The smooth homotopy category is a simultaneous enlargement of the usual homotopy category and of the category of smooth manifolds. Its structure can be described very simply and explicitly by a version of van Est's theorem.  It provides us with an  interpolation between topology and geometry (and with a toy model of derived algebraic geometry and motivic homotopy theory, though I shall not pursue those directions).  My talk will list some situations which the category seems to illuminate: one will be Kapranov's beautiful description of the Lie algebra of the 'group' of based loops in a manifold.
 

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature k<=0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k < 0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

I will describe a simple macroscopic model describing the evolution of a cloud of particles confined in a magneto-optical trap. The behavior of the particles is mainly driven by self--consistent attractive forces. In contrast to the standard model of gravitational forces, the force field does not result from a potential; moreover, the nonlinear coupling is more singular than the coupling based on the Poisson equation.  In addition to existence of uniqueness results of the model PDE, I will discuss the convergence of the  particles description towards the solution of the PDE system in the mean field regime.

While many cubic fourfolds are known to be rational, it is expected that the very general cubic fourfold is irrational (although none have been
proven to be so). There is a conjecture for precisely which cubics are rational, which can be expressed in Hodge-theoretic terms (by work of Hassett)
or in terms of derived categories (by work of Kuznetsov). The conjecture can be phrased as saying that one can associate a `noncommutative K3 surface' to any cubic fourfold, and the rational ones are precisely those for which this noncommutative K3 is `geometric', i.e., equivalent to an honest K3 surface. It turns out that the noncommutative K3 associated to a cubic fourfold has a conjectural symplectic mirror (due to  Batyrev-Borisov). In contrast to the algebraic side of the story, the mirror is always `geometric': i.e., it is always just an honest K3 surface equipped with an appropriate Kähler form. After explaining this background, I will state a theorem: homological mirror symmetry holds in this context (joint work with Ivan Smith).

This talk will report on the definition of some motivic cohomology classes and the proof that they satisfy the norm relations expected of Euler systems, emphasizing a connection with the local Gan-Gross-Prasad conjecture.

Let k be a field of characteristic zero and K=k((t)). Semi-algebraic sets over K are boolean combinations of algebraic sets and sets defined by valuative inequalities. The associated Grothendieck ring has been studied by Hrushovski and Kazhdan who link it via motivic integration to the Grothendieck ring of varieties over k. I will present a morphism from the former to the Grothendieck ring of motives of rigid analytic varieties over K in the sense of Ayoub. This allows to refine the comparison by Ayoub, Ivorra and Sebag between motivic Milnor fibre and motivic nearby cycle functor.
 

Dimer models with boundary were introduced in joint work with King and Marsh as a natural
generalisation of dimers. We use these to derive certain infinite dimensional algebras and
consider idempotent subalgebras w.r.t. the boundary.
The dimer models can be embedded in a surface with boundary. In the disk case, the
maximal CM modules over the boundary algebra are a Frobenius category which
categorifies the cluster structure of the Grassmannian.

 

We give a short reminder about central results of classical tilting theory, 
including the Brenner-Butler tilting theorem, and
homological properties of tilted and quasi-tilted algebras. We then discuss 
2-term silting complexes and endomorphism algebras of such objects,
and in particular show that some of these classical results have very natural 
generalizations in this setting.
(joint work with Yu Zhou)

G2-manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as 7-dimensional analogues of Calabi-Yau 3-folds.
In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2 manifolds (only four such manifolds were previously known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds. The metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how results in Calabi-Yau geometry about isolated singularities and their resolutions can be used to produce examples of complete G2-manifolds.