In this talk, we shall introduce various identities among partitions of integers, and how these can be expressed via formal power series. In particular, we shall look at the Rogers Ramanujan identities of power series, and discuss possible combinatorial proofs using partitions and Durfree squares.
This is a survey on recent advances in classical decidability issues for local and global fields and for some canonical infinite extensions of those.
In this informal talk, I describe the kinds of functions relevant to scattering amplitudes in perturbative, four-dimensional quantum field theories. In particular, I will argue that generic amplitudes are non-polylogarithmic (beyond one loop), but that there is an upper bound to their geometric complexity. Moreover, I show a veritable `bestiary' of examples which saturate this bound in complexity---including three, all-loop families of integrals defined in massless $\phi^4$ theory which can, at best, be represented as dilogarithms integrated over (2L-2)-dimensional Calabi-Yau manifolds.
This work deals with exploiting the low-dimensional hierarchical structure of speech signals towards the goal of improving acoustic modelling using deep neural networks (DNN). To this aim the work employ tools from sparsity aware signal processing under novel frameworks to enrich the acoustic information present in DNN posterior features.
We will discuss some recent results with Martin Bridson about
Sidki's construction X(G). In particular, if G is a finitely presented
group then X(G) is a finitely presented group. We will discuss as well the
result that if G has polynomial isoperimetric function and the maximal
metabelian quotient of G is virtually nilpotent then X(G) has polynomial
isoperimetric function. Part of the arguments we will use have homological
nature.
Convex and in particular semidefinite relaxations (SDP) for non-convex continuous quadratic optimisation can provide tighter bounds than traditional linear relaxations. However, using SDP relaxations directly in Branch&Cut is impeded by lack of warm starting and inefficiency when combined with other cut classes, i.e. the reformulation-linearization technique. We present a general framework based on machine learning for a strong linear outer-approximation that can retain most tightness of such SDP relaxations, in the form of few strong low dimensional linear cuts selected offline. The cut selection complexity is taken offline by using a neural network estimator (trained before installing solver software) as a selection device for the strongest cuts. Lastly, we present results of our method on QP/QCQP problem instances.
Recovery of reaction pathways under the DCSC framework.
Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods, each of which highlights the properties of graphs in a unique way. I will discuss a novel approach to study graphs: the simplex geometry (a simplex is a generalized triangle). This perspective, proposed by Miroslav Fiedler, introduces techniques from (simplex) geometry into the field of graph theory and conversely, via an exact correspondence. We introduce the graph-simplex correspondence, identify a number of basic connections between graph characteristics and simplex properties, and suggest some applications as example.
Reference: https://arxiv.org/abs/1807.06475
The study of isolated systems has been vastly successful in the context of vanishing cosmological constant, Λ=0. However, there is no physically useful notion of asymptotics for the universe we inhabit with Λ>0. The full non-linear framework is still under development, but some interesting results at the linearized level have been obtained. I will focus on the conceptual subtleties that arise at the linearized level and discuss the quadrupole formula for gravitational radiation as well as some recent developments.
A particularly active area of research in modern algebraic number theory is the study of a class of numbers, called periods. In their simplest form, periods are integrals of rational functions over domains defined by rational in equations. They form a ring, which encompasses all algebraic numbers, logarithms thereof and \pi. They arise in the study of modular forms, cohomology and quantum field theory, and conjecturally have a sort of Galois theory.
We will take a whirlwind tour of these numbers, before discussing non-periods. In particular, we will sketch the construction of an explicit non-period, often forgotten about.
We analyse under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction, occurs. I will discuss several regimes that appear in aggregation diffusion problems with homogeneous kernels. I will first concentrate in the fair competition case distinguishing among porous medium like cases and fast diffusion like ones. I will discuss the main qualitative properties in terms of stationary states and minimizers of the free energies. In particular, all the porous medium cases are critical while the fast diffusion are not. In the second part, I will discuss the diffusion dominated case in which this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrisation techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as time tends to infinity. This talk is based on works in collaboration with S. Hittmeir, B. Volzone and Y. Yao and with V. Calvez and F. Hoffmann.
If k is a field of characteristic zero, a theorem of Lurie and Pridham establishes an equivalence between formal moduli problems and differential graded Lie algebras over k. We generalise this equivalence in two different ways to arbitrary ground fields by using “partition Lie algebras”. These mysterious new gadgets are intimately related to the genuine equivariant topology of the partition complex, which allows us to access the operations acting on their homotopy groups (relying on earlier work of Dyer-Lashof, Priddy, Goerss, and Arone-B.). This is joint work with Mathew.
Unlike standard bivariate diffusion models, the rough Bergomi model by Bayer, Friz, and Gatheral (2016) allows to parsimoniously recover key stylized facts of market implied volatility surfaces such as the exploding power-law behaviour of the at-the-money volatility skew as time to maturity goes to zero. However, falling into the class of so-called rough stochastic volatility models sparked by Alo`s, Leo ́n, and Vives (2007); Fukasawa (2011, 2017); Gatheral, Jaisson, and Rosenbaum (2018), its non-Markovianity poses serious mathematical and computational challenges. To date, calibrating rough Bergomi remained prohibitively expensive since standard calibration routines rely on the repetitive evaluation of the map from model parameters to Black-Scholes implied volatility, which in the case of rough Bergomi involves a costly Monte Carlo simulation (Bennedsen, Lunde, & Pakkanen, 2017; McCrickerd & Pakkanen, 2018; Bayer et al., 2016; Horvath, Jacquier, & Muguruza, 2017). In this paper, we resolve the issue by combining a standard Levenberg-Marquardt calibration routine with a neural network regression, replacing expensive MC simulations with cheap forward runs of a network trained to approximate the implied volatility map. Some numerical results show the prowess of this approach.
TBC
In this talk, we consider the question of exact (boundary) controllability of wave equations: whether one can steer their solutions from any initial state to any final state using appropriate boundary data. In particular, we discuss new and fully general results for linear wave equations on time-dependent domains with moving boundaries. We also discuss the novel geometric Carleman estimates that are the main tools for proving these controllability results
The meeting on Monday 15th October will be on Impostor Syndrome. In this meeting we will discuss what impostor syndrome is, what might be the causes of it, and some advice for people who are struggling with it. For anyone who wants to read up on what it is and some different types of impostor syndrome in advance, we recommend this blog post: https://www.themuse.com/advice/5-different-types-of-imposter-syndrome-a…. If you have a smart phone or tablet that you can bring with you, we encourage you to, as we will have some anonymous voting, and the more of you that can join in, the better!
We hope to see many of you there again: Quillen Room (N3.12), Monday 1-2pm.
Superstring scattering amplitudes in genus one have a low-energy expansion in terms of certain real analytic modular forms, called modular graph functions (D'Hoger, Green, Gürdogan and Vanhove). I will sketch the proof that these functions belong to a family of iterated integrals of modular forms (a generalization of Eichler integrals), recently introduced by Francis Brown, which explains many of their properties. The main tools are elliptic multiple polylogarithms (Brown and Levin), single-valued versions thereof, and elliptic multiple zeta values (Enriquez).
Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. Current techniques for gossiping are designed to deal with worst-case scenarios, which is irrelevant in applications to distributed statistical learning and denoising in sensor networks. We design second-order gossip methods tailor-made for the case where the real values are i.i.d. samples from the same distribution. In some regular network structures, we are able to prove optimality of our methods, and simulations suggest that they are efficient in a wide range of random networks. Our approach of gossip stems from a new acceleration framework using the family of orthogonal polynomials with respect to the spectral measure of the network graph (joint work with Raphaël Berthier, and Pierre Gaillard).
What should you expect in intercollegiate classes? What can you do to get the most out of them? In this session, experienced class tutors will share their thoughts, and a current student will offer tips and advice based on their experience.
All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. (Students who attended the Part C/OMMS induction event will find significant overlap between the advice offered there and this session!)
In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has only been proved for degrees d=1, 2, 3. For each d we prove the hyperbolicity of all but (perhaps) finitely many Jensen polynomials. We obtain a general theorem which models such polynomials by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function. This result can be thought of as a proof of GUE for the Riemann zeta function in derivative aspect. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.
In my talk I will briefly introduce model-free approach to mathematical finance, which uses Vovk's outer measure. Then, using pathwise BDG inequality obtained by Beigbloeck and Siorpaes and modification of Vovk's measure, I will present and prove a model-free version of this inequality for continuous price paths. Finally, I will discuss possible applications, like the existence and uniqueness of solutions of SDEs driven by continuous, model-free price paths. The talk will be based on the joint work with Farai Mhlanga and Lesiba Galane (University of Limpopo, South Africa)
The desire to “freely suspend the constituents of matter” in order to study their behavior can be traced back over 200 years to the diaries of Lichtenberg. From radio-frequency ion traps to optical tweezing of colloidal particles, existing methods to trap matter in free space or solution rely on the use of external fields that often strongly perturb the integrity of a macromolecule in solution. We recently introduced the ‘electrostatic fluidic trap’, an approach that exploits equilibrium thermodynamics to realise stable, non-destructive confinement of single macromolecules in room temperature fluids, and represents a paradigm shift in a nearly century-old field. The spatio-temporal dynamics of a single electrostatically trapped object reveals fundamental information on its properties, e.g., size and electrical charge. We have demonstrated the ability to measure the electrical charge of a single macromolecule in solution with a precision much better than a single elementary charge. Since the electrical charge of a macromolecule in solution is in turn a strong function of its 3D conformation, our approach enables for the first time precise, general measurements of the relationship between 3D structure and electrical charge of a single macromolecule, in real time. I will present our most recent advances in this emerging area of molecular measurement and show how such high-precision measurement at the nanoscale may be able to unveil the presence of previously unexpected phenomena in intermolecular interactions in solution.
The present work concerns the approximation of the solution map associated to the parametric Helmholtz boundary value problem, i.e., the map which associates to each (real) wavenumber belonging to a given interval of interest the corresponding solution of the Helmholtz equation. We introduce a single-point Least Squares (LS) rational Padé-type approximation technique applicable to any meromorphic Hilbert space-valued univariate map, and we prove the uniform convergence of the Padé approximation error on any compact subset of the interval of interest that excludes any pole. We also present a simplified and more efficient version, named Fast LS-Padé, applicable to Helmholtz-type parametric equations with normal operators.
The LS-Padé techniques are then employed to approximate the frequency response map associated to various parametric time-harmonic wave problems, namely, a transmission/reflection problem, a scattering problem and a problem in high-frequency regime. In all cases we establish the meromorphy of the frequency response map. The Helmholtz equation with stochastic wavenumber is also considered. In particular, for Lipschitz functionals of the solution, and their corresponding probability measures, we establish weak convergence of the measure derived from the LS-Padé approximant to the true one. Two-dimensioanl numerical tests are performed, which confirm the effectiveness of the approximation method.As of the dates
Joint work with: Francesca Bonizzoni and Ilaria Perugia (Uni. Vienna), Davide Pradovera (EPFL)
Novel machine learning techniques based on deep learning, i.e., the data-driven manipulation of neural networks, have reported remarkable results in many areas such as image classification, game intelligence, or speech recognition. Driven by these successes, many scholars have started using them in areas which do not focus on traditional machine learning tasks. For instance, more and more researchers are employing neural networks to develop tools for the discretisation and solution of partial differential equations. Two reasons can be identified to be the driving forces behind the increased interest in neural networks in the area of the numerical analysis of PDEs. On the one hand, powerful approximation theoretical results have been established which demonstrate that neural networks can represent functions from the most relevant function classes with a minimal number of parameters. On the other hand, highly efficient machine learning techniques for the training of these networks are now available and can be used as a black box. In this talk, we will give an overview of some approaches towards the numerical treatment of PDEs with neural networks and study the two aspects above. We will recall some classical and some novel approximation theoretical results and tie these results to PDE discretisation. Afterwards, providing a counterpoint, we analyse the structure of network spaces and deduce considerable problems for the black box solver. In particular, we will identify a number of structural properties of the set of neural networks that render optimisation over this set especially challenging and sometimes impossible. The talk is based on joint work with Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok, Felix Voigtlaender, and Mones Raslan