We will introduce both the class of right-angled Artin groups (RAAG) and
the Nielsen realisation problem. Then we will discuss some recent progress
towards solving the problem.
The filtration on the infinite symmetric product of spheres by number of
factors provides a sequence of spectra between the sphere spectrum and
the integral Eilenberg-Mac Lane spectrum. This filtration has received a
lot of attention and the subquotients are interesting stable homotopy
types.
In this talk I will discuss the equivariant stable homotopy types, for
finite groups, obtained from this filtration for the infinite symmetric
product of representation spheres. The filtration is more complicated
than in the non-equivariant case, and already on the zeroth homotopy
groups an interesting filtration of the augmentation ideal of the Burnside
rings arises. Our method is by `global' homotopy theory, i.e., we study
the simultaneous behaviour for all finite groups at once. In this context,
the equivariant subquotients are no longer rationally trivial, nor even
concentrated in dimension 0.
A perfect obstruction theory for a commutative ring is a morphism from a perfect complex to the cotangent complex of the ring
satisfying some further conditions. In this talk I will present work in progress on how to associate in a functorial manner commutative
differential graded algebras to such a perfect obstruction theory. The key property of the differential graded algebra is that its zeroth homology
is the ring equipped with the perfect obstruction theory. I will also indicate how the method introduced can be globalized to work on schemes
without encountering gluing issues.
Both parts will deal with spherical objects in the bounded derived
category of coherent sheaves on K3 surfaces. In the first talk I will
focus on cycle theoretic aspects. For this we think of the Grothendieck
group of the derived category as the Chow group of the K3 surface (which
over the complex numbers is infinite-dimensional due to a result of
Mumford). The Bloch-Beilinson conjecture predicts that over number
fields the Chow group is small and I will show that this is equivalent to
the derived category being generated by spherical objects (which
I do not know how to prove). In the second talk I will turn to stability
conditions and show that a stability condition is determined by its
behavior with respect to the discrete collections of spherical objects.
Both parts will deal with spherical objects in the bounded derived
category of coherent sheaves on K3 surfaces. In the first talk I will
focus on cycle theoretic aspects. For this we think of the Grothendieck
group of the derived category as the Chow group of the K3 surface (which
over the complex numbers is infinite-dimensional due to a result of
Mumford). The Bloch-Beilinson conjecture predicts that over number
fields the Chow group is small and I will show that this is equivalent to
the derived category being generated by spherical objects (which
I do not know how to prove). In the second talk I will turn to stability
conditions and show that a stability condition is determined by its
behavior with respect to the discrete collections of spherical objects.
Many problems from combinatorics, number theory, quantum field theory and topology lead to power series of a special kind called q-hypergeometric series. Sometimes, like in the famous Rogers-Ramanujan identities, these q-series turn out to be modular functions or modular forms. A beautiful conjecture of W. Nahm, inspired by quantum theory, relates this phenomenon to algebraic K-theory.
In a different direction, quantum invariants of knots and 3-manifolds also sometimes seem to have modular or near-modular properties, leading to new objects called "quantum modular forms".
We consider problems of elastic plastic deformation with isotropic and kinematic hardening.
A dual formulation with stresses as principal variables is used.
We obtain several results on Sobolev space regularity of the stresses
and strains.
In particular, we obtain the existence of a full derivative of the
stress tensor up to the boundary of the basic domain.
Finally, we present an outlook for obtaining further regularity
results in connection with general nonlinear evolution problems.