I will explain how to prove the exactness of the homotopy sequence of overconvergent p-adic fundamental groups for a smooth and projective morphism in characteristic p. We do so by first proving a corresponding result for rigid analytic varieties in characteristic 0, following dos Santos in the algebraic case. In characteristic p we proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result. Joint work with Chris Lazda.

I will talk about a recent proof, joint with M. Gröchenig and D. Wyss, of a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The proof, inspired by an argument of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration. I will start with an introduction to Higgs bundles and their moduli spaces and then explain our argument.

The $r$-neighbour bootstrap process on a graph $G$ starts with an initial set of "infected" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ becomes infected during the process, then we say that the initial set percolates.

In this talk I will discuss the proof of a conjecture of Balogh and Bollobás: for fixed $r$ and $d\to\infty$, the minimum cardinality of a percolating set in the $d$-dimensional hypercube is $\frac{1+o(1)}{r}\binom{d}{r-1}$. One of the key ideas behind the proof exploits a connection between bootstrap percolation and weak saturation. This is joint work with Jonathan Noel.

Adjoint derivatives reveal the sensitivity of a computer program's output to changes in its inputs. These derivatives are useful as a building block for optimisation, uncertainty quantification, noise estimation, inverse design, etc., in many industrial and scientific applications that use PDE solvers or other codes.
Algorithmic differentiation (AD) is an established method to transform a given computation into its corresponding adjoint computation. One of the key challenges in this process is the efficiency of the resulting adjoint computation. This becomes especially pressing with the increasing use of shared-memory parallelism on multi- and many-core architectures, for which AD support is currently insufficient.
In this talk, I will present an overview of challenges and solutions for the differentiation of shared-memory-parallel code, using two examples: an unstructured-mesh CFD solver, and a structured-mesh stencil kernel, both parallelised with OpenMP. I will show how AD can be used to generate adjoint solvers that scale as well as their underlying original solvers on CPUs and a KNC XeonPhi. The talk will conclude with some recent efforts in using AD and formal verification tools to check the correctness of manually optimised adjoint solvers.
 

Structural optimization can be interpreted as the attempt to find the best mechanical structure to support specific load cases respecting some possible constraints. Within this context, topology optimization aims to obtain the connectivity, shape and location of voids inside a prescribed structural design domain. The methods for the design of stiff lightweight structures are well established and can already be used in a specific range of industries where such structures are important, e.g., in aerospace and automobile industries.

In this seminar, we will go through the basic engineering concepts used to quantify and analyze the computational models of mechanical structures. After presenting the motivation, the methods and mathematical tools used in structural topology optimization will be discussed. In our method, an implicit level set function is used to describe the structural boundaries. The optimization problem is approximated by linearization of the objective and constraint equations via Taylor’s expansion. Shape sensitivities are used to evaluate the change in the structural performance due to a shape movement and to feed the mathematical optimiser in an iterative procedure. Recent developments comprising multiscale and Multiphysics problems will be presented and a specific application proposal including acoustic-structure interaction will be discussed.

For a prime number p, we will consider completed group algebras, or Iwasawa algebras, of the form kG, for G a complete p-valued group of finite rank, k a field of characteristic p. Classifying the ideal structure of Iwasawa algebras has been an ongoing project within non-commutative algebra and representation theory, and we will discuss ideas related to this topic based on previous work and try to extend it. An important concept in studying ideals of group algebras is the notion of controlling ideals, where we say a closed subgroup H of G controls a right ideal I of kG if I is generated by a subset of kH. It was proved by Konstantin Ardakov in 2012 that for G nilpotent, every faithful prime ideal of kG is controlled by the centre of G, and it follows that the prime spectrum of kG can be realised as the disjoint union of commutative strata. I am hoping to extend this to a more general case, perhaps to when G is solvable. A key step in the proof is to consider a faithful prime ideal P in kG, and an automorphism of G, trivial mod centre, that fixes P. By considering the Mahler expansion of the automorphism, and approximating the coefficients, we can examine sequences of bounded k-linear functions of kG, and study their convergence. If we find that they converge to an appropriate quantized divided power, we can find proper open subgroups of G that control P. I have extended this notion to larger classes of automorphisms, not necessarily trivial mod centre, using which this proof can be replicated, and in some cases extended to when G is abelian-by-procyclic. I will give some examples, for G with small rank, for which these ideas yield results.