Past

The construction of permutation functions of a finite field is a task of great interest in cryptography and coding theory. In this talk we describe a method which combines Chebotarev density theorem with elementary group theory to produce permutation rational functions over a finite field F_q. Our method is entirely constructive and as a corollary we get the classification of permutation polynomials up to degree 4 over any finite field of odd characteristic.

This is a joint work with Andrea Ferraguti.
 

The talk originates from two studies on the dynamic properties of one-dimensional elastic quasicrystalline solids. The first one refers to a detailed investigation of scaling and self-similarity of the spectrum of an axial waveguide composed of repeated elementary cells designed by adopting the family of generalised Fibonacci substitution rules corresponding to the so-called precious means. For those, an invariant function of the circular frequency, the Kohmoto's invariant, governs self-similarity and scaling of the stop/pass band layout within defined ranges of frequencies at increasing generation index. The Kohmoto's invariant also explains the existence of particular frequencies, named canonical frequencies, associated with closed orbits on the geometrical three-dimensional representation of the invariant. The second part shows the negative refraction properties of a Fibonacci-generated quasicrystalline laminate and how the tuning of this phenomenon can be controlled by selecting the generation index of the sequence.

We consider calculation of VaR/TVaR capital requirements when the underlying economic scenarios are determined by simulatable risk factors. This problem involves computationally expensive nested simulation, since evaluating expected portfolio losses of an outer scenario (aka computing a conditional expectation) requires inner-level Monte Carlo. We introduce several inter-related machine learning techniques to speed up this computation, in particular by properly accounting for the simulation noise. Our main workhorse is an advanced Gaussian Process (GP) regression approach which uses nonparametric spatial modeling to efficiently learn the relationship between the stochastic factors defining scenarios and corresponding portfolio value. Leveraging this emulator, we develop sequential algorithms that adaptively allocate inner simulation budgets to target the quantile region. The GP framework also yields better uncertainty quantification for the resulting VaR/\TVaR estimators that reduces bias and variance compared to existing methods.  Time permitting, I will highlight further related applications of statistical emulation in risk management.
This is joint work with Jimmy Risk (Cal Poly Pomona). 
 

In this talk I will review recent results on the analysis of shock-capturing-type methods applied to convection-dominated problems. The method of choice is a variant of the Algebraic Flux-Correction (AFC) scheme. This scheme has received some attention over the last two decades due to its very satisfactory numerical performance. Despite this attention, until very recently there was no stability and convergence analysis for it. Thus, the purpose of the works reviewed in this talk was to bridge that gap. The first step towards the full analysis of the method is a rewriting of it as a nonlinear edge-based diffusion method. This writing makes it possible to present a unified analysis of the different variants of it. So, minimal assumptions on the components of the method are stated in such a way that the resulting scheme satisfies the Discrete Maximum Principle (DMP) and is convergence. One property that will be discussed in detail is the linearity preservation. This property has been linked to the good performance of methods of this kind. We will discuss in detail its role and the impact of it in the overall convergence of the method. Time permitting, some results on a posteriori error estimation will also be presented. 
This talk will gather contributions with A. Allendes (UTFSM, Chile), E. Burman (UCL, UK), V. John (WIAS, Berlin), F. Karakatsani (Chester, UK), P. Knobloch (Prague, Czech Republic), and 
R. Rankin (U. of Nottingham, China).

We consider the  symbiotic branching model, which describes a spatial population consisting of two types in terms of a coupled system of stochastic PDEs. One particularly important special case is Kimura's stepping stone model in evolutionary biology. Our main focus is a description of the interfaces between the types in the large scale limit of the system. As a new tool we will introduce a moment duality, which also holds for the limiting model. This also has implications for a classification of entrance laws of annihilating Brownian motions.

 The $n$-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus $g$ with $n$-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the $n=1$ case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively.  In this talk, I will describe the upper and lower bounds I have proved as a function of $g$ and $n$.

For energy functionals composed of competing short- and long-range interactions, minimizers are often conjectured to form essentially periodic patterns on some intermediate lengthscale. However,  not many detailed structural results or proofs of periodicity are known in dimensions larger than 1. We study a functional composed of  the attractive, local interfacial energy of charges concentrated on a hyperplane and the energy of the electric field generated by these charges in the full space, which can be interpreted as a repulsive, non-local functional of the charges. We follow the approach of Alberti-Choksi-Otto and prove that the energy of minimizers of this functional is uniformly distributed  on cubes intersecting the hyperplane, which are sufficiently large with respect to the intrinsic lengthscale.

This is a joint work with A. Julia and F. Otto.

Side channel leakage is no longer just a concern for industries that
traditionally have a high degree of awareness and expertise in
(implementing) cryptography. With the rapid growth of security
sensitive applications in other areas, e.g. smartphones, homes, etc.
there is a clear need for developers with little to no crypto
expertise to implement and instantiate cryptography securely on
embedded devices. In this talk, I explain what makes finding side
channel leaks challenging (in theory and in practice) and give an
update on our latest work to develop methods and tools to enable
non-domain experts to ‘get a grip’ on leakage in their
implementations.

Hall algebras of categories of quiver representations and coherent sheaves

on smooth projective curves over F_q recover interesting

representation-theoretic objects such as quantum groups and their

generalizations. I will define and describe the structure of the Hall

algebra of coherent sheaves on a projective variety over F_1, with P^2 as

the main example. Examples suggest that it should be viewed as a degenerate

q->1 limit of its counterpart over F_q.

We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold X and define DT4 invariants by integrating the Euler class of a tautological vector bundle against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when L corresponds to a smooth divisor on X. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples. Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by exp(M(q) − 1), where M(q) denotes the MacMahon function. This is joint work with Martijn Kool.

Forecasting a diagnosis of Alzheimer’s disease is a promising means of selection for clinical trials of Alzheimer’s disease therapies. A positive PET scan is commonly used as part of the inclusion criteria for clinical trials, but PET imaging is expensive, so when a positive scan is one of the trial inclusion criteria it is desirable to avoid screening failures. In this talk I will describe a scheme for pre-selecting participants using statistical learning methods, and investigate how brain regions change as the disease progresses.  As a means of generating features I apply the Chen path signature. This is a systematic way of providing feature sets for multimodal data that can probe the nonlinear interactions in the data as an extension of the usual linear features. While it can easily perform a traditional analysis, it can also probe second and higher order events for their predictive value. Combined with Lasso regularisation one can auto detect situations where the observed data has nonlinear information.

The Peter-Weyl idempotent of a parahoric subgroup is the sum of the idempotents of irreducible representations which have a nonzero Iwahori fixed vector. The associated convolution algebra is called a Peter-Weyl Iwahori algebra.  We show any Peter-Weyl Iwahori algebra is Morita equivalent to the Iwahori-Hecke algebra.  Both the Iwahori-Hecke algebra and a Peter-Weyl Iwahori algebra have a natural C*-algebra structure, and the Morita equivalence preserves irreducible hermitian and unitary modules.  Both algebras have another anti-involution denoted as •, and the Morita equivalence preserves irreducible and unitary modules for the • involution.   This work is joint with Dan Barbasch.
 

The latest CPUs by Intel and ARM support vectorised operations, where a single set of instructions (e.g. add, multiple, bit shift, XOR, etc.) are performed in parallel for small batches of data. This can provide great performance improvements if each parallel instruction performs the same operation, but carries the risk of performance loss if each needs to perform different tasks (e.g. if else conditions). I will present the work I have done so far looking into how to recover the full performance of the hardware, and some of the challenges faced when trading off between ever larger parallel tasks, risks of tasks diverging, and how certain coding styles might be modified for memory bandwidth limited applications. Examples will be taken from finance and Monte Carlo applications, inspecting some standard maths library functions and possibly random number generation.

Collective neural crest (NC) cell migration determines the formation of peripheral tissues during vertebrate development. If NC cells fail to reach a target or populate an incorrect location, improper cell differentiation or uncontrolled cell proliferation can occur. Therefore, knowledge of embryonic cell migration is important for understanding birth defects and tumour formation. However, the response of NC cells to different stimuli, and their ability to migrate to distant targets, are still poorly understood. Recently, experimental and computational studies have provided evidence that there are at least two subpopulations of NC cells, namely “leading” and “trailing” cells, with potential further differentiation between the cells in these subpopulations [1,2]. The main difference between these two cell types is the mechanism driving motility and invasion: the leaders follow the gradient of a chemoattractant, while the trailing cells follow “gradients” of the leaders. The precise mechanisms underlying these leader-follower interactions are still unclear.

We develop and apply innovative multi-scale modelling frameworks to analyse signalling effects on NC cell dynamics. We consider different potential scenarios and investigate them using an individual-based model for the cell motility and reaction-diffusion model to describe chemoattractant dynamics. More specifically, we use a discrete self-propelled particle model [3] to capture the interactions between the cells and incorporate volume exclusion. Streaming migration is represented using an off-lattice model to generate realistic cell arrangements and incorporate nonlinear behaviour of the system, for example the coattraction between cells at various distances. The simulations are performed using Aboria, which is a C++ library for the implementation of particle-based numerical methods [4]. The source of chemoattractant, the characteristics of domain growth, and types of boundary conditions are some other important factors that affect migration. We present results on how robust/sensitive cells invasion is to these key biological processes and suggest further avenues of experimental research.

[1] R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R.E. Baker, P. K. Maini and P. M. Kulesa. (2012). Multiscale mechanisms of cell migration during development: theory and experiment, Development, 139, 2935-2944.

[2] R. McLennan, L. J. Schumacher, J. A. Morrison, J. M. Teddy, D. A. Ridenour, A. C. Box, C. L. Semerad, H. Li, W. McDowell, D. Kay, P. K. Maini, R. E. Baker and P. M. Kulesa. (2015). Neural crest migration is driven by a few trailblazer cells with a unique molecular signature narrowly confined to the invasive front, Development, 142, 2014-2025.

[3] G. Grégoire, H. Chaté and Y Tu. (2003). Moving and staying together without a leader, Physica D: Nonlinear Phenomena, 181, 157-170.

[4] M. Robinson and M. Bruna. (2017). Particle-based and meshless methods with Aboria, SoftwareX, 6, 172-178. Online documentation https://github.com/martinjrobins/Aboria.

There have been enormous advances in both our ability to represent scattering amplitudes at the integrand-level (for an increasingly wide variety of quantum field theories), and also in our integration technology (and our understanding of the functions that result). In this talk, I review both sides of these recent developments. At the integrand-level, I describe the "prescriptive" refinement of generalized unitarity, and show how closed, integrand-level formulae can be given for all leading-weight contributions to any amplitude in any quantum field theory. Regarding integration, I describe some new results that could be summarized as "dual-conformal sufficiency": that all planar, ultraviolet-finite integrands can be regulated and computed directly in terms of manifestly dual-conformal integrals. I illustrate the power of having such representations, and discuss the role played by a (conjectural) cluster-algebraic structure for kinematic dependence. 

The story of human progress is often described as a succession of ‘eras’ or ‘ages’ that are characterised by their most dominant technologies (e.g., the bronze age, the industrial revolution or the information age). In modern times, the fast pace of technological progress has accelerated the succession of eras. In addition, the increasing complexity of inventions has made the task of determining when eras begin and end more challenging, as eras are less about the dominance of a single technology and more about the way in which different technologies are combined. We present a data-driven method to determine and uncover technological eras based on networks and patent classification data. We construct temporal networks of technologies that co-appear in patents. By analyzing the evolution of the core-periphery structure and centrality time-series in these networks, we identify periods of time dominated by technological combinations which we identify as distinct ‘eras’. We test the performance of our method using a database of patents in Great Britain spanning a century, and identify five distinct eras.

A well known result by Schaeffer (1973) shows that generic solutions to a scalar conservation law are piecewise smooth, containing a finite family of shock curves.

In this direction, it is of interest to find other classes of nonlinear hyperbolic equations where nearly all solutions (in a Baire category sense) are piecewise smooth, and classify their singularities.

The talk will mainly focus on conservative solutions to the nonlinear variational wave equation $u_{tt} - c(u)(c(u) u_x)_x = 0$. For an open dense set of $C^3$ initial data, it is proved that the conservative solution is piecewise smooth in the $t - x$ plane, while the gradient $u_x$ can blow up along  finitely  many characteristic curves. The analysis relies on a variable transformation which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.   

A detailed description of the solution profile can be given, in a neighborhood of every singular point and every singular curve.

Some results on structurally stable singularities have been obtained  also for dissipative solutions, of the above wave equation. Recent progress on the Burgers-Hilbert equation, and related open problems, will also be discussed.

These results are in collaboration with Geng Chen, Tao Huang, Fang Yu, and Tianyou Zhang.

A particle bouncing around inside a Euclidean polygon gives rise to a biinfinite "bounce sequence" (or "cutting sequence") recording the (labeled) sides encountered by the particle.  In this talk, I will describe recent work with Duchin, Erlandsson, and Sadanand, where we prove that the set of all bounce sequences---the "bounce spectrum"---essentially determines the shape of the polygon.  This is consequence of a technical result about Liouville currents associated to nonpositively curved Euclidean cone metrics on surfaces.  In the talk I will explain the objects mentioned above, how they relate to each other, and give some idea of how one determines the shape of the polygon from its bounce spectrum.

McKean-Vlasov SDEs are SDEs where  the coefficients depend on the law of the solution to the SDE. Their interest is in the links with nonlinear PDEs on one side (the SDE-related Fokker-Planck equation is nonlinear) and with interacting particles on the other side: the McKean-Vlasov SDE be approximated by a system of weakly coupled SDEs. In this talk we consider McKean-Vlasov SDEs with irregular drift: though well-posedness for this SDE is not known, we show a large deviation principle for the corresponding interacting particle system. This implies, in particular, that any limit point of the particle system solves the McKean-Vlasov SDE. The proof combines rough paths techniques and an extended Vanrdhan lemma.

This is a joint work with Thomas Holding.

 Consider dY(t)=f(X(t))dX(t), where X(t) is a pure jump Levy process with finite p-variation norm, 1<= p < 2, and f is a Lipchitz continuous function. Following the geometric solution construction of Levy-driven stochastic differential equations in (Williams 2001), we develop a class of epsilon-strong simulation algorithms that allows us to construct a probability space, supporting both the geometric solution Y and a fully simulatable process Y_epsilon, such that Y_epsilon is within epsilon distance from Y under the uniform metric on compact time intervals with probability 1. Moreover, the users can adaptively choose epsilon’ < epsilon, so that Y_epsilon’ can be constructed conditional on Y_epsilon. This tolerance-enforcement feature allows us to easily combine our algorithm with Multilevel Monte Carlo for efficient estimation of expectations, and adding as a benefit a straightforward analysis of rates of convergence. This is joint with Jose Blanchet, Fei He and Offer Kella.

Symplectic reduction is the natural quotient construction for symplectic manifolds. Given a free and proper action of a Lie group G on a symplectic manifold M, this process produces a new symplectic manifold of dimension dim(M) - 2 dim(G). For non-free actions, however, the result is usually fairly singular. But Sjamaar-Lerman (1991) showed that the singularities can be understood quite precisely: symplectic reductions by non-free actions are partitioned into smooth symplectic manifolds, and these manifolds fit nicely together in the sense that they form a stratification.

Symplectic reduction has an analogue in hyperkähler geometry, which has been a very important tool for constructing new examples of these special manifolds. In this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified
spaces whose strata are hyperkähler.

In this talk we will discuss non-Abelian T-duality as a solution generating technique in type II Supergravity, briefly reviewing its potential to motivate, probe or challenge classifications of supersymmetric solutions, and focusing on the open problem of providing the newly generated AdS brackgrounds with consistent dual superconformal field theories. These can be seen as renormalization fixed points of linear quivers of increasing rank. As illustrative examples, we consider the non-Abelian T-duals of AdS5xS5, the Klebanov-Witten background, and the IIA reduction of AdS4xS7, whose proposed quivers are, respectively, the four dimensional N=2 Gaiotto-Maldacena theories describing the worldvolume dynamics of D4-NS5 brane intersections, its N=1 mass deformations realized as D4-NS5-NS5’, and the three dimensional N=4 Gaiotto-Witten theories, corresponding to D3-D5-NS5. Based on 1705.09661 and 1609.09061.

The so-called moonshine phenomenon relates modular forms and finite group representations. After the celebrated monstrous moonshine, various new examples of moonshine connection have been discovered in recent years. The study of these new moonshine examples has revealed interesting connections to K3 surfaces, arithmetic geometry, and string theory.  In this colloquium I will give an overview of these recent developments. 
 

The derivation of so-called `effective descriptions' that explicitly incorporate microscale physics into a macroscopic model has garnered much attention, with popular applications in poroelasticity, and models of the subsurface in particular. More recently, such approaches have been applied to describe the physics of biological tissue. In such applications, a key feature is that the material is active, undergoing both elastic deformation and growth in response to local biophysical/chemical cues.

Here, two new macroscale descriptions of drug/nutrient-limited tissue growth are introduced, obtained by means of two-scale asymptotics. First, a multiphase viscous fluid model is employed to describe the dynamics of a growing tissue within a porous scaffold (of the kind employed in tissue engineering applications) at the microscale. Secondly, the coupling between growth and elastic deformation is considered, employing a morpho-elastic description of a growing poroelastic medium. Importantly, in this work, the restrictive assumptions typically made on the underlying model to permit a more straightforward multiscale analysis are relaxed, by considering finite growth and deformation at the pore scale.

In each case, a multiple scales analysis provides an effective macroscale description, which incorporates dependence on the microscale structure and dynamics provided by prototypical `unit cell-problems'. Importantly, due to the complexity that we accommodate, and in contrast to many other similar studies, these microscale unit cell problems are themselves parameterised by the macroscale dynamics.

In the first case, the resulting model comprises a Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. Stokes-type cell problems retain multiscale dependence, incorporating active cell motion [1]. Example numerical simulations indicate the influence of microstructure and cell dynamics on predicted macroscale tissue evolution. In the morpho-elastic model, the effective macroscale dynamics are described by a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection--reaction--diffusion equation [2].

[1] HOLDEN, COLLIS, BROOK and O'DEA. (2018). A multiphase multiscale model for nutrient limited tissue growth, ANZIAM (In press)

[2] COLLIS, BROWN, HUBBARD and O'DEA. (2017). Effective Equations Governing an Active Poroelastic Medium, Proceedings of the Royal Society A. 473, 20160755

In this talk we discuss the recent proof for the existence of $C^{1,1}$ isometric immersions of several classes of negatively curved surfaces into $\R^3$, including the Lobachevsky plane, metrics of helicoid type and a one-parameter family of perturbations of the Enneper surface. Our method, following Chen--Slemrod--Wang and Cao--Huang--Wang, is to transform the Gauss--Codazzi equations into a system of hyperbolic balance laws, and prove the existence of weak solutions by finding the invariant regions. In addition, we provide further characterisation of the $C^{1,1}$ isometrically immersed generalised helicoids/catenoids established in the literature.

I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of $\mathbb{R}^4$ to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with András Vasy.

The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e., a collection of intervals, to a finite metric space. Because of the nature of the invariant, barcodes are not well adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes, and these outputs can then be used as input to standard algorithms. I will explain how we can use tropical-like functions to coordinatize the space of persistence barcodes. These coordinates are stable with respect to the bottleneck and Wasserstein distances. I will also show how they can be used in practice.

Phase transitions and critical phenomena are ubiquitous in Nature. They permeate physics, chemistry, biology and complex systems in general, and are characterized by the role of correlations and fluctuations of many degrees of freedom. From a mathematical viewpoint, in the vicinity of a critical point, thermodynamic quantities exhibit singularities and scaling properties. Theoretical attempts to describe classical phase transitions using tools from differential topology and Morse theory provided strong arguments pointing that a phase transition may emerge as a consequence of topological changes in the configuration space around the critical point.

On the other hand, much work was done concerning the topology of networks which spontaneously emerge in complex systems, as is the case of the genome, brain, and social networks, most of these built intrinsically based on measurements of the correlations among the constituents of the system.

We aim to transpose the topological methodology previously applied in n-dimensional manifolds, to describe phenomena that emerge from correlations in a complex system, in which case Hamiltonian models are hard to invoke. The main idea is to embed the network onto an n-dimensional manifold and to study the equivalent to level sets of the network according to a filtration parameter, which can be the probability for a random graph or even correlations from fMRI measurements as height function in the context of Morse theory.  By doing so, we were able to find topological phase transitions either in random networks and fMRI brain networks.  Moreover, we could identify high-dimensional structures, in corroboration with the recent finding from the blue brain project, where neurons could form structures up to eleven dimensions.The efficiency and generality of our methodology are illustrated for a random graph, where its Euler characteristic can be computed analytically, and for brain networks available in the human connectome project.  Our results give strong arguments that the Euler characteristic, together with the distributions of the high dimensional cliques have potential use as topological biomarkers to classify brain Networks. The above ideas may pave the way to describe topological phase transitions in complex systems emerging from correlation data.

Quiver varieties, as first studied by Grojnowski and Nakajima, form an interesting class of geometric objects, which can be constructed by an array of different techniques (GIT, symplectic and Hyperkaehler reduction). In this talk, we will explain how to construct these varieties, and how their homology gives rise to a categorification of the representations of Kac-Moody Lie algebras

After reviewing old work with Teitelbaum, in which we constructed the character variety X of the additive group o_L in a finite extension L/Q_p and established the Fourier isomorphism for the distribution algebra of o_L, I will briefly report on more recent work with Berger and Xie, in which we establish the theory of (\varphi_L,\Gamma_L)-modules over X and relate it to Galois representations. Then I will discuss an ongoing project with Venjakob. Our goal is to use this theory over X for Iwasawa theory.

Joint work with Josè E. Figueroa-Lòpez, Washington University in St. Louis

Abstract: We consider a univariate semimartingale model for (the logarithm 
of) an asset price, containing jumps having possibly infinite activity. The 
nonparametric threshold estimator\hat{IV}_n of the integrated variance 
IV:=\int_0^T\sigma^2_sds proposed in Mancini (2009) is constructed using 
observations on a discrete time grid, and precisely it sums up the squared 
increments of the process when they are below a  threshold, a deterministic 
function of the observation step and possibly of the coefficients of X. All the
threshold functions satisfying given conditions allow asymptotically consistent 
estimates of IV, however the finite sample properties of \hat{IV}_n can depend 
on the specific choice of the threshold.
We aim here at optimally selecting the threshold by minimizing either the 
estimation mean squared error (MSE) or the conditional mean squared error 
(cMSE). The last criterion allows to reach a threshold which is optimal not in 
mean but for the specific  volatility and jumps paths at hand.

A parsimonious characterization of the optimum is established, which turns 
out to be asymptotically proportional to the Lévy's modulus of continuity of 
the underlying Brownian motion. Moreover, minimizing the cMSE enables us 
to  propose a novel implementation scheme for approximating the optimal 
threshold. Monte Carlo simulations illustrate the superior performance of the 
proposed method.

Most motile bacteria are equipped with multiple helical flagella, slender appendages whose rotation in viscous fluids allow the cells to self-propel. We highlight in this talk two consequences of hydrodynamics for bacteria. We first show how the swimming of cells with multiple flagella is enabled by an elastohydrodynamic instability. We next demonstrate how interactions between flagellar filaments mediated by the fluid govern the ability of the cells to reorient. 

The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed.

However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.

After giving a brief general introduction to reduced order models, we discuss developments in two different directions. In the first part, we discuss recent developments of reduced methods that conserve chosen invariants for nonlinear time-dependent problems. We pay particular attention to the development of reduced models for Hamiltonian problems and propose a greedy approach to build the basis. As we shall demonstrate, attention to the construction of the basis must be paid not only to ensure accuracy but also to ensure stability of the reduced model. Time permitting, we shall also briefly discuss how to extend the approach to include more general dissipative problems through the notion of port-Hamiltonians, resulting in reduced models that remain stable even in the limit of vanishing viscosity and also touch on extensions to Euler and Navier-Stokes equations.

The second part of the talk discusses the combination of reduced order modeling for nonlinear problems with the use of neural networks to overcome known problems of on-line efficiency for general nonlinear problems. We discuss the general idea in which training of the neural network becomes part of the offline part and demonstrate its potential through a number of examples, including for the incompressible Navier-Stokes equations with geometric variations.

This work has been done with in collaboration with B.F. Afkram (EPFL, CH), N. Ripamonti EPFL, CH) and S. Ubbiali (USI, CH).

Many problems arising in Physics can be posed as minimisation of energy functionals under linear partial differential constraints. For example, a prototypical example in the Calculus of Variations is given by functionals defined on curl-free fields, i.e., gradients. Most work done subject to more general constraints met significant difficulty due to the lack of associated potentials. We show that under the constant rank assumption, which holds true of almost all examples of constraints investigated in connection with lower-semicontinuity, linear constraints admit a potential in frequency space. As a consequence, the notion of A-quasiconvexity, which involves testing with periodic fields leading to difficulties in establishing sufficiency for weak sequential lower semi-continuity, can be tested against compactly supported fields. We will indicate how this can simplify the general framework.

Euler’s equation, the ‘most beautiful equation in mathematics’, startlingly connects the five most important constants in the subject: 1, 0, π, e and i. Central to both mathematics and physics, it has also featured in a criminal court case and on a postage stamp, and has appeared twice in The Simpsons. So what is this equation – and why is it pioneering?

Robin Wilson is an Emeritus Professor of Pure Mathematics at the Open University, Emeritus Professor of Geometry at Gresham College, London, and a former Fellow of Keble College, Oxford.

28 February 2018, 5pm-6pm, Mathematical Institute, Oxford

Please email @email to register

It is a classical theorem of Magnus that the word problem for one-relator groups is solvable; its precise complexity remains unknown. A geometric characterization of the complexity is given by the Dehn function. I will present joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of Dehn functions, including the Brady--Bridson snowflake groups on which our work relies.
 

It is a classical theorem of Magnus that the word problem for one-relator groups is solvable; its precise complexity remains unknown. A geometric characterization of the complexity is given by the Dehn function. I will present joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of Dehn functions, including the Brady--Bridson snowflake groups on which our work relies.

We present some recent results on the regularity criteria for weak solutions to the incompressible Navier--Stokes equations (NSE) in 3 dimensions. By the work of Constantin--Fefferman, it is known that the alignment of vorticity directions is crucial to the regularity of NSE in $\R^3$.  In this talk we show a boundary regularity theorem for NSE on curvilinear domains with oblique derivative boundary conditions. As an application, the boundary regularity of incompressible flows on balls, cylinders and half-spaces with Navier boundary condition is established, provided that the vorticity is coherently aligned up to the boundary. The effects of  vorticity alignment on the $L^q$, $1<q<\infty$ norm of the vorticity will also be discussed.

We develop a structure  theory for del Pezzo surfaces that are regular but geometrically non-normal, based on work of Reid, but now independence on the p-degree of the ground field. This leads to existence results, as well as non-existence results for ground fields  of p-degree one. In turn, we  settle questions arising from Koll'ar's analysis on the structure of Mori fiber spaces in dimension three. This is joint work with Andrea Fanelli.

Low-rank plus sparse matrices arise in many data-oriented applications, most notably in a foreground-background separation from a moving camera. It is known that low-rank matrix recovery from a few entries (low-rank matrix completion) requires low coherence (Candes et al 2009) as in the extreme cases when the low-rank matrix is also sparse, where matrix completion can miss information and be unrecoverable. However, the requirement of low coherence does not suffice in the low-rank plus sparse model, as the set of low-rank plus sparse matrices is not closed. We will discuss the relation of non-closedness of the low-rank plus sparse model to the notion of matrix rigidity function in complexity theory.

Let $F$ be a non-Archimedean local field with ring of integers $\mathcal O$ and maximal ideal $\mathfrak p$. T. Shintani and G. Hill independently introduced a large class of smooth representations of $GL_N(\mathcal O)$, called regular representations. Roughly speaking they correspond to elements in the Lie algebra $M_N(\mathcal O)$ which are regular mod $\mathfrak p$ (i.e, having centraliser of dimension $N$). The study of regular representations of $GL_N(\mathcal O)$ goes back to Shintani in the 1960s, and independently and later, Hill, who both constructed the regular representations with even conductor, but left the much harder case of odd conductor open. In recent simultaneous and independent work, Krakovski, Onn and Singla gave a construction of the regular representations of $GL_N(\mathcal O)$ when the residue characteristic of $\mathcal O$ is not $2$.

In this talk I will present a complete construction of all the regular representations of $GL_N(\mathcal O)$. The approach is analogous to, and motivated by, the construction of supercuspidal representations of $GL_N(F)$ due to Bushnell and Kutzko. This is joint work with Shaun Stevens.
 

The object of this talk is a class of generalised Newtonian fluids with implicit constitutive law.
Both in the steady and the unsteady case, existence of weak solutions was proven by Bul\'\i{}\v{c}ek et al. (2009, 2012) and the main challenge is the small growth exponent qq and the implicit law.
I will discuss the application of a splitting and regularising strategy to show convergence of FEM approximations to weak solutions of the flow. 
In the steady case this allows to cover the full range of growth exponents and thus generalises existing work of Diening et al. (2013). If time permits, I will also address the unsteady case.
This is joint work with Endre Suli.

In this talk, I will present the construction of a family of solutions to
vacuum Einstein equations which consist of an arbitrary number of high
frequency waves travelling in different directions. In the high frequency
limit, our family of solutions converges to a solution of Einstein equations
coupled to null dusts. This construction is an illustration of the so called
backreaction, studied by physicists (Isaacson, Burnet, Green, Wald...) : the
small scale inhomogeneities have an effect on the large scale dynamics in
the form of an energy impulsion tensor in the right-hand side of Einstein
equations. This is a joint work with Jonathan Luk (Stanford).

Protein interaction networks (PINs) allow the representation and analysis of biological processes in cells. Because cells are dynamic and adaptive, these processes change over time. Thus far, research has focused either on the static PIN analysis or the temporal nature of gene expression. By analysing temporal PINs using multilayer networks, we want to link these efforts. The analysis of temporal PINs gives insights into how proteins, individually and in their entirety, change their biological functions. We present a general procedure that integrates temporal gene expression information with a monolayer PIN to a temporal PIN and allows the detection of modular structure using multilayer modularity maximisation.