. Rossler related the degree zero part of the polylogarithm
on abelian schemes pol^0 with another object previously defined by V. Maillot and D.
Rossler. More precisely, they proved that the canonical class of currents constructed
by Maillot and Rossler provides us with the realization of pol^0 in analytic Deligne
cohomology.
I will show that, adding some properness conditions, it is possible to give a
refinement of Kings and Rossler’s result involving Deligne-Beilinson cohomology
instead of analytic Deligne cohomology.

The wall-crossing behaviour of Donaldson-Thomas invariants in CY3 categories is controlled by a beautiful formula involving the group of automorphisms of a symplectic algebraic torus. This formula invites one to solve a certain Riemann-Hilbert problem. I will start by explaining how to solve this problem in the simplest possible case (this is undergraduate stuff!). I will then talk about a more general class of examples of the wall-crossing formula involving moduli spaces of quadratic differentials.