2019 | Book

# 2017 MATRIX Annals

Editors: Prof. Jan de Gier, Prof. Cheryl E. Praeger, Prof. Terence Tao

Publisher: Springer International Publishing

Book Series : MATRIX Book Series

2019 | Book

Editors: Prof. Jan de Gier, Prof. Cheryl E. Praeger, Prof. Terence Tao

Publisher: Springer International Publishing

Book Series : MATRIX Book Series

MATRIX is Australia’s international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration. This book is a scientific record of the eight programs held at MATRIX in its second year, 2017:

- Hypergeometric Motives and Calabi–Yau Differential Equations

- Computational Inverse Problems

- Integrability in Low-Dimensional Quantum Systems

- Elliptic Partial Differential Equations of Second Order: Celebrating 40 Years of Gilbarg and Trudinger’s Book

- Combinatorics, Statistical Mechanics, and Conformal Field Theory

- Mathematics of Risk

- Tutte Centenary Retreat

- Geometric R-Matrices: from Geometry to Probability

The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on a topic related to the MATRIX program; the remaining contributions are predominantly lecture notes or short articles based on talks or activities at MATRIX.

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We investigate the use of the randomize-then-optimize (RTO) method as a proposal distribution for sampling posterior distributions arising in nonlinear, hierarchical Bayesian inverse problems. Specifically, we extend the hierarchical Gibbs sampler for linear inverse problems to nonlinear inverse problems by embedding RTO-MH within the hierarchical Gibbs sampler. We test the method on a nonlinear inverse problem arising in differential equations.

Optimal Bayesian sequential inference, or filtering, for the state of a deterministic dynamical system requires simulation of the Frobenius-Perron operator, that can be formulated as the solution of an initial value problem in the continuity equation on filtering distributions. For low-dimensional, smooth systems the finite-volume method is an effective solver that conserves probability and gives estimates that converge to the optimal continuous-time values. A Courant–Friedrichs–Lewy condition assures that intermediate discretized solutions remain positive density functions. We demonstrate this finite-volume filter (FVF) in a simulated example of filtering for the state of a pendulum, including a case where rank-deficient observations lead to multi-modal probability distributions.

A new approach was recently introduced for the task of estimation of parameters of chaotic dynamical systems. Here we apply the method for stochastic differential equation (SDE) systems. It turns out that the basic version of the approach does not identify such systems. However, a modification is presented that enables efficient parameter estimation of SDE models. We test the approach with basic SDE examples, compare the results to those obtained by usual state-space filtering methods, and apply it to more complex cases where the more traditional methods are no more available.

We propose and test a numerical method for the computation of the convex source support from single-measurement electrical impedance tomography data. Our technique is based on the observation that the convex source support is the unique minimum of an optimization problem in the space of all convex and compact subsets of the imaged body.

The calibration of volatility models from observable option prices is a fundamental problem in quantitative finance. The most common approach among industry practitioners is based on the celebrated Dupire’s formula, which requires the knowledge of vanilla option prices for a continuum of strikes and maturities that can only be obtained via some form of price interpolation. In this paper, we propose a new local volatility calibration technique using the theory of optimal transport. We formulate a time continuous martingale optimal transport problem, which seeks a martingale diffusion process that matches the known densities of an asset price at two different dates, while minimizing a chosen cost function. Inspired by the seminal work of Benamou and Brenier, we formulate the problem as a convex optimization problem, derive its dual formulation, and solve it numerically via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm. The solution effectively reconstructs the dynamic of the asset price between the two dates by recovering the optimal local volatility function, without requiring any time interpolation of the option prices.

We use likelihood informed dimension reduction (LIS) (Cui et al. Inverse Prob 30(11):114015, 28, 2014) for inverting vertical profile information of atmospheric methane from ground based Fourier transform infrared (FTIR) measurements at Sodankylä, Northern Finland. The measurements belong to the word wide TCCON network for greenhouse gas measurements and, in addition to providing accurate greenhouse gas measurements, they are important for validating satellite observations.LIS allows construction of an efficient Markov chain Monte Carlo sampling algorithm that explores only a reduced dimensional space but still produces a good approximation of the original full dimensional Bayesian posterior distribution. This in effect makes the statistical estimation problem independent of the discretization of the inverse problem. In addition, we compare LIS to a dimension reduction method based on prior covariance matrix truncation used earlier (Tukiainen et al., J Geophys Res Atmos 121:10312–10327, 2016).

Contour integrals in the complex plane are the basis of effective numerical methods for computing matrix functions, such as the matrix exponential and the Mittag-Leffler function. These methods provide successful ways to solve partial differential equations, such as convection–diffusion models. Part of the success of these methods comes from exploiting the freedom to choose the contour, by appealing to Cauchy’s theorem. However, the pseudospectra of non-normal matrices or operators present a challenge for these methods: if the contour is too close to regions where the norm of the resolvent matrix is large, then the accuracy suffers. Important applications that involve non-normal matrices or operators include the Black–Scholes equation of finance, and Fokker–Planck equations for stochastic models arising in biology. Consequently, it is crucial to choose the contour carefully. As a remedy, we discuss choosing a contour that is wider than it might otherwise have been for a normal matrix or operator. We also suggest a semi-analytic approach to adapting the contour, in the form of a parabolic bound that is derived by estimating the field of values.To demonstrate the utility of the approaches that we advocate, we study three models in biology: a monomolecular reaction, a bimolecular reaction and a trimolecular reaction. Modelling and simulation of these reactions is done within the framework of Markov processes. We also consider non-Markov generalisations that have Mittag-Leffler waiting times instead of the usual exponential waiting times of a Markov process.

Point set registration involves identifying a smooth invertible transformation between corresponding points in two point sets, one of which may be smaller than the other and possibly corrupted by observation noise. This problem is traditionally decomposed into two separate optimization problems: (1) assignment or correspondence, and (2) identification of the optimal transformation between the ordered point sets. In this work, we propose an approach solving both problems simultaneously. In particular, a coherent Bayesian formulation of the problem results in a marginal posterior distribution on the transformation, which is explored within a Markov chain Monte Carlo scheme. Motivated by Atomic Probe Tomography (APT), in the context of structure inference for high entropy alloys (HEA), we focus on the registration of noisy sparse observations of rigid transformations of a known reference configuration. Lastly, we test our method on synthetic data sets.

Optimization plays an important role in solving many inverse problems. Indeed, the task of inversion often either involves or is fully cast as a solution of an optimization problem. In this light, the mere non-linear, non-convex, and large-scale nature of many of these inversions gives rise to some very challenging optimization problems. The inverse problem community has long been developing various techniques for solving such optimization tasks. However, other, seemingly disjoint communities, such as that of machine learning, have developed, almost in parallel, interesting alternative methods which might have stayed under the radar of the inverse problem community. In this survey, we aim to change that. In doing so, we first discuss current state-of-the-art optimization methods widely used in inverse problems. We then survey recent related advances in addressing similar challenges in problems faced by the machine learning community, and discuss their potential advantages for solving inverse problems. By highlighting the similarities among the optimization challenges faced by the inverse problem and the machine learning communities, we hope that this survey can serve as a bridge in bringing together these two communities and encourage cross fertilization of ideas.

We prove the asymptotic large volume expression of diagonal form factors in integrable models by evaluating carefully the diagonal limit of a non-diagonal form factor in which we send the rapidity of the extra particle to infinity.

We consider a certain abstract of RNA secondary structures, which is closely related to so-called RNA shapes. The generating function counting the number of the abstract structures is obtained in three different ways, namely, by means of Narayana numbers, Chebyshev polynomials and Motzkin paths. We show that a combinatorial interpretation on 2-Motzkin paths explains a relation between Motzkin paths and RNA shapes and also provides an identity related to Narayana numbers and Motzkin polynomial coefficients.

We present a unitary transformation relating two apparently different supersymmetric lattice models in one dimension. The first (Fendley and Schoutens, J Stat Mech, P02017, 2007) describes semionic particles on a 1D ladder, with supersymmetry moving particles between the two legs. The second (de Gier et al., J Stat Mech, 023104, 2016) is a fermionic model with particle-hole symmetry and with supersymmetry creating or annihilating pairs of domain walls. The mapping we display features non-trivial phase factors that generalise the sign factors occurring in the Jordan-Wigner transformation.

Elementary proofs are presented for the factorization of the elliptic Boltzmann weights of the A n ( 1 ) $$A^{(1)}_n$$ face model, and for the sum-to-1 property in the trigonometric limit, at a special point of the spectral parameter. They generalize recent results obtained in the context of the corresponding trigonometric vertex model.

This short note is the announcement of a forthcoming work in which we prove a first general boundary regularity result for area-minimizing currents in higher codimension, without any geometric assumption on the boundary, except that it is an embedded submanifold of a Riemannian manifold, with a mild amount of smoothness ( C 3 , a 0 $$C^{3, a_0}$$ for a positive a 0 suffices). Our theorem allows to answer a question posed by Almgren at the end of his Big Regularity Paper. In this note we discuss the ideas of the proof and we also announce a theorem which shows that the boundary regularity is in general weaker that the interior regularity. Moreover we remark an interesting elementary byproduct on boundary monotonicity formulae.

In this note, we summarise some regularity results recently obtained for an optimal transport problem where the matter transported is either accelerated by an external force field, or self-interacting, at a given intermediate time.

We show that small energy curves under a particular sixth order curvature flow with generalised Neumann boundary conditions between parallel lines converge exponentially in the C ∞ topology in infinite time to straight line segments.

In this paper we describe the asymptotic behavior of the solutions to quasilinear parabolic equations with a Hardy potential. We prove that all the solutions have the same asymptotic behavior: they all tend to the solution of the original problem which satisfies a zero initial condition. Moreover, we derive estimates on the “distance” between the solutions of the evolution problem and the solutions of elliptic problems showing that in many cases (as for example the autonomous case) these last solutions are “good approximations” of the solutions of the original parabolic PDE.

We prove that the existence of corners in a class of planar domain, which includes all simply connected polygonal domains and all smoothly bounded domains, is a spectral invariant of the Laplacian with both Neumann and Robin boundary conditions. The main ingredient in the proof is a locality principle in the spirit of Kac’s “principle of not feeling the boundary,” but which holds uniformly up to the boundary. Albeit previously known for Dirichlet boundary condition, this appears to be new for Robin and Neumann boundary conditions, in the geometric generality presented here. For the case of curvilinear polygons, we describe how the same arguments using the locality principle are insufficient to obtain the analogous result. However, we describe how one may be able to harness powerful microlocal methods and combine these with the locality principles demonstrated here to show that corners are a spectral invariant; this is current work-in-progress (Nursultanov et al., Preprint).

Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate a histogram-type prior on the volatility with piecewise constant realisations on bins forming a partition of the time interval. The values on the bins are assigned an inverse Gamma Markov chain (IGMC) prior. Posterior inference is straightforward to implement via Gibbs sampling, as the full conditional distributions are available explicitly and turn out to be inverse Gamma. We also discuss in detail the hyperparameter selection for our method. Our nonparametric Bayesian approach leads to good practical results in representative simulation examples. Finally, we apply it on a classical data set in change-point analysis: weekly closings of the Dow-Jones industrial averages.

For a multivariate Lévy process satisfying the Cramér moment condition and having a drift vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by the multivariate ruin problem introduced in Avram et al. (Ann Appl Probab 18:2421–2449, 2008) in the two-dimensional case. Our solution relies on the analysis from Pan and Borovkov (Preprint. arXiv:1708.09605, 2017) for multivariate random walks and an appropriate time discretization.

This paper presents some new results on Parisian ruin under Lévy insurance risk process, where ruin occurs when the process has gone below a fixed level from the last record maximum, also known as the high-water mark or drawdown, for a fixed consecutive periods of time. The law of ruin-time and the position at ruin is given in terms of their joint Laplace transforms. Identities are presented semi-explicitly in terms of the scale function and the law of the Lévy process. They are established using recent developments on fluctuation theory of drawdown of spectrally negative Lévy process. In contrast to the Parisian ruin of Lévy process below a fixed level, ruin under drawdown occurs in finite time with probability one.

This paper gives a partial answer to a question asked by Pierre-Emmanuel Caprace at the Groups St Andrews conference at Birmingham (UK) in August 2017, and investigated at the ‘Tutte Centenary Retreat’ workshop held at MATRIX in November 2017. Caprace asked if there exists a 2-transitive permutation group P such that only finitely many simple groups act arc-transitively on a connected graph X with local action P (of the stabiliser of a vertex v on the neighbourhood of v). Some evidence is given to suggest that the answer is “No”, even when ‘2-transitive’ is replaced by ‘transitive’, and then by way of illustration, a follow-up question is answered by showing that all but finitely many alternating groups have such an action on a 6-valent connected graph with vertex-stabiliser A 6.

In this short note we describe a recently initiated research programme aiming to use a normal quotient reduction to analyse finite connected, oriented graphs of valency 4, admitting a vertex- and edge-transitive group of automorphisms which preserves the edge orientation. In the first article on this topic (Al-bar et al. Electr J Combin 23, 2016), a subfamily of these graphs was identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one ‘basic’ member. These basic members can be further divided into three types: quasiprimitive, biquasiprimitive and cycle type. The first and third of these types have been analysed in some detail. Recently, we have begun an analysis of the basic graphs of biquasiprimitive type. We describe our approach and mention some early results. This work is on-going. It began at the Tutte Memorial MATRIX Workshop.

Bill Tutte was born on May 14, 1917 in Newmarket, England. In 1935, he began studying at Trinity College, Cambridge reading natural sciences specializing in chemistry. After completing a master’s degree in chemistry in 1940, he was recruited to work at Bletchley Park as one of an elite group of codebreakers that included Alan Turing. While there, Tutte performed “one of the greatest intellectual feats of the Second World War.” Returning to Cambridge in 1945, he completed a Ph.D. in mathematics in 1948. Thereafter, he worked in Canada, first in Toronto and then as a founding member of the Department of Combinatorics and Optimization at the University of Waterloo. His contributions to graph theory alone mark him as arguably the twentieth century’s leading researcher in that subject. He also made groundbreaking contributions to matroid theory including proving the first excluded-minor theorems for matroids, one of which generalized Kuratowski’s Theorem. He extended Menger’s Theorem to matroids and laid the foundations for structural matroid theory. In addition, he introduced the Tutte polynomial for graphs and extended it and some relatives to matroids. This paper will highlight some of his many contributions focusing particularly on those to matroid theory.

For a simply connected, connected, semisimple complex algebraic group G, we define two geometric crystals on the A $$\mathscr A$$ -cluster variety of double Bruhat cell B −∩ Bw 0 B. These crystals are related by the ∗ duality. We define the graded Donaldson-Thomas correspondence as the crystal bijection between these crystals. We show that this correspondence is equal to the composition of the cluster chamber Ansatz, the inverse generalized geometric RSK-correspondence, and transposed twist map due to Berenstein and Zelevinsky.

Finite hypergeometric functions are functions of a finite field ?? q $${\mathbb F}_q$$ to ℂ $${\mathbb C}$$ . They arise as Fourier expansions of certain twisted exponential sums and were introduced independently by John Greene and Nick Katz in the 1980s. They have many properties in common with their analytic counterparts, the hypergeometric functions. One restriction in the definition of finite hypergeometric functions is that the hypergeometric parameters must be rational numbers whose denominators divide q − 1. In this note we use the symmetry in the hypergeometric parameters and an extension of the exponential sums to circumvent this problem as much as possible.

Integrals from Feynman diagrams with massive particles soon outgrow polylogarithms. We consider the simplest situation in which this occurs, namely for diagrams with two vertices in two space-time dimensions, with scalar particles of unit mass. These comprise vacuum diagrams, on-shell sunrise diagrams and diagrams obtained from the latter by cutting internal lines. In all these cases, the Feynman integral is a moment of n = a + b Bessel functions, of the form M ( a , b , c ) : = ∫ 0 ∞ I 0 a ( t ) K 0 b ( t ) t c d t $$M(a,b,c):=\int _0^\infty I_0^a(t) K_0^b(t)t^c\mathrm {d}t$$ . The corresponding L-series are built from Kloosterman sums over finite fields. Prior to the Creswick conference, the first author obtained empirical relations between special values of L-series and Feynman integrals with up to n = 8 Bessel functions. At the conference, the second author indicated how to extend these. Working together we obtained empirical relations involving Feynman integrals with up to 24 Bessel functions, from sunrise diagrams with up to 22 loops. We have related results for moments that lie beyond quantum field theory.

Mirror maps are power series which occur in Mirror Symmetry as the inverse for composition of q ( z ) = exp ( f ( z ) ∕ g ( z ) ) $$q(z)=\exp (f(z)/g(z))$$ , called local q-coordinates, where f and g are particular solutions of the Picard–Fuchs differential equations associated with certain one-parameter families of Calabi–Yau varieties. In several cases, it has been observed that such power series have integral Taylor coefficients at the origin. In the case of hypergeometric equations, we discuss p-adic tools and techniques that enable one to prove a criterion for the integrality of the coefficients of mirror maps. This is a joint work with T. Rivoal and J. Roques. This note is an extended abstract of the talk given by the author in January 2017 at the conference “Hypergeometric motives and Calabi–Yau differential equations” in Creswick, Australia.

We define a finite-field version of Appell–Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier et al. (Hypergeometric functions over finite fields, arXiv:1510.02575v2) in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite-field Appell–Lauricella functions to establish a finite-field analogue of Koike and Shiga’s cubic transformation (Koike and Shiga, J. Number Theory 124:123–141, 2007) for the Appell hypergeometric function F 1, proving a conjecture of Ling Long. We also prove a finite field analogue of Gauss’ quadratic arithmetic geometric mean. We use our multivariable period functions to construct formulas for the number of ?? p $$\mathbb {F}_p$$ -points on the generalized Picard curves. Lastly, we give some transformation and reduction formulas for the period functions, and consequently for the finite-field Appell–Lauricella functions.

It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. Here, we briefly indicate that the connection to modular forms persists for sequences associated to Brown’s cellular integrals and state a general conjecture concerning supercongruences.

We prove various supercongruences involving truncated hypergeometric sums. These include a strengthened version of a conjecture of van Hamme. Our method is to employ various hypergeometric transformation and evaluation formulae to convert the truncated sums to quotients of Γ-values. We then convert these to quotients of Γ p-values and use Taylor’s Theorem to make p-adic approximations. In the cases under consideration higher order coefficients often vanish leading to the supercongruences.

We apply the Guinand-Weil-Mestre explicit formula to resolve two questions about how a certain hypergeometric motive splits into two irreducible motives.

We discuss two related principles for hypergeometric supercongruences, one related to accelerated convergence and the other to the vanishing of Hodge numbers.

Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. Among the arithmetic correspondences predicted by mirror symmetry are correspondences between point counts over finite fields, and more generally between factors of their Zeta functions. In particular, we will discuss our results on a common factor for Zeta functions of alternate families of invertible polynomials. We will also explore closed formulas for the point counts for our alternate mirror families of K3 surfaces and their relation to their Picard–Fuchs equations. Finally, we will discuss how all of this relates to hypergeometric motives. This report summarizes work from two papers.

In this review article we show how the theory of Schwarzian differential equations leads to an interesting class of meromorphic functions on the upper-half plane ℍ $${\mathbb H}$$ named equivariant functions. These functions have the property that their Schwarz derivatives are weight 4 automorphic forms for a discrete subgroup Γ of PSL 2 ( ℝ ) $${\mbox{PSL}_2({\mathbb R})}$$ . It turns out that these functions must satisfy the relation f ( γ τ ) = ρ ( γ ) f ( τ ) , τ ∈ ℍ , γ ∈ Γ , $$\displaystyle f(\gamma \tau )\,=\, \rho (\gamma )f(\tau )\, ,\ \, \tau \in {\mathbb H}\;,\ \gamma \in {\varGamma }, $$ where ρ is a 2-dimensional complex representation of Γ and the matrix action on both sides is by linear fractional transformation. When ρ is the identity representation ρ(γ) = γ, the equivariant functions are parameterized by scalar automorphic forms, while if ρ is an arbitrary representation they are parameterized by vector-valued automorphic forms with multiplier ρ. If Γ is a modular subgroup we obtain important applications to modular forms for Γ as well as a description in terms of elliptic functions theory. We also prove the existence of equivariant functions for the most general case by constructing a vector bundle attached to the data (Γ, ρ) and applying the Kodaira vanishing theorem.

We discuss recent work of the authors in which we study the translation of classical hypergeometric transformation and evaluation formulas to the finite field setting.Our approach is motivated by the desire for both an algorithmic type approach that closely parallels the classical case, and an approach that aligns with geometry. In light of these objectives, we focus on period functions in our construction which makes point counting on the corresponding varieties as straightforward as possible.We are also motivated by previous work joint with Deines, Fuselier, Long, and Tu in which we study generalized Legendre curves using periods to determine a condition for when the endomorphism algebra of the primitive part of the associated Jacobian variety contains a quaternion algebra over ℚ $${\mathbb {Q}}$$ . In most cases this involves computing Galois representations attached to the Jacobian varieties using Greene’s finite field hypergeometric functions.

In this project, we establish the supercongruences for the 14 families of rigid hypergeometric Calabi–Yau threefolds conjectured by Roriguez-Villegas in 2003.

We study classical hypergeometric series as a p-adic function of its parameters inspired by a problem in the Monthly solved by D. Zagier.

For a multivariate Laurent polynomial f(x) with coefficients in a ring R we construct a sequence of matrices with entries in R whose reductions modulo p give iterates of the Hasse–Witt operation for the hypersurface of zeroes of the reduction of f(x) modulo p. We show that our matrices satisfy a system of congruences modulo powers of p. If the Hasse–Witt operation is invertible these congruences yield p-adic limit formulas, which conjecturally describe the Gauss–Manin connection and the Frobenius operator on the slope 0 part of a crystal attached to f(x). We also apply our results on congruences to integrality of formal group laws of Artin–Mazur kind.

We consider certain generalizations of modular curves arising from congruence subgroups of triangle groups.

We give an extended abstract regarding our talk, and the associated Magma implementation of Jacobi sums and Hecke Grössencharacters. This builds upon seminal work of Weil (Trans Am Math Soc 73:487–495, 1952), and makes his construction explicitly computable, inherently relying on his upper bound for the conductor. Moreover, we can go slightly further than Weil by additionally allowing Kummer twists of the Jacobi sums. We also note the correspondence of these (twisted) Jacobi sums to tame prime information for hypergeometric motives.Although our viewpoint and notation is derived from later work of Anderson, we do not use his formalism in any substantial way, and indeed the main thrust of all we do is already in Weil’s work.

Let X = X 0 6 ( 1 ) ∕ W 6 $$X=X_0^6(1)/W_6$$ be the quotient of the Shimura curve X 0 6 ( 1 ) $$X_0^6(1)$$ by all the Atkin-Lehner involutions. By realizing modular forms on X in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer’s formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over Q ¯ $$\overline Q$$ with complex multiplication.

This a write up of a talk given at the MATRIX conference at Creswick in 2017 (to be precise, on Friday, January 20, 2017). It reports on work in progress with P. Candelas and X. de la Ossa. The aim of that work is to determine, under certain conditions, the local Euler factors of the L-functions of the fibres of a family of varieties without recourse to the equations of the varieties in question, but solely from the associated Picard–Fuchs equation.

The Kalman filter is a data analysis method used in a wide range of engineering and applied mathematics problems. This paper presents a matrix-theoretic derivation of the method in the linear model, Gaussian measurement error case. Standard derivations of the Kalman filter make use of probabilistic notation and arguments, whereas we make use, primarily, of methods from numerical linear algebra. In addition to the standard Kalman filter, we derive an equivalent variational (optimization-based) formulation, as well as the extended Kalman filter for nonlinear problems.

Recent advances in biology, economics, engineering and physical sciences have generated a large number of mathematical models for describing the dynamics of complex systems. A key step in mathematical modelling is to estimate model parameters in order to realize experimental observations. However, it is difficult to derive the analytical density functions in the Bayesian methods for these mathematical models. During the last decade, approximate Bayesian computation (ABC) has been developed as a major method for the inference of parameters in mathematical models. A number of new methods have been designed to improve the efficiency and accuracy of ABC. Theoretical studies have also been conducted to investigate the convergence property of these methods. In addition, these methods have been applied to a wide range of deterministic and stochastic models. This chapter gives a brief review of the main ABC algorithms and various improvements.

Loop models are statistical ensembles of closed paths on a lattice. The most well-known among them has a variety of names such as the dense O(n) loop model, the Temperley-Lieb (TL) model. This note concerns the model in which the weight of the loop n = 1, and a local operator which changes the weight of all the loops that surround the position of the operator to some other value. A conjecture of the expectation value of the one-point function of this operator was formulated 15 years ago. In this note we sketch the proof.

In the present note we address the important problem of stability of blockchain systems. The so-called “double-spending attacks” (attempts to spend digital funds more than once) have been analyzed by several authors. We re-state these questions under more realistic assumptions than previously discussed and show that they can be formulated as an optimal stopping problem.

In this short note, we consider the optimization problem with probability distortion when the objective functional involves a running term which is given by an S-shaped function. A stochastic maximum principle is presented.

In this paper, we consider a classical risk model refracted at given level. We give an explicit expression for the joint density of the ruin time and the cumulative number of claims counted up to ruin time. The proof is based on solving some integro-differential equations and employing the Lagrange’s Expansion Theorem.

We show that the distribution of two-sided weighted Kolmogorov-Smirnov (wK-S) statistics can be obtained via the solution of the system of two Volterra type integral equations for corresponding boundary crossing probabilities for a diffusion process. Based on this result we propose a numerical approximation method for evaluating the distribution of wK-S statistics. We provide the numerical solutions to the system of the integral equations which were also verified via Monte Carlo simulations.

We present an overview of Univariate Extreme Value Theory (EVT) providing standard and new tools to model the tails of distributions. One of the main issues in the statistical literature of extremes concerns the tail index estimation, which governs the probability of extreme occurrences. This estimation relies heavily on the determination of a threshold above which a Generalized Pareto Distribution (GPD) can be fitted. Approaches to this estimation may be classified into two classes, one qualified as ‘supervised’, using standard Peak Over Threshold (POT) methods, in which the threshold to estimate the tail is chosen graphically according to the problem, the other class collects unsupervised methods, where the threshold is algorithmically determined.We introduce here a new and practically relevant method belonging to this second class. It is a self-calibrating method for modeling heavy tailed data, which we developed with N. Debbabi and M. Mboup. Effectiveness of the method is addressed on simulated data, followed by applications in neuro-science and finance. Results are compared with those obtained by more standard EVT approaches.Then we turn to the notion of dependence and the various ways to measure it, in particular in the tails. Through examples, we show that dependence is also a crucial topic in risk analysis and management. Underestimating the dependence among extreme risks can lead to serious consequences, as for instance those we experienced during the last financial crisis. We introduce the notion of copula, which splits the dependence structure from the marginal distribution, and show how to use it in practice. Taking into account the dependence between random variables (risks) allows us to extend univariate EVT to multivariate EVT. We only give the first steps of the latter, to motivate the reader to follow or to participate in the increasing research development on this topic.

We introduce a new measure of performance of investment strategies, the monotone Sharpe ratio. We study its properties, establish a connection with coherent risk measures, and obtain an efficient representation for using in applications.

We construct a sequential test for the sign of the drift of a fractional Brownian motion. We work in the Bayesian setting and assume the drift has a prior normal distribution. The problem reduces to an optimal stopping problem for a standard Brownian motion, obtained by a transformation of the observable process. The solution is described as the first exit time from some set, whose boundaries satisfy certain integral equations, which are solved numerically.

These lecture notes are based on Yang’s talk at the MATRIX program Geometric R-Matrices: from Geometry to Probability, at the University of Melbourne, Dec. 18–22, 2017, and Zhao’s talk at Perimeter Institute for Theoretical Physics in January 2018. We give an introductory survey of the results in Yang and Zhao (Quiver varieties and elliptic quantum groups, 2017. arxiv1708.01418). We discuss a sheafified elliptic quantum group associated to any symmetric Kac-Moody Lie algebra. The sheafification is obtained by applying the equivariant elliptic cohomological theory to the moduli space of representations of a preprojective algebra. By construction, the elliptic quantum group naturally acts on the equivariant elliptic cohomology of Nakajima quiver varieties. As an application, we obtain a relation between the sheafified elliptic quantum group and the global affine Grassmannian over an elliptic curve.