Spacetimes formed from the cartesian product of Minkowski space and a flat torus play an important role as toy models for theories of supergravity and string theory. In this talk I will discuss an upcoming work with Huneau and Stingo showing the nonlinear stability of such a Kaluza-Klein spacetime. The result is also connected to a claim of Witten.
Despite their straightforward definition, the homeomorphism groups of surfaces are far from straightforward. Basic algebraic and dynamical problems are wide open for these groups, which is a far cry from the closely related and much better understood mapping class groups of surfaces. With Jonathan Bowden and Sebastian Hensel, we introduced the fine curve graph as a tool to study homeomorphism groups. Like its mapping class group counterpart, it is Gromov hyperbolic, and can shed light on algebraic properties such as scl, via geometric group theoretic techniques. This brings us to the enticing question of how much of Thurston's theory (e.g. Nielsen--Thurston classification, invariant foliations, etc.) for mapping class groups carries over to the homeomorphism groups. We will describe new phenomena which are not encountered in the mapping class group setting, and meet some new connections with topological dynamics, which is joint work with Bowden, Hensel, Kathryn Mann and Emmanuel Militon. I will survey what's known, describe some of the new and interesting problems that arise with this theory, and give an idea of what's next.
It was recently shown by Aidekon and Da Silva how to construct a growth fragmentation process from a planar Brownian excursion. I will explain how this same growth fragmentation process arises in another setting: when one decorates a certain “critical Liouville quantum gravity random surface” with a conformal loop ensemble of parameter 4. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun.
We consider the varifolds associated to phase transitions whose first variation of Allen-Cahn energy is $L^p$ integrable with respect to the energy measure. We can see that the Dirichlet and potential part of the energy are almost equidistributed. After passing to the phase field limit, one can obtain an integer rectifiable varifold with bounded $L^p$ mean curvature. This is joint work with Huy Nguyen.
The data revolution has changed the landscape of computational mathematics and has increased the demand for new numerical linear algebra tools to handle the vast amount of data. One crucial task is data compression to capture the inherent structure of data efficiently. Tensor-based approaches have gained significant traction in this setting by exploiting multilinear relationships in multiway data. In this talk, we will describe a matrix-mimetic tensor algebra that offers provably optimal compressed representations of high-dimensional data. We will compare this tensor-algebraic approach to other popular tensor decomposition techniques and show that our approach offers both theoretical and numerical advantages.
I will present results from arXiv:2202.13924, where we studied the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. The notion of complexity of interest to us will be Nielsen’s complexity applied to the time-dependent evolution operator of the quantum systems. I will review Nielsen’s complexity, discuss the difficulties associated with this definition and introduce a simplified approach which appears to retain non-trivial information about the integrable properties of the dynamical systems.
Let F be a p-adic local field and let G be a connected split reductive group over F. Let H be the pro-p Iwahori-Hecke algebra of the p-adic group G(F), with coefficients in an algebraically closed field k of characteristic p. The module theory over H (or a certain derived version thereof) is of considerable interest in the so-called mod p local Langlands program for G(F), whose aim is to relate the smooth modular representation theory of G(F) to modular representations of the absolute Galois group of F. In this talk, we explain a possible construction of a certain moduli space for those Galois representations into the Langlands dual group of G over k which are semisimple. We then relate this space to the geometry of H. This is a work in progress with Cédric Pépin.
Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude. (Joint work with Arnur Nigmetov.)
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.
Chair: Ian Hewitt (Associate HoD (People))
Panel:
James Sparks (Head of Department)
Helen Byrne (winner of MPLS Outstanding Supervisor Awards for 2022)
Ali Goodall (Head of Faculty Services and HR)
Matija Tapuskovic (EPSRC Postdoctoral Research Fellow and JRF at Corpus Christi)
Scientific discussions with colleagues, at conferences and seminars, during supervisions and collaborations, are a crucial part of our research process. How can we ensure our academic discussions are fruitful, respectful, and a positive experience for everyone involved? What factors and power dynamics can impact our conversations? How can we make sure everyone’s voice is heard and respected? This panel discussion will probe these questions and encourage us all to reflect on how we approach our academic discussions.
The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Cancer cells can arise from any type of cell in the body; cancers can grow in or around any tissue or organ making the disease highly complex. My research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modelling. In this talk I shall present a 3D individual-based force-based model for tumour growth and development in which we simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent is fully realised, for example, cells are described as viscoelastic sphere with radius and centre given within the off-lattice model. Interactions are primarily governed by mechanical forces between elements. However, as well as he mechanical interactions we also consider chemical interactions, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells, as well as intercellular aspects such as cell phenotypes.
Quantum toroidal algebras $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ are certain Drinfeld quantum affinizations of quantum groups associated to affine Lie algebras, and can therefore be thought of as `double affine quantum groups'.
In particular, they contain (and are generated by) a horizontal and vertical copy of the affine quantum group.
Utilising an extended double affine braid group action, Miki obtained in type $A$ an automorphism of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ which exchanges these subalgebras. This has since played a crucial role in the investigation of its structure and representation theory.
In this talk I shall present my recent work -- which extends the braid group action to all types and generalises Miki's automorphism to the ADE case -- as well as potential directions for future work in this area.
Given a finite set A of integer lattice points in d dimensions, let NA denote the N-fold iterated sumset (i.e. the set comprising all sums of N elements from A). In 1992 Khovanskii observed that there is a fixed polynomial P(N), depending on A, such that the size of the sumset NA equals P(N) exactly (once N is sufficiently large, that is). In addition to this 'size stability', there is a related 'structural stability' property for the sumset NA, which Granville and Shakan recently showed also holds for sufficiently large N. But what does 'sufficiently large' mean in practice? In this talk I will discuss some perspectives on these questions, and explain joint work with Granville and Shakan which proves the first explicit bounds for all sets A. I will also discuss current work with Granville, which gives a tight bound 'up to logarithmic factors' for one of these properties.
Skeletal joints are often modelled as two adjacent layers of poroviscoelastic cartilage that are permitted to slide past each other. The talk will begin by outlining a mathematical model that may be used, focusing on two unusual features of the model: (i) the solid component of the poroviscoelastic body has a charged surface that ionises the fluid within the pores, generating a swelling pressure; and (ii) appropriate conditions are required at the interface between the two adjacent layers of cartilage. The remainder of the talk will then address various theoretical and practical issues in computing a finite element solution of the governing equations.
We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modeled by an interaction potential. Without the diffusion term, we consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution’s support. With the diffusion term, the attractive interaction and the diffusion compete. We provide the conditions of the attractive interaction for each part to win.
By definition, planktonic organisms drift with the water flows. But these millimetric organisms are not necessarily passive; many can swim and can sense the surrounding flow through mechanosensory hairs. But how useful can be flow sensing in a turbulent environment? To address this question, we show two examples of smart planktonic behavior: (1) we develop a model where plantkters choose a swimming direction as a function of the velocity gradient to "surf on turbulence" and move efficiently upwards; (2) we show how a plankter measuring the velocity gradient may track the position of a swimming target from its hydrodynamic signature.
Ernst Haeckel, Kunstformen der Natur (1904), Copepoda
Christophe is Professor of Fluid Mechanics at Centrale Marseille. His research activity is carried out at the IRPHE institute in Marseille.
'His research addresses various fundamental problems of fluid and solid mechanics, including fluid-structure interactions, hydrodynamic instabilities, animal locomotion, aeroelasticity, rotating flows, and plant biomechanics. It generally involves a combination of analytical modeling, experiments, and numerical work.' (Taken from his website here: https://www.irphe.fr/~eloy/).'
Given a mathematical object, what can you compute about it? In some settings, you cannot say very much. Given an arbitrary group presentation, for example, there is no procedure to decide whether the group is trivial. In 3-manifolds, however, algorithms are a fruitful and active area of study (and some of them are even implementable!). One of the main tools in this area is normal surface theory, which allows us to describe interesting surfaces in a 3-manifold with respect to an arbitrary triangulation. I will discuss some results in this area, particularly around Seifert fibered spaces.
We will discuss the pros and cons of targets vs quotas in increasing diversity in Mathematics.
By Jewett-Krieger theorems minimal dynamical systems on the Cantor set are topological analogous of ergodic systems on probability Lebesgue spaces. In this analogy and to study a Cantor minimal system, indicator functions of clopen sets (as continuous integer or real valued functions) are considered while they are mod out by the subgroup of all co-boundary functions. That is how dimension group which is an operator algebraic object appears in dynamical systems. In this talk, I try to explain a bit more about dimension groups from dynamical point of view and how it relates to topological factoring and spectrum of Cantor minimal systems.
Bounded cohomology is a functional-analytic analogue of ordinary cohomology that has become a fundamental tool in many fields, from rigidity theory to the geometry of manifolds. However it is infamously hard of compute, and the lack of very basic examples makes the overall picture still hard to grasp. I will report on recent progress in this direction, and draw attention to some natural questions that remain open.
Random greedy algorithms became ubiquitous in Combinatorics after Rödl's nibble (semi-random method), which was repeatedly refined for various applications, such as iterative graph colouring algorithms (Molloy-Reed) and lower bounds for the Ramsey number $R(3,t)$ via the triangle-free process (Bohman-Keevash / Fiz Pontiveros-Griffiths-Morris). More recently, when combined with absorption, they have played a key role in many existence and approximate counting results for combinatorial structures, following a paradigm established by my proofs of the Existence of Designs and Wilson's Conjecture on the number of Steiner Triple Systems. Here absorption (converting approximate solutions to exact solutions) is generally the most challenging task, which has spurred the development of many new ideas, including my Randomised Algebraic Construction method, the Kühn-Osthus Iterative Absorption method and Montgomery's Addition Structures (for attacking the Ryser-Brualdi-Stein Conjecture). The design and analysis of a suitable guiding mechanism for the random process can also come with major challenges, such as in the recent proof of Erdős' Conjecture on Steiner Triple Systems of high girth (Kwan-Sah-Sawhney-Simkin). This talk will survey some of this background and also mention some recent results on the Queens Problem (Bowtell-Keevash / Luria-Simkin / Simkin) and the Existence of Subspace Designs (Keevash-Sah-Sawhney). I may also mention recent solutions of the Talagrand / Kahn-Kalai Threshold Conjectures (Frankston-Kahn-Narayanan-Park / Park-Pham) and thresholds for Steiner Triple Systems / Latin Squares (Keevash / Jain-Pham), where the key to my proof is constructing a suitable spread measure via a guided random process.
Perverse sheaves are an indispensable tool in representation theory. Their stalks often encode important representation theoretic information such as composition multiplicities or canonical bases. For the nilpotent cone, there is an algorithm that computes these stalks, known as the Lusztig-Shoji algorithm. In this talk, we discuss how this algorithm can be modified to compute stalks of perverse sheaves on more general varieties. As an application, we obtain a new algorithm for computing canonical bases in certain quantum groups as well as composition multiplicities for standard modules of the affine Hecke algebra of $\mathrm{GL}_n$.
In mathematical modelling, data and solutions are often represented as measurable functions, and their quality is being captured by their membership to a certain function space. One of the core questions arising in applications of this approach is the comparison of the quality of the data and that of the solution. A particular attention is being paid to optimality of the results obtained. A delicate choice of scales of suitable function spaces is required in order to balance the expressivity (the ability to capture fine mathematical properties of the model) and the accessibility (the level of its technical difficulty) for a practical use. We will give an overview of the research area which grew out of these questions and survey recent results obtained in this direction as well as challenging open questions. We will describe a development of a powerful method based on the so-called reduction principles and demonstrate its use on specific problems including the continuity of Sobolev embeddings or boundedness of pivotal integral operators such as the Hardy - Littlewood maximal operator and the Laplace transform.
We ask to what extend the SL(2,C)-character variety of the
fundamental group of the complement of a knot in S^3 determines the
knot. Our methods use results from group theory, classical 3-manifold
topology, but also geometric input in two ways: the geometrisation
theorem for 3-manifolds, and instanton gauge theory. In particular this
is connected to SU(2)-character varieties of two-component links, a
topic where much less is known than in the case of knots. This is joint
work with Michel Boileau, Teruaki Kitano, and Steven Sivek.
We consider an investor who is dynamically informed about the future evolution of one of the independent Brownian motions driving a stock's price fluctuations. The resulting rough semimartingale dynamics allow for strong arbitrage, but with linear temporary price impact the resulting optimal investment problem with exponential utility turns out to be well posed. The dynamically revealed Brownian path segment makes the problem infinite-dimensional, but by considering its convex-analytic dual problem, we show that it still can be solved explicitly and we give some financial-economic insights into the optimal investment strategy and the properties of maximum expected utility. (This is joint work with Yan Dolinsky, Hebrew University of Jerusalem).
It is well understood by works of Fukaya and Teleman that the Fukaya category of a symplectic reduction at a regular value of the moment map can be computed before taking the quotient as an equivariant Fukaya category. Informed by mirror calculations, we will give a new geometric interpretation of the equivariant Fukaya category corresponding to a singular value of the moment map where the equivariance is traded with wrapping.
Joint work in progress with Ed Segal.
DeFi: Data-Driven Characterisation of Uniswap v3 Ecosystem & an Ideal Crypto Law for Liquidity Pools
Uniswap is a Constant Product Market Maker built around liquidity pools, where pairs of tokens are exchanged subject to a fee that is proportional to the size of transactions. At the time of writing, there exist more than 6,000 pools associated with Uniswap v3, implying that empirical investigations on the full ecosystem can easily become computationally expensive. Thus, we propose a systematic workflow to extract and analyse a meaningful but computationally tractable sub-universe of liquidity pools.
Leveraging on the 34 pools found relevant for the six-months time window January-June 2022, we then investigate the related liquidity consumption behaviour of market participants. We propose to represent each liquidity taker by a suitably constructed transaction graph, which is a fully connected network where nodes are the liquidity taker’s executed transactions, and edges contain weights encoding the time elapsed between any two transactions. We extend the NLP-inspired graph2vec algorithm to the weighted undirected setting, and employ it to obtain an embedding of the set of graphs. This embedding allows us to extract seven clusters of liquidity takers, with equivalent behavioural patters and interpretable trading preferences.
We conclude our work by testing for relationships between the characteristic mechanisms of each pool, i.e. liquidity provision, consumption, and price variation. We introduce a related ideal crypto law, inspired from the ideal gas law of thermodynamics, and demonstrate that pools adhering to this law are healthier trading venues in terms of sensitivity of liquidity and agents’ activity. Regulators and practitioners could benefit from our model by developing related pool health monitoring tools.
A Random Neural Network Approach to Pricing SPDEs for Rough Volatility
We propose a novel machine learning-based scheme for solving partial differential equations (PDEs) and backward stochastic partial differential equations (BSPDE) stemming from option pricing equations of Markovian and non-Markovian models respectively. The use of the so-called random weighted neural networks (RWNN) allows us to formulate the optimisation problem as linear regression, thus immensely speeding up the training process. Furthermore, we analyse the convergence of the RWNN scheme and are able to specify error estimates in terms of the number of hidden nodes. The performance of the scheme is tested on Black-Scholes and rBergomi models and shown to have superior training times with accuracy comparable to existing deep learning approaches.
A 2d SCFT given as a non-linear sigma model of a Ricci-flat Kahler target
space is not a rational CFT in general; only special points in the moduli
space of the target-space metric, the 2d SCFTs are rational.
Gukov-Vafa's paper in 2002 hinted at a possibility that such special points
may be characterized by the property "complex multiplication" of the target space,
which has its origin in number theory. We revisit the idea, refine the Conjecture,
and prove it in the case the target space is T^4.
This presentation is based on arXiv:2205.10299 and 2212.13028 .
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.
Speaker: Dr Aleksander Horawa (North Wing)
Title: Bitcoin, elliptic curves, and this building
Abstract:
We will discuss two motivations to work on Algebraic Number Theory: applications to cryptography, and fame and fortune. For the first, we will explain how Bitcoin and other companies use Elliptic Curves to digitally sign messages. For the latter, we will introduce two famous problems in Number Theory: Fermat's Last Theorem, worth a name on this building, and the Birch Swinnerton--Dyer conjecture, worth $1,000,000 according to some people in this building (Clay Mathematics Institute).
Speaker: Dr Jemima Tabeart (South Wing)
Title: Numerical linear algebra for weather forecasting
Abstract:
The quality of a weather forecast is strongly determined by the accuracy of the initial condition. Data assimilation methods allow us to combine prior forecast information with new measurements in order to obtain the best estimate of the true initial condition. However, many of these approaches require the solution an enormous least-squares problem. In this talk I will discuss some mathematical and computational challenges associated with data assimilation for numerical weather prediction, and show how structure-exploiting numerical linear algebra approaches can lead to theoretical and computational improvements.
Simulating diffusion computationally allows to predict the diffusivity of materials, understand diffusion mechanisms, and to tailor-make materials such as solid-state electrolytes with desired properties aiming at developing new batteries. By studying the geometry and topology of 3-periodic scalar fields (e.g. the potential of ions in the electrolyte), we develop a cost-efficient multi-scale model for diffusion in crystalline materials. This project is a typical example of a collaboration in the overlap of topology and materials science that started as a persistent homology project and turned into something else.
Nanopore sequencing is a method to infer the sequence of nucleotides in DNA or RNA molecules from small variations in ionic current during transit through a nanoscale pore. We will give an introduction to nanopore sequencing and some of its applications and then explore simple models of the signal generation process. These can provide insight to guide optimisation of the system and inform the design of more flexible neural network models, capable of extracting the rich contextual information required for accurate sequence inference.
The crepant resolutions of a singular threefold are related by a finite sequence of birational maps called flops. In the simplest cases, this network of flops is governed by simple combinatorics. I will begin the talk with an overview of flops and crepant resolutions. I will then move on to explain how their underlying combinatorial structure can be abstracted to define the notion of a cluster category.
Toward a characterization of modularity using shatter functions, we show that an o-minimal expansion of the real ordered additive group $(\mathbb{R}; 0, +,<)$ does not define restricted multiplication if and only if the shatter function of every definable family is asymptotic to a polynomial. Our result implies that vc-density can only take integer values in $(\mathbb{R}; 0, +,<)$ confirming a special case of a conjecture by Chernikov. (Joint with Abdul Basit.)
Oxford Mathematics Public Lecture
Cascading Principles - Conrad Shawcross, Martin Bridson and James Sparks with Fatos Ustek
Thursday 23 February, 2023
5pm - 6.15pm Andrew Wiles Building, Mathematical Institute, Oxford
Cascading Principles is an exhibition of nearly 40 stunning, mathematically inspired sculptures which are living alongside the mathematicians that inspired them in the Andrew Wiles Building, home to Oxford Mathematics. In this 'lecture', chaired by exhibition curator Fatos Ustek, Conrad will talk about what motivates his work, and how the possibilities and uncertainties of science inform his art. In turn, mathematicians Martin Bridson and James Sparks will describe how a mathematician responds to art motivated by their subject.
There will be an opportunity to view the exhibition from 4pm on the day of the lecture.
Conrad Shawcross specialises in mechanical sculptures based on philosophical and scientific ideas. He is the youngest living member of the Royal Academy of Arts. James Sparks is Professor of Mathematical Physics and Head of the Mathematical Institute in Oxford. Martin Bridson is Whitehead Professor of Pure Mathematics in Oxford and President of the Clay Mathematics Institute. Fatos Ustek is a curator and writer and a leading voice in contemporary art.
Please email @email to register.
The Oxford Mathematics Public Lectures and the Conrad Shawcross Exhibition are generously supported by XTX Markets.
Assuming the Riemann Hypothesis, Soundararajan established almost sharp upper bounds for all positive moments of the Riemann zeta-function. This result was later improved by Harper, who proved upper bounds of the right order of magnitude. I will describe some of the ideas in their proofs, and then discuss recent joint work with Alexandra Florea, where we consider negative moments of the Riemann zeta-function. For example, we can obtain asymptotic formulas for negative moments when the shift in the zeta function is large enough, confirming a conjecture of Gonek. We also obtain an upper bound for the average of the generalised Mobius function.
In this talk, we first review the de Rham complex and the finite element exterior calculus, a cohomological framework for structure-preserving discretisation of PDEs. From de Rham complexes, we derive other complexes with applications in elasticity, geometry and general relativity. The derivation, inspired by the Bernstein-Gelfand-Gelfand (BGG) construction, also provides a general machinery to establish results for tensor-valued problems (e.g., elasticity) from de Rham complexes (e.g., electromagnetism and fluid mechanics). We discuss some applications and progress in this direction, including mechanics models and the construction of bounded homotopy operators (Poincaré integrals) and finite elements.
In this presentation, I will discuss the construction and results of numerical simulations quantifying flows around several species of soft corals. In the first project, the flows near the tentacles of xeniid soft corals are quantified for the first time. Their active pulsations are thought to enhance their symbionts' photosynthetic rates by up to an order of magnitude. These polyps are approximately 1 cm in diameter and pulse at frequencies between approximately 0.5 and 1 Hz. As a result, the frequency-based Reynolds number calculated using the tentacle length and pulse frequency is on the order of 10 and rapidly decays as with distance from the polyp. This introduces the question of how these corals minimize the reversibility of the flow and bring in new volumes of fluid during each pulse. We estimate the Péclet number of the bulk flow generated by the coral as being on the order of 100–1000 whereas the flow between the bristles of the tentacles is on the order of 10. This illustrates the importance of advective transport in removing oxygen waste. In the second project, the flows through the elaborate branching structures of gorgonian colonies are considered. As water moves through the elaborate branches, it is slowed, and recirculation zones can form downstream of the colony. At the smaller scale, individual polyps that emerge from the branches expand their tentacles, further slowing the flow. At the smallest scale, the tentacles are covered in tiny pinnules where exchange occurs. We quantified the gap to diameter ratios for various gorgonians at the scale of the branches, the polyp tentacles and the pinnules. We then used computational fluid dynamics to determine the flow patterns at all three levels of branching. We quantified the leakiness between the branches, tentacles and pinnules over the biologically relevant range of Reynolds numbers and gap-to-diameter ratios, and found that the branches and tentacles can act as either leaky rakes or solid plates depending upon these dimensionless parameters. The pinnules, in contrast, mostly impede the flow. Using an agent-based modeling framework, we quantified plankton capture as a function of the gap-to diameter ratio of the branches and the Reynolds number. We found that the capture rate depends critically on both morphology and Reynolds number.
Laura Miller is Professor of Mathematics. Her research group, 'investigate[s] changes in the fluid dynamic environment of organisms as they grow or shrink in size over evolutionary or developmental time.' (Taken from her group website here: https://sites.google.com/site/swimflypump/home?authuser=0)
The Lott-Sturm-Villani curvature-dimension condition CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It was proved by Juillet that the CD(K,N) condition is not satisfied in a large class of sub-Riemannian manifolds, for every choice of the parameters K and N. In a joint work with Tommaso Rossi, we extended this result to the setting of almost-Riemannian manifolds and finally it was proved in full generality by Rizzi and Stefani. In this talk I present the ideas behind the different strategies, discussing in particular their possible adaptation to the sub-Finsler setting. Lastly I show how studying the validity of the CD condition in sub-Finsler Carnot groups could help in proving rectifiability of CD spaces.
Mike will briefly describe the scope and shape of science within the Met Office and of his career there. He will also outline the coordination of the development of the NEMO ocean model, which he leads, and work to ensure the marine systems at the Met Office work efficiently on modern High Performance Computers (HPCs). In the second half of the talk, Mike will focus on two of his current scientific interests: accurate calculation of horizontal pressure forces in models with steeply sloping coordinates; and dynamical interpretations of meridional overturning circulations and ocean heat uptake.
Mathematicians frequently present their own work in a diachronic fashion, e.g. by comparing their "modern" methods to those supposedly of the "Ancients," or by situating their latest theories as an "abstract" counterpart to more "classical" ones. The construction of such contrasts entangle mathematical labour and cultural life writ large. Indeed, it involves on the part of mathematicians the shaping up of correspondences between their technical achievements and intellectual discussions taking place on a much broader stage, such as those surrounding the concept of modernity, its relation to an imagined ancient past, or the characterisation of scientific progress as an increase in abstraction. This talk will explore the creation and use of such mathematical diachronies, the focus being on the works of Felix Klein, Hieronymus Zeuthen, and Hermann Schubert.
Stable commutator length (scl) is a measure of homological complexity in groups that has attracted attention for its various connections with geometric topology and group theory. In this talk, I will introduce scl and discuss the (hard) problem of computing scl in surface groups. I will present some results concerning isometric embeddings of free groups for scl, and how they generalise to surface groups for the relative Gromov seminorm.
The joint spectral radius of a finite family S of matrices measures the rate of exponential growth of the maximal norm of an element from the product set S^n as n grows. This notion was introduced by Rota and Strang in the 60s. It arises naturally in a number of contexts in pure and applied mathematics. I will discuss its basic properties and focus on a formula of Berger and Wang and results of J. Bochi that extend to several matrices the classical for formula of Gelfand that relates the growth rate of the powers of a single matrix to its spectral radius. I give new proofs and derive explicit estimates with polynomial dependence on the dimension, refining these results. If time permits I will also discuss connections with the Tits alternative, the notion of joint spectrum, and a geometric version of these results regarding groups acting on non-positively curved spaces.
In 1977, Milnor formulated the following conjecture: every discrete group of affine transformations acting properly on the affine space is virtually solvable. We now know that this statement is false; the current goal is to gain a better understanding of the counterexamples to this conjecture. Every group that violates this conjecture "lives" in a certain algebraic affine group, which can be specified by giving a linear group and a representation thereof. Representations that give rise to counterexamples are said to be non-Milnor. We will talk about the progress made so far towards classification of these non-Milnor representations.
At the heart of all quasi-Newton methods is an update rule that enables us to gradually improve the Hessian approximation using the already available gradient evaluations. Theoretical results show that the global performance of optimization algorithms can be improved with higher-order derivatives. This motivates an investigation of generalizations of quasi-Newton update rules to obtain for example third derivatives (which are tensors) from Hessian evaluations. Our generalization is based on the observation that quasi-Newton updates are least-change updates satisfying the secant equation, with different methods using different norms to measure the size of the change. We present a full characterization for least-change updates in weighted Frobenius norms (satisfying an analogue of the secant equation) for derivatives of arbitrary order. Moreover, we establish convergence of the approximations to the true derivative under standard assumptions and explore the quality of the generated approximations in numerical experiments.
A regular bipartite tournament is an orientation of a complete balanced bipartite graph $K_{2n,2n}$ where every vertex has its in- and outdegree both equal to $n$. In 1981, Jackson conjectured that any regular bipartite tournament can be decomposed into Hamilton cycles. We prove this conjecture for sufficiently large $n$. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs.