Abstract: The accurate estimation of the separation between two signals is at the core of many modern technologies. I will show a bunch of new inspired schemes able to estimate that separation at the quantum limit. They have been implemented in the spatial and in the time domain, achieving enormous improvements over the best classical methods. This opens the way to practical implementations of quantum metrological protocols.
Abstract: A bilevel optimization problem is a special class of optimization problem partly constrained by another optimization problem. The problem can be traced back to the habilitation thesis of German economist Heinrich von Stackelberg, which was completed in 1934. Hence, the problem is referred to by economists and game theorists as a Stackelberg game. The bilevel optimization problem, which clearly has a hierarchical structure with two levels of decision-making, respectively controlled by a leader and a follower, represents a powerful tool for modelling the interactions between various engineering, economic, and human systems. The problem was introduced in the field of optimization in the early 1970s and interest in the subject has grown exponentially since then, driven largely by the wide range of applications. Initial works on the optimization side were significantly influenced by developments in the closely related area of mathematical programs with equilibrium or complementarity constraints. Recently though, the more natural transformation known as the lower-level value function reformulation has also been at the center of attentions, and will represent the main focus of our analysis in this talk. In particular, we will discuss some recent attempts to design Newton-type methods for bilevel optimization problems, covering the theoretical framework for these types of methods and relevant challenges. We will conclude the talk with some promising numerical results. This talk is based on the papers [1, 2, 3].
References
[1] Fischer, A., Zemkoho, A.B. and Zhou, S., 2021. Semismooth Newton-type method for bilevel optimization: Global convergence and extensive numerical experiments. Optimization Methods and Software, doi:10.1080/10556788.2021.1977810
[2] Fliege, J., Tin, A. and Zemkoho, A., 2021. Gauss–Newton-type methods for bilevel optimization. Computational Optimization and Applications, 78(3), pp.793-824.
[3] Zemkoho, A.B. and Zhou, S., 2021. Theoretical and numerical comparison of the Karush–Kuhn–Tucker and value function reformulations in bilevel optimization. Computational Optimization and Applications, 78(2), pp.625-674.
Abstract: I will give an introduction to quantum computing for mathematicians and non quantum-tech experts. I aim to make clear links among quantum physics, mathematics, and computer sciences and expect that it leads to my recent research topics at the end.
Abstract:
Dynamical systems are used to model how quantities change over time. They have enormous numbers of applications across a wide variety of topics spanning physics, chemistry, biology, cosmology and celestial mechanics, social systems, and so on. In its simplest form, a dynamical system consists of a function that is applied repeatedly to some initial condition in order to produce a sequence of values that we call an orbit, representing how the quantity changes over time.
Some dynamical systems display a special type of behaviour called a period-doubling cascade, in which we observe cyclic behaviour whose period (i.e., the time taken to repeat) doubles repeatedly as we vary some experimental parameter. What is most striking and surprising is that certain features of these "doubling cascades" turn out to be universal; the same qualitative and even quantitative behaviour is observed across a wide variety of (apparently very different) systems.
An explanation for this universality rests on the existence of a fixed point to a "renormalisation operator" that acts on dynamical systems to simplify them (in the sense of halving the doubled periods).
We show how rigorous computer-assisted techniques can be used to prove the existence of such renormalisation fixed points, to bound the spectrum of the derivative of the operator at the fixed point, which helps to explain universal behaviour, and to bound the eigenfunctions and eigenvalues that determine the universal properties of systems with noise.
Our computations use multi-precision interval arithmetic with rigorous directed rounding modes to bound tightly the coefficients of the relevant power series and their high-order terms, and the corresponding universal constants.
Abstract: The study of nonlinear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi et al. I will introduce a family of such systems which consist of chains of N harmonically coupled particles with the nonlinearity introduced by confining the motion of each individual particle to a billiard with hard walls. Those chains exhibit a mixed phase space for any finite value of N for the simplest choice of the billiard (a one-dimensional box). This case is further related to the integrable sinh Gordon field theory. In this talk I will present the model and discuss its main thermodynamic properties.
Abstract: Sport, exercise and health science disciplines have benefitted tremendously from a range of research advances which have collectively facilitated the wellbeing and performance of elite athletes as well as the wider population. These scientific advances have benefitted from the introduction of new technologies, which have led to the availability of large data sets that sport and health organisations can utilise to further enhance wellbeing and performance. However, understanding of the most appropriate methods to analyse these data is still in its infancy in a number of disciplines. One particular research sub-discipline, which will be the focus of this presentation, is research that has studied the gaze patterns of people during different behavioural contexts. Firstly, there will be consideration of the research on sport contexts, including instances of visual anticipation – for example, what are the gaze characteristics of elite performers when returning a tennis serve or saving a penalty kick? Secondly, an overview of the literature on the broader domain of visually controlled locomotion will be provided. This work is of particular interest given evidence highlighting importance of gaze patterns during many day-to-day settings that appear critical to enhancing current understanding related to falls prevention interventions. It is hoped that the presentation will provide an opportunity to discuss potential opportunities for collaboration both related to the study of gaze patterns but also across other sport, health and exercise science domains.
Abstract: As parts of the Arctic Ocean become increasingly ice-free for parts of the year shipping of all kinds - for transport, tourism or resource extraction - are growing. This increases the chance of accidents and, in such a hostile environment, the risks of loss of life. Search and Rescue (SAR) facilities are therefore of growing importance. This work, which is part of a larger Ph.D. project on Goal Programming, demonstrates how a mixed-integer Extended Goal Programming model can be used to optimise the location of SAR operations throughout the region. It is capable of taking into account, among a number of other often conflicting factors, a mix of SAR methods and the complexities of cross-border cooperation. The results of a sensitivity analysis highlight the relative importance of some of these goals.
Abstract: Changing individuals' travel behaviours, particularly in a car-dependent region, is a challenge as travel behaviours are often habitual and require personalised alternatives to the current behaviours. Understanding the important factors influencing mode choice and travel behaviour play an essential role in providing transport alternatives. The Solent Future Transport Zone, a trial programme funded by the Department for Transport (DfT), has been awarded £28.8m to run numerous tests and trials of innovative approaches to transport across the Solent region to help make journeys easier, smarter and greener. This might be achieved through Solent's new Mobility as a Service (MaaS) app. MaaS encourages travel behaviour change towards the use of sustainable modes of transport as it provides personalised journey planning, booking, and payment on a single platform. This study presents a set of qualitative and qualitative approaches that can be used to collect the longitudinal travel data to analyse the travel behaviour change caused by using a MaaS app. These approaches enable scholars, practitioners, and policy-makers to measure and understand the rationale behind the travel behaviour change caused by a mobility app.
Abstract:
The dynamics of objects with internal structure is considerably more complex than, and qualitatively different from, that of point-like entities. The main culprit for the increased complexity is the coupling between orbital motion and internal dynamics.
This applies to big classical objects, such as two asteroids orbiting each other while travelling through space - a binary asteroid system - just as to tiny quantum entities, such as electrons. This might seem odd at first, since electrons lack any classically defined structure: as far as we can tell, they are point-like. They possess however a quantum structure, that is, a series of quantum numbers characterising their state. In the vacuum, this is limited to their intrinsic spin.
On the contrary, in a solid they generally acquire numerous emerging properties - in rough terms, they stop being naked electrons, and become dressed quasi-electrons. The coupling between their resulting internal pseudospin degrees of freedom and orbital motion can have critical consequences from zero up to room temperature, and in particular in Dirac systems, where orbit and pseudospin are "locked".
Abstract: We introduce the concept of maximal monotonicity of a point-to-set map from R^n to R^n. We then introduce maximal strong monotonicity of the point-to-set map which is a stronger concept than maximal monotonicity. An example of a maximal monotone operator is the subdifferential of convex function which is proper and lower semicontinuous, and if the function is furthermore strongly convex, then the subdifferential is maximal strongly monotone. With the concepts of maximal monotonicity and maximal strong monotonicity, we consider the problem to find zeros of the sum of two maximal monotone operators. Solving this problem has wide applicability, as the problem appears in many areas, such as in machine learning for example. Often in practice, the problem can be framed such that both operators are maximal strongly monotone. The relaxed Peaceman-Rachford splitting method is one of the known methods to solve the problem. We provide motivation through this solution method for the advantage to solve the problem when both operators are maximal strongly monotone. For the rest of the presentation, we then give theoretical results on the convergence and the convergence rate of the method to find the zero of the sum of two maximal strongly monotone operators for different values of the relaxation parameter.
Abstract: We give a brief introduction to current practices in cafe-style espresso extraction. Making the tastiest cup is considered an art rather than a science, however, processes can be made more systematic by better understanding the physical processes underlying brewing. We develop a physics-based mathematical model for the extraction process. Owing to the disparity in lengthscales between that of a grain within the "puck" and the depth of the whole "puck", the model can be systematically reduced via asymptotic homogenisation. The reduced model can then be solved by numerical methods. We will show that the model is able to reproduce some of the trends that are observed in experimental data and in practice. We also discuss possible strategies to make espresso more efficiently and more reproducibly.
Abstract
Starting from the Riccati equation and the second order element of the Riccati chain as the simplest examples of linearizable equations, by introducing a suitable change of variables, it is shown how the Schwarzian derivative represents a key tool in the construction of solutions of the Painleve XXV-Ermakov equation. Two families of Bäcklund transformations which link the linear and nonlinear equations under investigation are obtained. The talk is based on the joint work with S. Carillo, A. Chichurin and F. Zullo and is partially based on the arxiv preprint https://arxiv.org/abs/2201.02267.
Abstract: As the spatial diffusion of a human language is triggered by the contact among individuals and human populations, geographic distribution of dialects usually displays a strong spatial autocorrelation. In this seminar, I will present a mathematical model of Bayesian inference in order to analyze the spatial data of dialect-level variation of linguistic features recorded at multiple survey locations. To take into consideration the mutual influence of human groups, we consider a network, whose nodes represent populations, and model the spread of linguistic features among them. In addition, the model incorporates methods borrowed from the field of biological/linguistic phylogenetics and considers how the present variants of linguistic features have split and descended from the common ancestor. Posterior distributions yielded by this Bayesian model gives insight into the rate at which the focal linguistic feature changes into another variant, how often these features are transmitted among different populations, and the phylogenic relationship of linguistic features. As an example, I will introduce my ongoing project, in which I aim to reconstruct the phylogeny tree of accent systems of Japanese dialects by applying this model to the accent map over Japan’s mainland.
Abstract: How do you prevent an AI from taking over the world? In this talk, Dr. Nira Chamberlain discusses how mathematics is providing crucial answers. Mathematical modelling is the most creative side of applied mathematics which itself connects pure maths with science and technology. Mathematical models look into the real world, translate it into mathematics, solve those mathematics and then apply the solution back into the real world.
Looking through a mathematical model of the competition Dr. Nira will show how the same model can be used to investigate how to minimize the probability of an artificial intelligence takeover. The late great Professor Stephen Hawking once said: “The development of full artificial intelligence could spell the end of the human race.”
He also went on to state that he advocated research into precautionary measures to ensure future super-intelligent machines remain under human control. However, the AI apocalypse is not necessarily robots marching down the street, there are a number of examples subtler than this. So what is the risk of AI apocalypse and can we calculate this probability?
Abstract: Crew scheduling has been a significant part of the decision-making process on efficiently operating and maintaining the tasks in the offshore supply vessels due to crew costs being a significant proportion of operational costs. Additionally, the sophisticated requirements for crew members’ assignments on a global scale, the sudden changes, and uncertainties in the maritime settings make the assignment of the crew scheduling problem harder and these sudden changes cause higher costs. In this study, the allocation of crew members to the offshore supply vessels with recovery schedules was considered. A case study, crew scheduling in offshore supply vessels which is used for specific operations of a global scaled company in the oil and gas industry was analysed and two different (task-based and time-windows) MINLP models were proposed to formulate this problem. To solve the model efficiently, heuristic and decomposition methods were designed and implemented in C++. An extensive computational study was used to compare the performance efficiency between the proposed heuristic and the exact solution method by Xpress-Ive. Based on this computational study, the generation of more economic schedules within a short period of time by using the heuristic was observed.
Abstract: Squeezing is a well established resource to achieve quantum-enhanced sensitivity in linear optical networks [1, 2, 3]. Compared with approaches that exploit entanglement as quan- tum resource, these protocols show a number of advantages, such as feasibility of the probes employed and of the measurement performed, and robustness against decoherence. Despite the advantages, squeezing-based strategies found in recent literature still exhibit experimental challenges that require to be overcome in order to make the squeezing approach feasible in most experimental situation { e.g. strict limitations on the range of values of the unknown pa- rameter ' to be measured, which is usually assumed to be small, and on the form of the linear network which encodes it. Here we address these limitations in detail, while reviewing several protocols that manage to alleviate, if not solve, the impracticalities in different scenarios { e.g. estimation of a single distributed parameter [4, 5, 6], or of a function of multiple independent parameters [7]. In particular, we find that it is always possible to reach Heisenberg-scaling sensitivity $\delta \varphi = O(N^{-1})$, where $N$ is the average number of photons in the probe employed, for any given M-channel linear network which encodes in an arbitrary way a single unknown parameter or a function of unknown parameters. This is possible by employing only a single squeezed probe and homodyne measurements.
[1] Alex Monras. Optimal phase measurements with pure gaussian states. Phys. Rev. A, 73:033821.
[2] Teruo Matsubara, Paolo Facchi, Vittorio Giovannetti, and Kazuya Yuasa. Optimal gaussian metrology for generic multimode interferometric circuit. New Journal of Physics, 21(3):033014.
[3] Lorenzo Maccone and Alberto Riccardi. Squeezing metrology: a unified framework. Quantum, 4:292.
[4] Giovanni Gramegna, Danilo Triggiani, Paolo Facchi, Frank A. Narducci, and Vincenzo Tamma. Typicality of heisenberg scaling precision in multimode quantum metrology. Phys. Rev. Re- search, 3:013152.
[5] Giovanni Gramegna, Danilo Triggiani, Paolo Facchi, Frank A Narducci, and Vincenzo Tamma. Heisenberg scaling precision in multi-mode distributed quantum metrology. New Journal of Physics, 23(5):053002.
[6] Danilo Triggiani, Paolo Facchi, and Vincenzo Tamma. Non-adaptive heisenberg-limited metrol- ogy with multi-channel homodyne measurements, 2021. arXiv:2110.03582.
[7] Danilo Triggiani, Paolo Facchi, and Vincenzo Tamma. Heisenberg scaling precision in the estimation of functions of parameters, 2021. arXiv:2103.08564.
Abstract: Developing new data storage technologies that meet the demand at reduced cost and power consumption, have prompted the acceleration of research into solid-state memories, which are fast replacing magnetic hard discs in consumer electronics and portable devices. In this lecture I present the first experimental demonstration of a novel solid-state memory effect in anti-ferroelectric materials [1-3]. The initial experimental demonstration of the memory effect in anti-ferroelectric ceramics shows, remarkably, that the proposed technology encodes data in both ferroelectric sublattices of an anti-ferroelectric medium. This results in a 4-state non-volatile memory capable of storing two digital bits simultaneously, unlike all other volatile and non-volatile Random Access Memory (RAM) technologies that have 2-memory states and are capable of storing one digital bit per cell. On the basis of the newly discovered memory effect, a new RAM memory chip technology is proposed, which uses anti-ferroelectric capacitors to store digital information in four possible memory states. The non-volatile 4-state anti-ferroelectric random access memory was termed AFRAM by the author [1]. It will be shown that the AFRAM memory cell functions using a similar FRAM architecture with one transistor - one capacitor (1T-1C) per cell, it requires a more complex operation protocol, but it retains all the advantages of FRAM, while doubling the memory capacity in the same volume. The discovery of a 4-state memory effect in anti-ferroelectric materials opens new avenues for research and developments in less known anti-ferroelectric solids.
References:
[1]. M. Vopson, X. Tan, 4-state anti-ferroelectric random access memory, Electron Device Letters (2016).
[2]. M. Vopson, G. Caruntu, X. Tan, Polarization reversal and memory effect in anti-ferroelectric materials, Scripta Mater. 128, 61-64 (2017).
[3]. M. Vopson, X. Tan, Nonequilibrium polarization dynamics in antiferroelectrics, Physical Review B 96 (1), 014104 (2017).
Abstract: Error feedback (EF), also known as error compensation, is an immensely popular convergence stabilization mechanism in the context of distributed training of supervised machine learning models enhanced by the use of contractive communication compression mechanisms, such as Top-k. First proposed by Seide et al (2014) as a heuristic, EF resisted any theoretical understanding until recently [Stich et al., 2018, Alistarh et al., 2018]. However, all existing analyses either i) apply to the single node setting only, ii) rely on very strong and often unreasonable assumptions, such global boundedness of the gradients, or iterate-dependent assumptions that cannot be checked a-priori and may not hold in practice, or iii) circumvent these issues via the introduction of additional unbiased compressors, which increase the communication cost. In this work we fix all these deficiencies by proposing and analyzing a new EF mechanism, which we call EF21, which consistently and substantially outperforms EF in practice. Our theoretical analysis relies on standard assumptions only, works in the distributed heterogeneous data setting, and leads to better and more meaningful rates. In particular, we prove that EF21 enjoys a fast $O(1/T)$ convergence rate for smooth nonconvex problems, beating the previous bound of $O(1/T^{2/3})$, which was shown under a bounded gradients assumption. We further improve this to a fast linear rate for PL functions, which is the first linear convergence result for an EF-type method not relying on unbiased compressors. Since EF has a large number of applications where it reigns supreme, we believe that our 2021 variant, EF21, can have a large impact on the practice of communication efficient distributed learning.
Abstract: We consider a Hamiltonian system obtained by Takasaki [Tak], related by an algebraic transformation to the fourth Painlevé equation PIV under the so-called Painlevé-Calogero correspondence. The Hamiltonian in question is associated to the rank-one case of an extended Calogero system with rational potential [IM, Ino].
The algebraic transformation gives a map from this system to the Okamoto Hamiltonian form [Ok80] of PIV, which possesses the Painlevé property in the sense that all solutions are single-valued about movable singularities. This property is closely related to the existence of a space of initial conditions for the system, first constructed by Okamoto [Ok79], which is a bundle of open rational surfaces of which the flow of the system defines a uniform foliation. However, the algebraic map to the Okamoto system is not birational and the Takasaki Hamiltonian system does not enjoy the Painlevé property, so does not possess a space of initial conditions in the sense of Okamoto.
We show that it is still possible to associate a bundle of rational surfaces to the Takasaki Hamiltonian system on which it admits certain regularising transformations. This detects and recovers the algebraic transformation to the Okamoto system, and we discuss how this realises the surface from the Okamoto Hamiltonian as a quotient of that from the Takasaki system. Many properties of the Okamoto Hamiltonian system can be explained in terms of the geometry of the associated surfaces, so if time permits we will discuss what the surfaces we have constructed can tell us about the Takasaki system, beginning with symmetries.
This talk is based on joint work with Galina Filipuk (University of Warsaw).
[IM] V. I. Inozemtsev and D.V. Meshcheryakov, Extension of the class of integrable dynamical systems connected with semisimple Lie algebras, Lett. Math. Phys. 9 (1985), no. 1, 13–18.
[Ino] V. I. Inozemtsev, Lax representation with spectral parameter on a torus for integrable particle systems, Lett. Math. Phys. 17 (1989), no. 1, 11–17.
[Ok79] K. Okamoto, Sur les feuilletages associés aux équations du second ordre á points critiques fixes de P. Painlevé (French) [On foliations associated with second-order Painlevé equations with fixed critical points], Japan. J. Math. (N.S.) 5 (1979), no. 1, 1–79.
[Ok80] K.Okamoto, Polynomial Hamiltonians associated with Painlevé equations, I, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 264–268.
[Tak] K. Takasaki, Painlevé-Calogero correspondence revisited, J. Math. Phys. 42 (2001), no. 3, 1443–1473
Abstract: This talk is a guided tour of how language evolution can be studied using statistical physics. I will begin with the mating calls of the Puget Sound white-crowned sparrow, which exhibit distinctive spatially defined dialects. These calls may be viewed as spin states in the Ising model of magnetism, with death rate playing the role of thermodynamic temperature, and dialect zones analogous to magnetic domains, whose edges exhibit surface tension. I will then generalize to humans by defining a modified time dependent Ginzburg-Landau equation for speech variants. I will show how this equation, and modifications of it, can be used to predict language patterns in England, Italy, USA, Germany and the Saami region. I will conclude the first part of my tour by considering a linguistic analogue of Kimura's stepping-stone model of neutral genetic evolution, which is an alternative to the surface tension model, but belonging to a different universality class. I will present evidence against the genetic model of language, by exploiting the properties of the correlation functions of the two models. In the final part of the tour, I will investigate how the acoustic properties of individual speech sounds can be studied using physics. I will show that a popular theory of language acquisition, when applied to vowel sounds, may be formulated as a colloidal mixture of words in a box which coagulate to form distinctive vowel systems. Having seen the predictions of this model, we will think more carefully about how sounds have been traditionally embedded in space by linguists, and how convolutional neural networks can be used to perform this embedding in a more rigorous way.
Abstract: Metric spaces are easy objects to understand, they are just a set and a way of measuring a distance between the elements in the non-negative reals. This is closely linked to the idea of a topology, in fact topology is often taught as an abstraction of metric spaces. However, a key observation by Lawvere demonstrated that a metric space is nothing more than an enriched category, and so it is possible to have distance functions that take values in places other than the non-negative reals. Thus it makes sense to consider if this link to topology survives generalising to these other enriched categories. On the other hand, topological data analysis (TDA) involves applying topological techniques to point clouds which are themselves just metric spaces. From this point-of-view this becomes the problem of axiomatising a class of enriched categories for which the techniques of TDA are extended beyond the classical metric setting. These two questions form the sides of the same coin. By starting with classical definition of a metric space and taking careful steps to abstract this notion, we will define a class of quantales as a solution to these questions. In particular, we will examine a natural construction of the product of two classical metric spaces. This, perhaps surprisingly, will differ from a classical metric space in that the metric will take values in a specific quantale, rather than the non-negative reals. This construction is then equivalent to the topological construction of a product, but has constructively different potential.