Fri, 15 Jun 2018

14:15 - 15:15
C3

The particulars of particulates

Nathalie Vriend
(Cambridge)
Abstract

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.

Tue, 30 Jan 2018
14:30
L6

Embedding simply connected 2-complexes in 3-space

Johannes Carmesin
(Cambridge)
Abstract

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.

 

Mon, 05 Feb 2018

14:15 - 15:15
L5

On symplectic stabilisations and mapping classes

Ailsa Keating
(Cambridge)
Abstract

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.

 

Tue, 31 Oct 2017

15:45 - 16:45
L4

Orbital degeneracy loci and applications

Sara Filippini
(Cambridge)
Abstract

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.

Mon, 23 Oct 2017

14:15 - 15:15
L5

Cubic fourfolds, K3 surfaces, and mirror symmetry

Nicholas Sheridan
(Cambridge)
Abstract

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).

 

Tue, 30 May 2017

15:45 - 16:45
L4

Symmetries in monotone Lagrangian Floer theory

Jack Smith
(Cambridge)
Abstract

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.

Tue, 06 Jun 2017
14:30
L6

Monochromatic Infinite Sumsets

Paul Russell
(Cambridge)
Abstract

It is well known that there is a finite colouring of the natural numbers such that there is no infinite set X with X+X (the pairwise sums from X, allowing repetition) monochromatic. It is easy to extend this to the rationals. Hindman, Leader and Strauss showed that there is also such a colouring of the reals, and asked if there exists a space 'large enough' that for every finite colouring there does exist an infinite X with X+X monochromatic. We show that there is indeed such a space. Joint work with Imre Leader.

Tue, 23 May 2017

15:45 - 16:45
L4

On Short Time Existence of Lagrangian Mean Curvature Flow

Tom Begley
(Cambridge)
Abstract

The goal of this talk will be to give an overview of recent work, joint with Kim Moore, on a short time existence problem in Lagrangian mean curvature flow. More specifically, we consider a compact initial Lagrangian submanifold with a finite number of singularities, each asymptotic to a pair of transversely intersecting planes. We show it is possible to construct a smooth Lagrangian mean curvature flow, existing for positive times, that attains the singular Lagrangian as its initial condition in a suitable weak sense.  The construction uses a family of smooth solutions whose initial conditions approximate the singular Lagrangian. In order to appeal to compactness theorems and produce the desired solution, it is necessary to first establish uniform curvature estimates on the approximating family. As time allows I hope to focus in particular on the proof of these estimates, and their role in the proof of the main theorem.

Thu, 25 May 2017
16:00
L6

Reduction of dynatomic curves

Holly Krieger
(Cambridge)
Abstract

Dynatomic curves parametrize n-periodic orbits of a one-parameter family of polynomial dynamical systems. These curves lack the structure of their arithmetic-geometric analogues (modular curves of level n) but can be studied dynamically.  Morton and Silverman conjectured a dynamical analogue of the uniform boundedness conjecture (theorems of Mazur, Merel), asserting uniform bounds for the number of rational periodic points for such a family.  I will discuss recent work towards the function field version of their conjecture, including results on the reduction mod p of dynatomic curves for the quadratic polynomial family z^2+c.

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