Past

The vast majority of the world's electricity is generated by converting thermal energy into electric energy by use of a steam turbine. Siemens are one of the worlds leading manufacturers of such
turbines, and aim to design theirs to be as efficient as possible. Using an internally built software, Siemens can estimate the efficiency which would result from a turbine design. In this presentation, we present the approaches that have been taken to improve turbine design using mathematical optimisation software. In particular, we focus on the failings of the approach currently taken, the obstacles in place which make solving this problem difficult, and the approach we intend to take to find a locally optimal solution.

This talk will review the main Tikhonov and Bayesian smoothing formulations of inverse problems for dynamical systems with partially observed variables and parameters. The main contenders: strong-constraint, weak-constraint and penalty function formulations will be described. The relationship between these formulations and associated optimisation problems will be revealed.  To close we will indicate techniques for maintaining sparsity and for quantifying uncertainty.

Calcination describes the heat treatment of anthracite particles in a furnace to produce a partially-graphitised material which is suitable for use in electrodes and for other met- allurgical applications. Electric current is passed through a bed of anthracite particles, here referred to as a coke bed, causing Ohmic heating and high temperatures which result in the chemical and structural transformation of the material.

Understanding the behaviour of such mechanisms on the scale of a single particle is often dealt with through the use of computational models such as DEM (Discrete Element Methods). However, because of the great discrepancy between the length scale of the particles and the length scale of the furnace, we can exploit asymptotic homogenisation theory to simplify the problem.  

In this talk, we will present some results relating to the electrical and thermal conduction through granular material which define effective quantities for the conductivities by considering a microscopic representative volume within the material. The effective quantities are then used as parameters in the homogenised macroscopic model to describe calcination of anthracite. 

Neural networks underpin both biological intelligence and modern AI systems, yet there is relatively little theory for how the observed behavior of these networks arises. Even the connectivity of neurons within the brain remains largely unknown, and popular deep learning algorithms lack theoretical justification or reliability guarantees.  In this talk, we consider paths towards a more rigorous understanding of neural networks. We characterize and, where possible, prove essential properties of neural algorithms: expressivity, learning, and robustness. We show how observed emergent behavior can arise from network dynamics, and we develop algorithms for learning more about the network structure of the brain.

In this work with P.--E. Jabin, we are interested in quantitative estimates for advective equations with an anelastic constraint in presence of vacuum. More precisely, we derive a stability estimate and obtain the existence of renormalized solutions. The method itself introduces weights which sole a dual equation and allow to propagate appropriatly weighted norms on the initial solution. In a second time, a control on where those weights may vanish allow to deduce global and precise quantitative regularity estimates.

We present a state-sum construction of TFTs on r-spin surfaces which
uses a combinatorial model of r-spin structures. We give an example of
such a TFT which computes the Arf invariant for r even. We use the
combinatorial model and this TFT to calculate diffeomorphism classes of
r-spin surfaces with parametrized boundary.

I will explain our recent description of the fundamental degrees of freedom underlying a generalized Kahler structure. For a usual Kahler
structure, it is well-known that the geometry is determined by a complex structure, a Kahler class, and the choice of a positive(1,1)-form in this class, which depends locally on only a single real-valued function: the Kahler potential. Such a description for generalized Kahler geometry has been sought since it was discovered in1984. We show that a generalized Kahler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a
holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on
only a single real-valued function, which we call the generalized Kahler potential. To solve the problem we make use of, and generalize,
two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds;
the second is Donaldson's interpretation of a Kahler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle.

The coefficients of the low energy expansion of closed string amplitudes transform as automorphic functions under En(Z) U-duality groups.
 The seminar will give an overview of some features of the coefficients of low order terms in this expansion, which involve a fascinating interplay between multiple zeta values and certain elliptic and hyperelliptic generalisations, Langlands Eisenstein series for the En groups, and the ultraviolet behaviour of maximally supersymmetric supergravity. 

Claudia Scheimbauer

Title: Quantum field theory meets higher categories

Abstract: Studying physics has always been a driving force in the development of many beautiful pieces of mathematics in many different areas. In the last century, quantum field theory has been a central such force and there have been several fundamentally different approaches using and developing vastly different mathematical tools. One of them, Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories, provides a beautiful link between the geometry of "spacetimes” (mathematically described as cobordisms) and algebraic structures. Combining this approach with the physical notion of "locality" led to the introduction of the language of higher categories into the topic. The Cobordism Hypothesis classifies "fully local" topological field theories and gives us a recipe to construct examples thereof by checking certain algebraic conditions generalizing the existence of the dual of a vector space. I will give an introduction to the topic and very briefly mention on my own work on these "extended" topological field theories.

Alberto Paganini

Title: Shape Optimization with Finite Elements

Abstract: Shape optimization means looking for a domain that minimizes a target cost functional. Such problems are commonly solved iteratively by constructing a minimizing sequence of domains. Often, the target cost functional depends on the solution to a boundary value problem stated on the domain to be optimized. This introduces the difficulty of solving a boundary value problem on a domain that changes at each iteration. I will suggest how to address this issue using finite elements and conclude with an application from optics.

I will discuss some properties of delay differential equations in which the delay is not prescribed a-priori but is determined from a threshold condition. Sometimes the delay depends on the solution of the differential equation and its history. A scenario giving rise to a threshold type delay is that larval insects sometimes experience halting or slowing down of development, known as diapause, perhaps as a consequence of intra-specific competition among larvae at higher densities. Threshold delays can result in population dynamical models having some unusual properties, for example, if the model has an Allee effect then diapause may cause extinction in some parameter regimes even where the initial population is high.

Please  note that this talk is only suitable for Mathematicians.

Persistent homology is an algebraic tool for quantifying topological features of shapes and functions, which has recently found wide applications in data and shape analysis. In the first and introductory part of this talk I recall the underlying ideas and basic concepts of this very active field of research. In the second part, I plan to sketch a concrete application of this concept to digital image processing. 

We will briefly revisit Voronoi summation in its classical form and mention some of its many applications in number theory. We will then show how to use the global Whittaker model to create Voronoi type formulae. This new approach allows for a wide range of weights and twists. In the end we give some applications to the subconvexity problem of degree two $L$-functions. 

The Witten-Reshetikhin-Turaev invariant Z(X,K) of a closed oriented three-manifold X containing a knot K, was originally introduced by Witten in order to extend the Jones polynomial of knots  in terms of Chern-Simons theory. Classically, the Jones polynomial is defined for a knot inside the three-sphere in  a combinatorial manner. In Witten's approach, the Jones polynomial J(K) emerge as the expectation value of a certain observable in Chern-Simons theory, which makes sense when K is embedded in any closed oriented three-manifold X. Moreover; he proposed that these invariants should be extendable to so-called topological quantum field theories (TQFT's). There is a catch; Witten's ideas relied on Feynman path integrals, which made them unrigorous from a mathematical point of view. However; TQFT's extending the Jones polynomial were subsequently constructed mathematically through combinatorial means by Reshetikhin and Turaev. In this talk, I shall expand slightly on the historical motivation of WRT invariants, introduce the formalism of TQFT's, and present some of the open problems concerning WRT invariants. The guiding motif will be the analogy between TQFT and quantum field theory.

We present a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, and bounds on the distribution of a sum of dependent random variables. As an application we focus on the problem of risk aggregation under model uncertainty. The talk is based on joint work with Stephan Eckstein and Mathias Pohl.

Statements in media about record wave heights being measured are more and more common, the latest being about a record wave of almost 24m in the Southern Ocean on 9 May 2018. We will review some of these wave measurements and the various techniques to measure waves. Then we will explain the various mechanisms that can produce extreme waves both in wave tanks and in the ocean. We will conclude by providing the mechanism that, we believe, explains some of the famous extreme waves. Note that extreme waves are not necessarily rogue waves and that rogue waves are not necessarily extreme waves.

In the past few decades, power grids across the world have become dependent on markets that aim to efficiently match supply with demand at all times via a variety of pricing and auction mechanisms. These markets are based on models that capture interactions between producers, transmission and consumers. Energy producers typically maximize profits by optimally allocating and scheduling resources over time. A dynamic equilibrium aims to determine prices and dispatches that can be transmitted over the electricity grid to satisfy evolving consumer requirements for energy at different locations and times. Computation allows large scale practical implementations of socially optimal models to be solved as part of the market operation, and regulations can be imposed that aim to ensure competitive behaviour of market participants.

Questions remain that will be outlined in this presentation.

Firstly, the recent explosion in the use of renewable supply such as wind, solar and hydro has led to increased volatility in this system. We demonstrate how risk can impose significant costs on the system that are not modeled in the context of socially optimal power system markets and highlight the use of contracts to reduce or recover these costs. We also outline how battery storage can be used as an effective hedging instrument.

Secondly, how do we guarantee continued operation in rarely occuring situations and when failures occur and how do we price this robustness?

Thirdly, how do we guarantee appropriate participant behaviour? Specifically, is it possible for participants to develop strategies that move the system to operating points that are not socially optimal?

Fourthly, how do we ensure enough transmission (and generator) capacity in the long term, and how do we recover the costs of this enhanced infrastructure?
 

In this talk I will start with a brief overview of the Cauchy problem for the Einstein equations of general relativity, and in particular the nonlinear stability of the trivial Minkowski solution in wave gauge as shown by Lindblad and Rodnianski. I will then discuss the Kaluza Klein spacetime of the form $R^{1+3} \times K$ where $K$ is the $n-$torus with the flat metric.  An interesting question to ask is whether this solution to the Einstein equations, viewed as an initial value problem, is stable to small perturbations of the initial data. Motivated by this problem, I will outline how the proof of stability in a restricted class of perturbations in fact follows from the work of Lindblad and Rodnianski, and discuss the physical justification behind this restriction. 

Elements of a finitely generated group have a natural notion of length: namely the length of a shortest word over the generators that represents the element. This allows us to study the growth of such groups by considering the size of spheres with increasing radii. One current area of interest is the rationality or otherwise of the formal power series whose coefficients are the sphere sizes. I will describe a combinatorial way to study this series for the class of virtually abelian groups, introduced by Benson in the 1980s, and then outline its applications to other types of growth series.

This lecture will give a brief review of the theory of non-abelian reciprocity maps and their applications to Diophantine geometry, and pose some questions for model-theorists.
 

Given a smooth variety X containing a smooth divisor Y, the relative Gromov-Witten invariants of (X,Y) are defined as certain counts of algebraic curves in X with specified orders of tangency to Y. Their intrinsic interest aside, they are an important part of any Gromov-Witten theorist’s toolkit, thanks to their role in the celebrated “degeneration formula.” In recent years these invariants have been significantly generalised, using techniques in logarithmic geometry. The resulting “log Gromov-Witten invariants” are defined for a large class of targets, and in particular give a rigorous definition of relative invariants for (X,D) where D is a normal crossings divisor. Besides being more general, these numbers are  intimately related to constructions in Mirror Symmetry, via the Gross-Siebert program. In this talk, we will describe a recursive formula for computing the invariants of (X,D) in genus zero. The result relies on a comparison theorem which expresses the log Gromov-Witten invariants as classical (i.e. non log-geometric) objects.
 

In this talk we address the numerical approximation of Mean Field Games with local couplings. For finite difference discretizations of the Mean Field Game system, we follow a variational approach, proving that the schemes can be obtained as the optimality system of suitably defined optimization problems. In order to prove the existence of solutions of the scheme with a variational argument, the monotonicity of the coupling term is not used, which allow us to recover general existence results. Next, assuming next that the coupling term is monotone, the variational problem is cast as a convex optimization problem for which we study and compare several proximal type methods. These algorithms have several interesting features, such as global convergence and stability with respect to the viscosity parameter. We conclude by presenting numerical experiments assessing the performance of the proposed methods. In collaboration with L. Briceno-Arias (Valparaiso, CL) and F. J. Silva (Limoges, FR).

Let $\mathfrak g$ be a semisimple Lie algebra.  A $\mathfrak g$-algebra is an associative algebra $R$ on which $\mathfrak g$ acts by derivations.  There are several significant examples.  Let $V$ a finite dimensional $\mathfrak g$ module and take  $R=\mathrm{End} V$ or $R=D(V)$ being the ring of derivations on  $V$ . Again take $R=U(\mathfrak g)$.   In all these cases  $ S=U(\mathfrak g)\otimes R$ is again a $\mathfrak g$-algebra.  Finally let $T$ denote the subalgebra of invariants of $S$.
 
For the first choice of $R$ above the representation theory of $T$ can be rather explicitly described in terms of Kazhdan-Lusztig polynomials.  In the second case the simple $T$ modules can be described in terms of the simple $D(V)$ modules.  In the third case it is shown that all simple $T$ modules are finite dimensional, despite the fact that $T$ is not a PI ring,  except for the case $\mathfrak  g =\mathfrak {sl}(2)$.

Semidefinite convex optimization problems often have low-rank solutions that can be represented with O(p)-storage. However, semidefinite programming methods require us to store the matrix decision variable with size O(p^2), which prevents the application of virtually all convex methods at large scale.

Indeed, storage, not arithmetic computation, is now the obstacle that prevents us from solving large- scale optimization problems. A grand challenge in contemporary optimization is therefore to design storage-optimal algorithms that provably and reliably solve large-scale optimization problems in key scientific and engineering applications. An algorithm is called storage optimal if its working storage is within a constant factor of the memory required to specify a generic problem instance and its solution.

So far, convex methods have completely failed to satisfy storage optimality. As a result, the literature has largely focused on storage optimal non-convex methods to obtain numerical solutions. Unfortunately, these algorithms have been shown to be provably correct only under unverifiable and unrealistic statistical assumptions on the problem template. They can also sacrifice the key benefits of convexity, as they do not use key convex geometric properties in their cost functions.

To this end, my talk introduces a new convex optimization algebra to obtain numerical solutions to semidefinite programs with a low-rank matrix streaming model. This streaming model provides us an opportunity to integrate sketching as a new tool for developing storage optimal convex optimization methods that go beyond semidefinite programming to more general convex templates. The resulting algorithms are expected to achieve unparalleled results for scalable matrix optimization problems in signal processing, machine learning, and computer science.

Cascading phenomena are prevalent in natural and social-technical complex networks. We study the persistent cascade-recovery dynamics on random networks which are robust against small trigger but may collapse for larger one. It is observed that depending on the relative intensity of triggering and recovery, the network belongs one of the two dynamical phases: collapsing or active phase. We devise an analytical framework which characterizes not only the critical behaviour but also the temporal evolution of network activity in both phases. Results from agent-based simulations show good agreement with theoretical calculations. This work is an important attempt in understanding networked systems gradually evolving into a state of critical transition, with many potential applications.
 

I will discuss a classical six-dimensional superconformal field theory containing a non-abelian tensor multiplet which we recently constructed in arXiv:1712.06623.

This theory satisfies many of the properties of the mysterious (2,0)-theory: non-abelian 2-form potentials, ADE-type gauge structure, reduction to Yang-Mills theory and reduction to M2-brane models. There are still some crucial differences to the (2,0)-theory, but our action seems to be a key stepping stone towards a potential classical formulation of the (2,0)-theory.

I will review in detail the underlying mathematics of categorified gauge algebras and categorified connections, which make our constructions possible.

In the first part of this talk the two-dimensional Landau-de Gennes energy with several elastic constants, subject to general k-radial symmetric boundary conditions, will be analysed. It will be shown that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case k=2. Analysis of the associated harmonic map type problem arising in the limit of small elastic constants allows to identify three types of radial profiles: with two, three or full five components. In the second part of the talk different paths for emergency of non-radially symmetric solutions and their analytical structure in 2D as well as 3D cases will be discussed. These results is a joint work with Jonathan Robbins, Valery Slastikov and Arghir Zarnescu.
 

Based on classical invariant theory, I describe a complete set of elements of the signature that is invariant to the general linear group, rotations or permutations.

A geometric interpretation of some of these invariants will be given.

Joint work with Jeremy Reizenstein (Warwick).

The problem of "unbounded rank expanders" asks 
whether we can endow a system of generators with a sequence of 
special linear groups whose degrees tend to infinity over quotient rings 
of Z such that the resulting Cayley graphs form an expander family.
Kassabov answered this question in the affirmative. Furthermore, the 
completely satisfactory solution to this question was given by 
Ershov and Jaikin--Zapirain (Invent. Math., 2010);  they proved
Kazhdan's property (T) for elementary groups over non-commutative 
rings. (T) is equivalent to the fixed point property with respect to 
actions on Hilbert spaces by isometries.

We provide a new framework to "upgrade" relative fixed point 
properties for small subgroups to the fixed point property for the 
whole group. It is inspired by work of Shalom (ICM, 2006). Our 
main criterion is stated only in terms of intrinsic group structure 
(but *without* employing any form of bounded generation). 
This, in particular, supplies a simpler (but not quantitative) 
alternative proof of the aforementioned result of Ershov and 
Jaikin--Zapirain.  

If time permits, we will discuss other applications of our result.

We present an algebraic formulation for the flow of a differential equation driven by a path in a Lie group. The formulation is motivated by formal differential equations considered by Chen.

Lagrangian Floer cohomology can be enriched by using local coefficients to record some homotopy data about the boundaries of the holomorphic disks counted by the theory. In this talk I will explain how one can do this under the monotonicity assumption and when the Lagrangians are equipped with local systems of rank higher than one. The presence of holomorphic discs of Maslov index 2 poses a potential obstruction to such an extension. However, for an appropriate choice of local systems the obstruction might vanish and, if not,
one can always restrict to some natural unobstructed subcomplexes. I will showcase these constructions with some explicit calculations for the Chiang Lagrangian in CP^3 showing that it cannot be disjoined from RP^3 by a Hamiltonian isotopy, answering a question of Evans-Lekili. Time permitting, I will also discuss some work-in-progress on the topology of monotone Lagrangians in CP^3, part of which follows from more general joint work with Jack Smith on the topology of monotone Lagrangians of maximal Maslov number in
projective spaces.

 I will summarise old and recent developments on the classification and solution of Rational Conformal Field Theories in 2 dimensions using the method of Modular Differential Equations. Novel and exotic theories are found with small numbers of characters and simple fusion rules, one of these being the Baby Monster CFT. Correlation functions for many of these theories can be computed using crossing-symmetric differential equations.

Details to follow

A Surstseyan eruption is a particular kind of volcanic eruption which involves the bulk interaction of water and hot magma. Surtsey Island was born during such an eruption process in the 1940s. I will talk about mathematical modelling of the flashing of water to steam inside a hot erupted lava ball called a Surtseyan bomb. The overall motivation is to understand what determines whether such a bomb will fragment or just quietly fizzle out...
Partial differential equations model transient changes in temperature and pressure in Surtseyan ejecta. We have used a highly simplified approach to the temperature behaviour, to separate temperature from pressure. The resulting pressure diffusion equation was solved numerically and asymptotically to derive a single parametric condition for rupture of ejecta. We found that provided the permeability of the magma ball is relatively large, steam escapes rapidly enough to relieve the high pressure developed at the flashing front, so that rupture does not occur. This rupture criterion is consistent with existing field estimates of the permeability of intact Surtseyan bombs, fizzlers that have survived.
I describe an improvement of this model that allows for the fact that pressure and temperature are in fact coupled, and that the process is not adiabatic. A more systematic reduction of the resulting coupled nonlinear partial differential equations that arise from mass, momentum and energy conservation is described. We adapt an energy equation presented in G.K. Batchelor's book {\em An Introduction to Fluid Dynamics} that allows for pressure-work. This is work in progress.  Work done with Emma Greenbank, Ian Schipper and Andrew Fowler 

The local volatility model is a celebrated model widely used for pricing and hedging financial derivatives. While the model’s main appeal is its capability of reproducing any given surface of observed option prices—it provides a perfect fit—the essential component of the model is a latent function which can only be unambiguously determined in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested involving steps of interpolation and extrapolation, most often of parametric form and with point-estimates as result. We seek to look at the calibration problem in a probabilistic framework with a nonparametric approach based on Gaussian process priors. This immediately gives a way of encoding prior believes about the local volatility function, and a hypothesis model which is highly flexible whilst being prone to overfitting. Besides providing a method for calibrating a (range of) point-estimate, we seek to draw posterior inference on the distribution over local volatility to better understand the uncertainty attached with the calibration. Further, we seek to understand dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this gives us means of inferring predictive distributions not only locally, but also for entire surfaces forward in time.

This talk focuses on algebraic and combinatorial-topological problems motivated by neuroscience. Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus,where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets?

This talk describes how to use tools from combinatorics and commutative algebra to uncover a variety of signatures of convex and non-convex codes.

This talk is based on joint works with Aaron Chen and Florian Frick, and with Carina Curto, Elizabeth Gross, Jack Jeffries, Katie Morrison, Mohamed Omar, Zvi Rosen, and Nora Youngs.

Archimedes, who famously jumped out of his bath shouting "Eureka", also invented $\pi$. 

Euler invented $e$ and had fun with his formula $e^{2\pi i} = 1$

The world is full of important numbers waiting to be invented. Why not have a go ?

Michael Atiyah is one of the world's foremost mathematicians and a pivotal figure in twentieth and twenty-first century mathematics. His lecture will be followed by an interview with Sir John Ball, Sedleian Professor of Natural Philosophy here in Oxford where Michael will talk about his lecture, his work and his life as a mathematician.

Please email @email to register.

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

It is an old problem in number theory to count number fields of a fixed degree and having a fixed Galois group for its Galois closure, ordered by their absolute discriminant, say. In this talk, I shall discuss some background of this problem, and then report a recent work with Stanley Xiao. In our paper, we considered quartic $D_4$-fields whose ring of integers has a certain nice algebraic property, and we counted such fields by their conductor.

The Grojnowski-Nakajima theorem states that the direct sum of the homologies of the Hilbert schemes on n points on an algebraic surface is an irreducible highest weight representation of an infinite-dimensional Heisenberg superalgebra. We present an idea to rederive the Grojnowski-Nakajima theorem using Halpern-Leistner's categorical Kirwan surjectivity theorem and Joyce's theorem that the homology of a moduli space of sheaves is a vertex algebra. We compute the homology of the moduli stack of perfect complexes of coherent sheaves on a smooth quasi-projective variety X, identify it as a (modified) lattice vertex algebra on the Lawson homology of X, and explain its relevance to the aforementioned problem.

High-frequency realized variance approaches offer great promise for 
estimating asset prices’ covariation, but encounter difficulties 
connected to the Epps effect. This paper models the Epps effect in a 
stochastic volatility setting. It adds dependent noise to a factor 
representation of prices. The noise both offsets covariation and 
describes plausible lags in information transmission. Non-synchronous 
trading, another recognized source of the effect, is not required. A 
resulting estimator of correlations and betas performs well on LSE 
mid-quote data, lending empirical credence to the approach.

The peeling of an elastic sheet away from thin layer of viscous fluid is a simply-stated and generic problem, that involves complex interactions between flow and elastic deformation on a range of length scales. 

I will illustrate the possibilities by considering theoretically and experimentally the injection and spread of viscous fluid beneath a flexible elastic lid; the injected fluid forms a blister, which spreads by peeling the lid away at the  perimeter of the blister. Among the many questions to be considered are the mechanisms for relieving the elastic analogue of the contact-line problem, whether peeling is "by bending" or "by pulling", the stability of the peeling front, and the effects of a capillary meniscus when peeling is by air injection. The result is a plethora of dynamical regimes and asymptotic scaling laws.

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.

Thompson's group F is a group of homeomorphisms of the unit interval which exhibits a strange mix of properties; on the one hand it has some self-similarity type properties one might expect of a really big group, but on the other hand it is finitely presented. I will give a proof of finite generation by expressing elements as pairs of binary trees.

The Fujita equation $u_{t}=\Delta u+u^{p}$, $p>1$, has been a canonical blow-up model for more than half a century. A great deal is known about the singularity formation under a variety of conditions. In particular we know that blow-up behaviour falls broadly into two categories, namely Type I and Type II. The former is generic and stable while the latter is rare and highly unstable. One of the central results in the field states that in the Sobolev subcritical regime, $1<p<\frac{n+2}{n-2}$, $n\geq 3$, only type I is possible whenever the domain is \emph{convex} in $\mathbb{R}^n$. Despite considerable effort the requirement of convexity has not been lifted and it is not clear whether this is an artefact of the methodology or whether the geometry of the domain may actually affect the blow-up type. In my talk I will discuss how the question of the blow-up type for non-convex domains is intimately related to the validity of some Li-Yau-Hamilton inequalities.