Cambridge

Advances in manufacturing technologies, most prominently in additive manufacturing or 3d printing, are making it possible to fabricate highly optimised products with increasing geometric and hierarchical complexity. This talk will introduce our ongoing work on design optimisation that combines CAD-compatible geometry representations, multiresolution geometry processing techniques and immersed finite elements with classical shape and topology calculus. As example applications,the shape optimisation of mechanical structures and electromechanical components, and the topology optimisation of lattice-skin structures will be discussed.

When studying a group, it is natural and often useful to try to cut it up 
onto simpler pieces. Sometimes this can be done in an entirely canonical 
way analogous to the JSJ decomposition of a 3-manifold, in which the 
collection of tori along which the manifold is cut is unique up to isotopy. 
It is a theorem of Brian Bowditch that if the group acts nicely on a metric 
space with a negative curvature property then a canonical decomposition can 
be read directly from the large-scale geometry of that space. In this talk 
we shall explore an algorithmic consequence of this relationship between 
the large-scale geometry of the group and is algebraic decomposition.

Recent tools make it possible to partition the space of rational Dehn 
surgery slopes for a knot (or in some cases a link) in a 3-manifold into 
domains over which the Heegaard Floer homology of the surgered manifolds 
behaves continuously as a function of slope. I will describe some 
techniques for determining the walls of discontinuity separating these 
domains, along with efforts to interpret some aspects of this structure 
in terms of the behaviour of co-oriented taut foliations. This talk 
draws on a combination of independent work, previous joint work with 
Jake Rasmussen, and work in progress with Rachel Roberts.

A granular material forms a distinct and fascinating phase in physics -- sand acts as a fluid as grains flow through your fingers, the fallen grains form a solid heap on the floor or may suspend in the wind like a gas.

The main challenge of studying granular materials is the development of constitutive models valid across scales, from the micro-scale (collisions between individual particles), via the meso-scale (flow structures inside avalanches) to the macro-scale (dunes, heaps, chute flows).

In this talk, I am highlighting three recent projects from my laboratory, each highlighting physical behavior at a different scale. First, using the property of birefringence, we are quantifying both kinetic and dynamic properties in an avalanche of macroscopic particles and measure rheological properties. Secondly, we explore avalanches on an erodible bed that display an intriguing dynamic intermittency between regimes. Lastly, we take a closer look at aqueous (water-driven) dunes in a novel rotating experiment and resolve an outstanding scaling controversy between migration velocity and dune dimension.

We characterise the embeddability of simply connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski’s characterisation of graph planarity, by excluded minors. This answers questions of Lovász, Pardon and Wagner.

 

In real dimension two, the symplectic mapping class group of a surface agrees with its `classical' mapping class group, whose properties are well-understood. To what extend do these generalise to higher-dimensions? We consider specific pairs of symplectic manifolds (S, M), where S is a surface, together with collections of Lagrangian spheres in S and in M, say v_1, ...,v_k and V_1, ...,V_k, that have analogous intersection patterns, in a sense that we will make precise. Our main theorem is that any relation between the Dehn twists in the V_i must also hold between Dehn twists in the v_i. Time allowing, we will give some corollaries, such as embeddings of certain interesting groups into auto-equivalence groups of Fukaya categories.

We consider a generalization of degeneracy loci of morphisms between vector bundles based on orbit closures of algebraic groups in their linear representations. Using a certain crepancy condition on the orbit closure we gain some control over the canonical sheaf in a preferred class of examples. This is notably the case for Richardson nilpotent orbits and partially decomposable skew-symmetric three-forms in six variables. We show how these techniques can be applied to construct Calabi-Yau manifolds and Fano varieties of dimension three and four.

This is a joint work with Vladimiro Benedetti, Laurent Manivel and Fabio Tanturri.

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).

Lagrangian Floer cohomology groups are extremely hard compute in most situations. In this talk I’ll describe two ways to extract information about the self-Floer cohomology of a monotone Lagrangian possessing certain kinds of symmetry, based on the closed-open string map and the Oh spectral sequence. The focus will be on a particular family of examples, where the techniques can be combined to deduce some unusual properties.