We will see some results and conjectures on the zeta and multizeta values in the function field context, and see how they relate to homological-homotopical objects, such as t-motives, iterated extensions, and to Hopf algebras, big Galois representations.
In this talk, I will present an incomplete equilibrium model to determine the price of an annuity. A finite number of agents receive stochastic income streams and choose between consumption and investment in the traded annuity. The novelty of this model is its ability to handle running consumption and general income streams. In particular, the model incorporates mean reverting income, which is empirically relevant but historically too intractable in equilibrium. The model is set in a Brownian framework, and equilibrium is characterized and proven to exist using a system of fully coupled quadratic BSDEs. This work is joint with Gordan Zitkovic.
Conical symplectic resolutions are one of the main objects in the contemporary mix of algebraic geometry and representation theory,
known as geometric representation theory. They cover many interesting families of objects such as quiver varieties and hypertoric
varieties, and some simpler such as Springer resolutions. The last findings [Braverman, Finkelberg, Nakajima] say that they arise
as Higgs/Coulomb moduli spaces, coming from physics. Most of the gadgets attached to conical symplectic resolutions are rather
algebraic, such as their quatizations and $\mathcal{O}$-categories. We are rather interested in the symplectic topology of them, in particular
finding smooth exact Lagrangians that appear in the central fiber of the (defining) resolution, as they are objects of the Fukaya category.
Many types of patterns emerging spontaneously can be observed in systems involving thin elastic plates and subjected to external or internal stresses (compression, differential growth, shearing, tearing, etc.). These mechanical systems can sometime be seen as model systems for more complex natural systems and allow to study in detail elementary emerging patterns. One of the simplest among such systems is a bilayer composed of a thin plate resting on a thick deformable substrate. Upon slight compression, periodic undulations (wrinkles) with a well-defined wavelength emerge at the level of the thin layer. We will show that, as the compression increases, this periodic state is unstable and that a second order transition to a localized state (fold) occurs when the substrate is a dense fluid.
Since the legendary 1972 encounter of H. Montgomery and F. Dyson at tea time in Princeton, a statistical correspondence of the non-trivial zeros of the Riemann Zeta function with eigenvalues of high-dimensional random matrices has emerged. Surrounded by many deep conjectures, there is a striking analogyto the energy levels of a quantum billiard system with chaotic dynamics. Thanks
to extensive calculation of Riemann zeros by A. Odlyzko, overwhelming numerical evidence has been found for the quantum analogy. The statistical accuracy provided by an enormous dataset of more than one billion zeros reveals distinctive finite size effects. Using the physical analogy, a precise prediction of these effects was recently accomplished through the numerical evaluation of operator determinants and their perturbation series (joint work with P. Forrester and A. Mays, Melbourne).
We will consider an extension of the Eisenberg-Noe model of financial contagion to allow for time dynamics in both discrete and continuous time. Mathematical results on existence and uniqueness of firm wealths under discrete and continuous-time will be provided. The financial implications of time dynamics will be considered, with focus on how the dynamic clearing solutions differ from those of the static Eisenberg-Noe model.
In this talk I will present the recent developments in the topic of existence of solutions to the two-fluid systems. I will discuss the application of approach developed by P.-L. Lions and E. Feireisl and explain the limitations of this technique in the context of multi-component flow models. A particular example of such a model is two-fluids Stokes system with single velocity field and two densities, and with an algebraic pressure law closure. The existence result that uses the compactness criterion introduced for the Navier-Stokes system by D. Bresch and P.-E. Jabin will be presented. I will also mention an innovative construction of solutions relying on the G. Crippa and C. DeLellis stability estimates for the transport equation.
Graph products are a class of groups that 'interpolate' between direct and free products, and generalise the notion of right-angled Artin groups. Given a property that free products (and maybe direct products) are known to satisfy, a natural question arises: do graph products satisfy this property? For instance, it is known that graph products act on tree-like spaces (quasi-trees) in a nice way (acylindrically), just like free products. In the talk we will discuss a construction of such an action and, if time permits, its relation to solving systems of equations over graph products.
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.