Geometry and Topology

Topological complexity of the Klein bottle

The notion of topological complexity of a space has been introduced by M. Farber in order to give a topological measure of the complexity of the motion planning problem in robotics. Surprisingly, the determination of this invariant for non-orientable surfaces has turned out to be difficult. A. Dranishnikov has recently established that the topological complexity of the non-orientable surfaces of genus at least 4 is maximal. In this talk, we will determine the topological complexity of the Klein bottle and extend Dranishnikov's result to all the non-orientable surfaces of genus at least 2.

Mirror symmetry for Nahm branes

Using the Dirac–Higgs bundle, we consider a new class of space-filling (BBB)-branes on moduli spaces of Higgs bundles, given by a generalized Nahm transform of a stable Higgs bundle. We then use the Fourier–Mukai–Nahm transform to describe its dual brane, which is checked to be a (BAA)-brane supported on a complex Lagrangian multisection of the Hitchin fibration.

Donaldson-Uhlenbeck compactification for the moduli space of framed sheaves on a Deligne-Mumford stack

Framed sheaves over surfaces first appeared as a generalisation of framed SU(r) instantons on S4. Later, Uhlenbeck provided a compactification for the moduli space of framed SU(r) instantons which through the Donaldson correspondence and the work of Li and Morgan produced the so called Donaldson-Uhlenbeck compactification of the moduli space of framed sheaves. We want to focus on a further generalisation to the moduli space of framed sheaves on Deligne-Mumford stacks and construct an analogous compactification. This is work in progress with U. Bruzzo.

12, 24 and Beyond

Symplectic geometry and combinatorics are strongly intertwined due to the existence of Hamiltonian torus actions. These actions are associated with a special map (called the moment map) which "transforms" a compact symplectic manifold into a convex polytope. We will concentrate on the  special class of reflexive polytopes which was introduced by Batyrev in the context of mirror symmetry and  has attracted much attention recently. In particular, we will see how the famous "12 and 24" properties in dimension 2 and 3 can be generalized with the help of symplectic geometry.

Branes on moduli spaces of sheaves

Branes are special submanifolds of hyperkähler manifolds that play an important role in string theory, particularly in the Kapustin–Witten approach to the geometric Langlands program, but which also are of intrinsic geometric interest. More precisely, a brane is a submanifold of a hyperkähler manifold which is either complex or Lagrangian with respect to each of the three complex structures or Kähler forms composing the hyperkähler structure.

The Sard conjecture on Martinet surfaces

Given a totally nonholonomic distribution of rank two ∆ on a three-dimensional manifold M, it is natural to investigate the size of the set of points Xx that can be reached by singular horizontal paths starting from a same point x ∈ M. In this setting, the Sard conjecture states that Xx should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.

The moduli space of Higgs bundles over a real curve and the real Abel-Jacobi map

The moduli space M_C of Higgs bundles over a complex curve X admits a hyperkaehler metric: a Riemannian metric which is Kaehler with respect to three different complex structures I, J, K, satisfying the quaternionic relations. If X admits an anti-holomorphic involution, then there is an induced involution on M_C which is anti-holomorphic with respect to I and J, and holomorphic with respect to K. The fixed point set of this involution, M_R, is therefore a real Lagrangian submanifold with respect to I and J, and complex symplectic with respect to K, making it a so called AAB-brane.

Stability of the Poincaré bundle

Let X be a smooth projective curve of genus at least two. Let G be an almost simple affine algebraic group. The universal principal G-bundle is called the Poincaré bundle. It is a principal G-bundle on the product of the curve and the moduli stack of principal G-bundles. We prove that it is stable with respect to any polarization (joint work with I. Biswas and N. Hoffmann).

L-infinity algebras, central extentions, and M-theory

In this introduction for geometers and topologists, we explain the role that central extensions of L-infinity algebras, a concept we will define, plays in physics. This connection first appeared with the work of physicists D’Auria and Fré in 1982, but is beautifully captured by the “brane bouquet” of Fiorenza, Sati and Schreiber which shows how physical objects such as “strings”, “D-branes” and “M- branes” can be classified by taking successive central extensions of an especially simple L-infinity algebra called the “supertranslation algebra”.

Some remarks on the Mumford-Welters connection

Mumford introduced in the 1960ies an algebraic approach to the construction of (almost) canonical bases of sections of ample line bundles on abelian varieties that permitted him to construct quasi-projective moduli spaces. His construction was later re-interpreted by Welters as a flat projective connection before being generalized by Hitchin to the non-abelian setting.

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