After giving an introduction to functorial field theories I will explain a natural generalization thereof, called "twisted" field theories by Stolz-Teichner. The definition uses the notion of lax or oplax natural transformations of strong functors of higher categories for which I will sketch a framework. I will discuss the fully extended case, which gives a comparison to Freed-Teleman's "relative" boundary field theories. Finally, I will explain some examples, one of which explicitly arises from factorization homology and whose target is the higher Morita category of E_n-algebras, bimodules, bimodules of bimodules etc.

An efficient way to descibe binary operations which are associative only up to coherent homotopy is via simplicial spaces. 2-Segal spaces were introduced independently by Dyckerhoff--Kapranov and G\'alvez-Carrillo--Kock--Tonks to encode spaces carrying multivalued, coherently associative products. For example, the Waldhausen S-construction of an abelian category is a 2-Segal space. It describes a multivalued product on the space of objects given in terms of short exact sequences. 
The main motivation to study spaces carrying multivalued products is that they can be linearised, producing algebras in the usual sense of the word. For the preceding example, the linearisation yields the Hall algebra of the abelian category. One can also extract tensor categories using a categorical linearisation procedure.
In this talk I will discuss double 2-Segal spaces, that is, bisimplicial spaces which satisfy the 2-Segal condition in each variable. Such bisimplicial spaces give rise to multivalued bialgebras. The second iteration of the Waldhausen S-construction is a double 2-Segal space whose linearisation is the bialgebra structure given by Green's Theorem. The categorial linearisation produces categorifications of Zelevinsky's positive, self-adjoint Hopf algebras.
 

Knots and links naturally appear in the neighbourhood of the singularity of a complex curve; this creates a bridge between algebraic geometry and differential topology. I will discuss a topological approach to the study of 1-parameter families of singular curves, using correction terms in Heegaard Floer homology. This is joint work with József Bodnár and Daniele Celoria.

Each year Prof. Trefethen gives the Problem Solving Squad a sequence of problems with no hints, one a week, where the solution of each problem is a single real number to be computed by any method available.  We will present this year's three problems, involving (1) an S-shaped bifurcation curve, (2) shortest path around a random web, and (3) switching a time-varying system to maximize a matrix norm.

The 14 students this year are Simon Vary plus InFoMM cohort 2: Matteo Croci, Davin Lunz, Michael McPhail, Tori Pereira, Lindon Roberts, Caoimhe Rooney, Ian Roper, Thomas Roy, Tino Sulzer, Bogdan Toader, Florian Wechsung, Jess Williams, and Fabian Ying.  The presentations will be by (1) Lindon Roberts, (2) Florian Wechsung, and (3) Thomas Roy.

We present some dimensionality reduction techniques for global optimization algorithms, in order to increase their scalability. Inspired by ideas in machine learning, and extending the approach of random projections in Zhang et al (2016), we present some new algorithmic approaches for global optimisation with theoretical guarantees of good behaviour and encouraging numerical results.
 

Given a representation of Gal_{Q_p} with coefficients in a p-adically complete local ring R, Fukaya and Kato have conjectured the existence of a canonical trivialization of the determinant of a certain cohomology complex.  When R=Z_p and the representation is a lattice in a de Rham representation, this trivialization should be related to the \varepsilon-factor of the corresponding Weil--Deligne representation.  Such a trivialization has been constructed for certain crystalline Galois representations, by the work of a number of authors. I will explain how to extend these trivializations to certain families of crystalline Galois representations.  This is joint work with Otmar Venjakob.

In this talk, I will give a brief overview of the Langlands program and Langlands functoriality with reference to the examples of Galois representations attached to cusp forms and the Jacquet-Langlands correspondence for $\mathrm{GL}_2$. I will then explain how one can generalise this idea, sketching a proof of a Jacquet-Langlands type correspondence from $\mathrm{U}_n(B)$, where $B$ is a quaternion algebra, to $\mathrm{Sp}_{2n}$ and showing that one can attach Galois representations to regular algebraic cuspidal automorphic representations of $\mathrm{Sp}_{2n}$.
 

Dedalus is a new open-source framework for solving general partial differential equations using spectral methods.  It is designed for maximum extensibility and incorporates features such as symbolic equation entry, custom domain construction, and automatic MPI parallelization.  I will briefly describe key algorithmic features of the code, including our sparse formulation and support for general tensor calculus in curvilinear domains.  I will then show examples of the code’s capabilities with various applications to astrophysical and geophysical fluid dynamics, including a compressible flow benchmark against a finite volume code, and direct numerical simulations of turbulent glacial melting

In my research I model three components of the Earth system: the ice sheets, the ocean, and the solid Earth. In the first half of this talk I will describe the traditional approach that is used to model the impact of ice sheet growth and decay on global sea-level change and solid Earth deformation. I will then go on to explain how collaboration across the fields of glaciology, geodynamics and seismology is providing exciting new insight into feedbacks between ice dynamics and solid Earth deformation.