King's College London

TBA

We discuss timelike surfaces of finite size in general relativity and the initial boundary value problem. We consider obstructions with the standard Dirichlet problem, and conformal version with improved properties. The ensuing dynamical features are discussed with general cosmological constant.

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.

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. 

Let K be a number field and let L/K be an abelian extension. The genus field of L/K is the largest extension of L which is unramified at all places of L and abelian as an extension of K. The genus group is its Galois group over L, which is a quotient of the class group of L, and the genus number is the size of the genus group. We study the quantitative behaviour of genus numbers as one varies over abelian extensions L/K with fixed Galois group. We give an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time. This is joint work with Christopher Frei and Daniel Loughran.

Representations of finite reductive groups have a rich, well-understood structure, first explored by Deligne--Lusztig. In joint work with Anne-Marie Aubert and Dan Ciubotaru, we show a way to lift some of this structure to representations of p-adic groups. In particular, we consider the relation between Lusztig's nonabelian Fourier transform and a certain involution we define on the level of p-adic groups. This talk will be an introduction to these ideas with a focus on examples.

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

Liouville conformal field theory models two-dimensional gravity with a cosmological constant and conformal matter. In its timelike regime, it reproduces the characteristic negative kinetic term of the conformal factor of the metric in the Einstein-Hilbert action, the sign which infamously makes the gravity path integral ill-defined. In this talk, I will first discuss the perturbative computation of the timelike Liouville partition function around the sphere saddle and propose an all-orders result. I will then turn to the disk and present the bulk 1-point functions of this CFT, and discuss possible interpretations in terms of boundary conditions.

Characterising the statistical properties of high dimensional random functions has been one of the central focus of the theory of disordered systems, and notably spin glasses, over the last decades. Applications to machine learning via deep neural network has seen a resurgence of interest towards this problem in recent years. The simplest yet non-trivial quantity to characterise these landscapes is the annealed total complexity, i.e. the rate of exponential growth of the average number of stationary points (or equilibria) with the dimension of the underlying space. A paradigmatic model for such random landscape in the $N$-dimensional Euclidean space consists of an isotropic harmonic confinement and a Gaussian random function, with rotationally and translationally invariant covariance [1]. The total annealed complexity in this model has been shown to display a ”topology trivialisation transition”: for weak confinement, the number of stationary points is exponentially large (positive complexity) while for strong confinement there is typically a single stationary point (zero complexity).

In this talk, I will present recent results obtained for a distinct exactly solvable model of random lanscape in the $N$-dimensional Euclidean space where the random Gaussian function is replaced by a superposition of $M > N$ random plane waves [2]. In this model, we compute the total annealed complexity in the limit $N\rightarrow\infty$ with $\alpha = M/N$ fixed and find, in contrast to the scenario exposed above, that the complexity remains strictly positive for any finite value of the confinement strength. Hence, there is no ”topology trivialisation transition” for this model, which seems to be a representative of a distinct class of universality.

References:

[1] Y. V. Fyodorov, Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices, Phys. Rev. Lett. 92, 240601 (2004) Erratum: Phys. Rev. Lett. 93, 149901(E) (2004).

[2] B. Lacroix-A-Chez-Toine, S. Belga-Fedeli, Y. V. Fyodorov, Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity, arXiv preprint arXiv:2202.03815, submitted to J. Math. Phys.

In this talk I will present network-theoretic tools [1-2] to filter information in large-scale datasets and I will show that these are powerful tools to study complex datasets. In particular I will introduce correlation-based information filtering networks and the planar filtered graphs (PMFG) and I will show that applications to financial data-sets can meaningfully identify industrial activities and structural market changes [3-4].

It has been shown that by making use of the 3-clique structure of the PMFG a clustering can be extracted allowing dimensionality reduction that keeps both local information and global hierarchy in a deterministic manner without the use of any prior information [5-6]. However, the algorithm so far proposed to construct the PMFG is numerically costly with O(N3) computational complexity and cannot be applied to large-scale data. There is therefore scope to search for novel algorithms that can provide, in a numerically efficient way, such a reduction to planar filtered graphs. I will introduce a new algorithm, the TMFG (Triangulated Maximally Filtered Graph), that efficiently extracts a planar subgraph which optimizes an objective function. The method is scalable to very large datasets and it can take advantage of parallel and GPUs computing [7]. Filtered graphs are valuable tools for risk management and portfolio optimization too [8-9] and they allow to construct probabilistic sparse modeling for financial systems that can be used for forecasting, stress testing and risk allocation [10].

Filtered graphs can be used not only to extract relevant and significant information but more importantly to extract knowledge from an overwhelming quantity of unstructured and structured data. I will provide a practitioner example by a successful Silicon Valley start-up, Yewno. The key idea underlying Yewno’s products is the concept of the Knowledge Graph, a framework based on filtered graph research, whose goal is to extract signals from evolving corpus of data. The common principle is that a methodology leveraging on developments in Computational linguistics and graph theory is used to build a graph representation of knowledge [11], which can be automatically analysed to discover hidden relations between components in many different complex systems. This Knowledge Graph based framework and inference engine has a wide range of applications, including finance, economics, biotech, law, education, marketing and general research.

[1] T. Aste, T. Di Matteo, S. T. Hyde, Physica A 346 (2005) 20.

[2] T. Aste, Ruggero Gramatica, T. Di Matteo, Physical Review E 86 (2012) 036109.

[3] M. Tumminello, T. Aste, T. Di Matteo, R. N. Mantegna, PNAS 102, n. 30 (2005) 10421.

[4] N. Musmeci, Tomaso Aste, T. Di Matteo, Journal of Network Theory in Finance 1(1) (2015) 1-22.

[5] W.-M. Song, T. Di Matteo, and T. Aste, PLoS ONE 7 (2012) e31929.

[6] N. Musmeci, T. Aste, T. Di Matteo, PLoS ONE 10(3): e0116201 (2015).

[7] Guido Previde Massara, T. Di Matteo, T. Aste, Journal of Complex networks 5 (2), 161 (2016).

[8] F. Pozzi, T. Di Matteo and T. Aste, Scientific Reports 3 (2013) 1665.

[9] N. Musmeci, T. Aste and T. Di Matteo, Scientific Reports 6 (2016) 36320.

[10] Wolfram Barfuss, Guido Previde Massara, T. Di Matteo, T. Aste, Phys.Rev. E 94 (2016) 062306.

[11] Ruggero Gramatica, T. Di Matteo, Stefano Giorgetti, Massimo Barbiani, Dorian Bevec and Tomaso Aste, PLoS One (2014) PLoS ONE 9(1): e84912.