Past

We use entropy methods to show that the heat equation with Dirichlet boundary conditions on the complement of a compact set in R^d shows a self-similar behaviour much like the usual heat equation on R^d, once we account for the loss of mass due to the boundary. Giving good lower bounds for the fundamental solution on these sets is surprisingly a relatively recent result, and we find some improvements using some advances in logarithmic Sobolev inequalities. In particular, we are able to give optimal asymptotic bounds for large times for the fundamental solution with an explicit approach rate in dimensions larger than 2, and some new bounds in dimension 2.

This is a work in collaboration with Alejandro Gárriz and Fernando Quirós.

In this talk we will study the graph structure of supersingular isogeny graphs. These graphs are known to have very few loops and multi-edges. We formalize this idea by studying and finding bounds for their number of loops and multi-edges. We also find conditions under which these graphs are simple. To do so, we introduce a method of counting the total number of collisions (which are special endomorphisms) based on a trace formula of Gross and a known formula of Kronecker, Gierster and Hurwitz. 

The method presented in this talk can be used to study many kinds of collisions in supersingular isogeny graphs. As an application, we will see how this method was used to estimate a certain number of collisions and then show that isogeny graphs do not satisfy a certain cryptographic property that was falsely believed (and proven!) to hold.

I shall begin with a brief history of the problem of trying to understand infinite groups knowing only their finite quotients. I'll then focus on 3-manifold groups, describing the prominent role that they have played in advancing our understanding of this problem in recent years. The story for 3-manifold groups involves a rich interplay of algebra, geometry, and arithmetic. I shall describe arithmetic Kleinian groups that are profinitely rigid in the absolute sense -- ie they can be distinguished from all other finitely generated, residually finite groups by their set of finite quotients. I shall then explain more recent work involving products of Seifert fibered manifolds -- here we find groups that are profinitely rigid in the class of finitely presented groups but not in the class of finitely generated groups. This is joint work with McReynolds, Reid, and Spitler.

We provide a framework for modelling risk and quantifying payment shortfalls in cleared markets with multiple central counterparties (CCPs). Building on the stylised fact that clearing membership is shared among CCPs, we show how this can transmit stress across markets through multiple CCPs. We provide stylised examples to lay out how such stress transmission can take place, as well as empirical evidence to illustrate that the mechanisms we study could be relevant in practice. Furthermore, we show how stress mitigation mechanisms such as variation margin gains haircutting by one CCP can have spillover effects on other CCPs. The framework can be used to enhance CCP stress-testing, which currently relies on the “Cover 2” standard requiring CCPs to be able to withstand the default of their two largest clearing members. We show that who these two clearing members are can be significantly affected by higher-order effects arising from interconnectedness through shared clearing membership. Looking at the full network of CCPs and shared clearing members is therefore important from a financial stability perspective.

This is joint work with Iñaki Aldasoro.

BIS Working Paper No 1052: https://www.bis.org/publ/work1052.pdf

First we recall the mirror symmetry identification of the coordinate ring of certain very stable upward flows in the Hitchin system and the Kirillov algebra for the minuscule representation of the Langlands dual group via the equivariant cohomology of the cominuscule flag variety (e.g. complex Grassmannian). In turn we discuss a conjectural extension of this picture to non-very stable upward flows in terms of a big commutative subalgebra of the Kirillov algebra, which also ringifies the equivariant intersection cohomology of the corresponding affine Schubert variety.

In recent progress on the black hole information paradox, Page curves consistent with unitarity have been obtained in 2d models and in bottom-up braneworld models using the notion of double holography. In this talk we discuss top-down models realizing 4d black holes coupled to a bath in Type IIB string theory and obtain Page curves. We make the ideas behind double holography precise in these models and address causality puzzles which have arisen in the bottom-up models, leading to a refinement of their interpretation.
 

In this session, we will hold a panel discussion on how to best give an academic talk. Among other topics, we will focus on techniques for engaging your audience, for determining the level and technical details of the talk, and for giving both blackboard and slide presentations. The discussion will begin with a directed panel discussion before opening up to questions from the audience.

Approaches to personalized diagnosis and treatment in oncology are heavily reliant on computer models that use molecular and clinical features to
characterize an individual patient’s disease. Most of these models use genome and/or gene expression sequences to develop classifiers of a patient’s
tumor. However, in order to fully model the behavior and therapy response of a tumor, dynamic models are desirable that can act like a Digital Twin of
the cancer patient allowing prognostic and predictive simulations of disease progression, therapy responses and development of resistance. We are
constructing Digital Twins of cancer patients in order to perform dynamic and predictive simulations that improve patient stratification and
facilitate the design of individualized therapeutic strategies. Using a hybrid approach that combines artificial intelligence / machine learning
with dynamic mechanistic modelling we are developing a computational framework for generating Digital Twins. This framework can integrate
different types of data (multiomics, clinical, and existing knowledge) and produces personalized computational models of a patient’s tumor. The
computational models are validated and refined by experimental work and in retrospective patient studies. We present some of the results of the dynamic
Digital Twins simulations in neuroblastoma. They include (i) identification on non-MYCN amplified high risk patients; (ii) prediction of individual
patients’ responses to chemotherapy; and (iii) identification of new drug targets for personalized therapy. Digital Twin models allow the dynamic and
mechanistic simulation of disease progression and therapy response. They are useful for the stratification of patients and the design of personalized
therapies.

Perverse sheaves are an indispensable tool in geometric representation theory that can be used to construct representations and compute composition multiplicities. These ‘sheaves’ live in a certain $\ell$-adic derived category. In this talk we will discuss a beautiful construction of this category based on the pro-étale topology and explore some applications in representation theory.

In his pioneering and celebrated 1968 paper on the elementary theory of finite fields Ax asked if the theory of the class of all the finite rings $\mathbb{Z}/m\mathbb{Z}$, for all $m>1$, is decidable. In that paper, Ax proved that the existential theory of this class is decidable via his result that the theory of the class of all the rings $\mathbb{Z}/p^n\mathbb{Z}$ (with $p$ and $n$ varying) is decidable. This used Chebotarev’s Density Theorem and model theory of pseudo-finite fields.

I will talk about a recent solution jointly with Angus Macintyre of Ax’s Problem using model theory of the ring of adeles of the rational numbers.

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.

Learning dynamical models from data plays a vital role in engineering design, optimization, and predictions. Building models describing the dynamics of complex processes (e.g., weather dynamics, reactive flows, brain/neural activity, etc.) using empirical knowledge or first principles is frequently onerous or infeasible. Therefore, system identification has evolved as a scientific discipline for this task since the 1960ies. Due to the obvious similarity of approximating unknown functions by artificial neural networks, system identification was an early adopter of machine learning methods. In the first part of the talk, we will review the development in this area until now.

For complex systems, identifying the full dynamics using system identification may still lead to high-dimensional models. For engineering tasks like optimization and control synthesis as well as in the context of digital twins, such learned models might still be computationally too challenging in the aforementioned multi-query scenarios. Therefore, it is desirable to identify compact approximate models from the available data. In the second part of this talk, we will therefore exploit that the dynamics of high-fidelity models often evolve in lowdimensional manifolds. We will discuss approaches learning representations of these lowdimensional manifolds using several ideas, including the lifting principle and autoencoders. In particular, we will focus on learning state-space representations that can be used in classical tools for computational engineering. Several numerical examples will illustrate the performance and limitations of the suggested approaches.

This talk will explore a series of topics and example applications at the interface of graph theory and dynamics, from synchronization and diffusion dynamics on networks, to graph-based data clustering, to graph convolutional neural networks. The underlying links are provided by concepts in spectral graph theory.

Group cohomology is a powerful tool which has found many applications in modern group theory. It can be calculated and interpreted through geometric, algebraic and topological means, and as such it encodes the relationships between these different aspects of infinite groups. The aim of this talk is to introduce a circle of ideas which link group cohomology with the theory of BNS invariants, and the property of being subgroup separable. No prior knowledge of any of these topics will be assumed.

In the operator algebraic approach to quantum field theory, the DHR category is a braided tensor category describing topological point defects of a theory with at least 1 (+1) dimensions. A single von Neumann algebra with no extra structure can be thought of as a 0 (+1) dimensional quantum field theory. In this case, we would not expect a braided tensor category of point defects since there are not enough dimensions to implement a braiding. We show, however, that one can think of central sequence algebras as operators localized ``at infinity", and apply the DHR recipe to obtain a braided tensor category of bimodules of a von Neumann algebra M, which is a Morita invariant. When M is a II_1 factor, the braided subcategory of automorphic objects recovers Connes' chi(M) and Jones' kappa(M). We compute this for II_1 factors arising naturally from subfactor theory and show that any Drinfeld center of a fusion category can be realized. Based on joint work with Quan Chen and Dave Penneys.

In recent years, duality approaches have yielded new results about the high-dimensional cohomology of several groups and moduli spaces, such as $\operatorname{SL}_n(\mathbb{Z})$ and $\mathcal{M}_g$. I will explain the general strategy of these approaches and survey results that have been obtained so far. To give an example, I will first explain how Borel-Serre duality can be used to show that the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes near its virtual cohomological dimension. This is based on joint work with Miller-Patzt-Sroka-Wilson and builds on results by Church-Farb-Putman. I will then put this into a more general context by giving an overview of analogous results for mapping class groups of surfaces, automorphism groups of free groups and further arithmetic groups such as $\operatorname{SL}_n(\mathcal{O}_K)$ and $\operatorname{Sp}_{2n}(\mathbb{Z})$.

We investigate the theoretical properties of subsampling and hashing as tools for approximate Euclidean norm-preserving embeddings for vectors with (unknown) additive Gaussian noises. Such embeddings are called Johnson-Lindenstrauss embeddings due to their celebrated lemma. Previous work shows that as sparse embeddings, if a comparable embedding dimension to the Gaussian matrices is required, the success of subsampling and hashing closely depends on the $l_\infty$ to $l_2$ ratios of the vectors to be mapped. This paper shows that the presence of noise removes such constrain in high-dimensions; in other words, sparse embeddings such as subsampling and hashing with comparable embedding dimensions to dense embeddings have similar norm-preserving dimensionality-reduction properties, regardless of the $l_\infty$ to $l_2$ ratios of the vectors to be mapped. The key idea in our result is that the noise should be treated as information to be exploited, not simply a nuisance to be removed. Numerical illustrations show better performances of sparse embeddings in the presence of noise.

The triangle removal lemma of Ruzsa and Szemerédi is a fundamental result in extremal graph theory; very roughly speaking, it says that if a graph is "far" from triangle-free, then it contains "many" triangles. Despite decades of research, there is still a lot that we don't understand about this simple statement; for example, our understanding of the quantitative dependencies is very poor.

In this talk, I will discuss asymmetric versions of the triangle removal lemma, where in some cases we can get almost optimal quantitative bounds. The proofs use a mix of ideas coming from graph theory, number theory, probabilistic combinatorics, and Ramsey theory.

Based on joint work with Lior Gishboliner and Asaf Shapira.

In the last few years, major advances have been made in our understanding of the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. Four key tools which are central to this progress are highest weight theory, reduction to the principal block, wall-crossing functors, and tilting modules. When considering instead the representation theory of the Lie algebras of these algebraic groups, more subtleties arise. If we look at those modules whose p-character is in so-called standard Levi form we are able to recover the four tools mentioned above, but they have been less well-studied in this setting. In this talk, we will explore the similarities and differences which arise when employing these tools for the Lie algebras rather than the algebraic groups. This research is funded by a research fellowship from the Royal Commission for the Exhibition of 1851.

In this talk we are presenting a new approach for compatible finite element discretisations for atmospheric flows on a terrain following mesh. In classical compatible finite element discretisations, the H(div)-velocity space involves the application of Piola transforms when mapping from a reference element to the physical element in order to guarantee normal continuity. In the case of a terrain following mesh, this causes an undesired coupling of the horizontal and vertical velocity components. We are proposing a new finite element space, that drops the Piola transform. For solving the equations we introduce a hybridisable formulation with trace variables supported on horizontal cell faces in order to enforce the normal continuity of the velocity in the solution. Alongside the discrete formulation for various fluid equations we discuss solver approaches that are compatible with them and present our latest numerical results.

In this talk I will discuss Onsager's conjecture for energy conservation. Moreover, in 1949 Onsager conjectured that weak solutions to the incompressible Euler equations, that were Hölder continuous with Hölder exponent greater than 1/3, conserved kinetic energy. Onsager also conjectured that there were weak solutions that were Hölder continuous with Hölder exponent less than 1/3 that didn't conserve kinetic energy. I will discuss the results regarding the former, focusing mainly on the case where the spacial domain is bounded with C^2 boundary, as proved by Bardos and Titi.

This talk will be devoted to our recent developments in the analysis of emerging models for complex flows. I will start from presenting a general PDE system describing two-fluid flows, for which we prove existence of global in time weak solutions for arbitrary large initial data. I will explain where the famous approach of Lions developed for the compressible Navier-Stokes equations fails and how to use a more direct, weighted Kolmogorov criterion to prove compactness of approximating sequences of solutions. Through a formal limit, I will link the two-fluid model to the constrained two-phase models. Applications of such models include modelling of granular flows, crowd motion, or shallow water flow through a channel. The last part of my talk will focus on the rigorous derivation of these models from the compressible Navier-Stokes equations via the vanishing singular pressure or viscosity limit.

In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor
function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is
to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their
conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this
talk, we'll discuss arithmetic problems over F_q[T] and their connections to matrix integrals, focusing on variations on
the divisor function problem with symplectic and orthogonal distributions. Joint work with Matilde Lalín.

In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this talk, we'll discuss arithmetic problems over F_q[T] and their connections to matrix integrals, focusing on variations on the divisor function problem with symplectic and orthogonal distributions. Joint work with Matilde Lalín.

I shall discuss my recent work showing that the Bogomolov-Tschinkel universality conjecture holds if and only if the mapping class groups of a punctured surface is large. One consequence of this result is that all genus 2 surface-by-surface (and all genus 2 surface-by-free) groups are virtually algebraically fibered. Moreover, I will explain why simple curve homology does not always generate homology of finite covers of closed surface. I will also mention my work with O. Tosic regarding the Putman-Wieland conjecture, and explain the partial solution to the Prill's problem about algebraic curves.

 

In this talk I will present an exclusion process with different types of particles: A, B and C. This last type can be understood as holes. Two scaling limits will be discussed: hydrodynamic limits in the boundary driven setting; and equilibrium fluctuations for an evolution on the torus. In the later case, we distinguish several cases, that depend on the choice of the jump rates, for which we get in the limit either the stochastic Burgers equation or the Ornstein-Uhlenbeck equation. These results match with predictions from non-linear fluctuating hydrodynamics. 
(Joint work with G. Cannizzaro, A. Occelli, R. Misturini).

Associated to any smooth projective curve C is its degree d Jacobian variety, parametrising isomorphism classes of degree d line bundles on C. Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves $\overline{\mathcal{M}}_{g,n}$, depending on the choice of a stability condition. In this talk I will introduce these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). This talk is based on arXiv:2210.11457.

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. We develop a deep transport map that is suitable for sampling concentrated distributions defined by an unnormalised density function. We approximate the target distribution as the push-forward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a tensor train decomposition. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. We propose two bridging strategies suitable for wide use: tempering the target density with a sequence of increasing powers, and smoothing of an indicator function with a sequence of sigmoids of increasing scales. The latter strategy opens the door to efficient computation of rare event probabilities in Bayesian inference problems.

Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.
 

We study supersymmetric gauge theories with 8 supercharges in d=3,4,5,6. For these theories, one can perform Higgsings by turning on VEVs of scalar fields. However, this process can often be difficult when dealing with superconformal field theories (SCFTs) where the Lagrangian is often not known. Using techniques of magnetic quivers and a new algorithm we call "Inverted Quiver Subtraction", we show how one can easily obtain the SCFT(s) after Higgsing. This technique can be equally well applied to SCFTs in d=3,4,5,6. 

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

Reaction-diffusion waves have long been used to describe the growth and spread of populations undergoing a spatial range expansion. Such waves are generally classed as either pulled, where the dynamics are driven by the very tip of the front and stochastic fluctuations are high, or pushed, where cooperation in growth or dispersal results in a bulk-driven wave in which fluctuations are suppressed. These concepts have been well studied experimentally in populations where the cooperation leads to a density-dependent growth rate. By contrast, relatively little is known about experimental populations that exhibit a density-dependent dispersal rate.

Using bacteriophage T7 as a test organism, we present novel experimental measurements that demonstrate that the diffusion of phage T7, in a lawn of host E. coli, is hindered by steric interactions with host bacteria cells. The coupling between host density, phage dispersal and cell lysis caused by viral infection results in an effective density-dependent diffusion rate akin to cooperative behavior. Using a system of reaction-diffusion equations, we show that this effect can result in a transition from a pulled to pushed expansion. Moreover, we find that a second, independent density-dependent effect on phage dispersal spontaneously emerges as a result of the viral incubation period, during which phage is trapped inside the host unable to disperse. Our results indicate both that bacteriophage can be used as a controllable laboratory population to investigate the impact of density-dependent dispersal on evolution, and that the genetic diversity and adaptability of expanding viral populations could be much greater than is currently assumed.

Néron models are mathematical objects which play a very important role in contemporary arithmetic geometry. However, they usually behave badly, particularly in respect of exact sequences and base change, which makes most problems regarding their behaviour very delicate. Chai introduced the base change conductor, a rational number associated with a semiabelian variety $G$ which measures the failure of the Néron model of $G$ to commute with (ramified) base change. Moreover, Chai conjectured that this invariant is additive in certain exact sequences. We shall introduce a new method to study the Néron models of Jacobians of proper (possibly singular) curves, and sketch a proof of Chai's conjecture for semiabelian varieties which are also Jacobians. 

In order for one to infer reasonable predictions from a model, it must be calibrated to reproduce observations in the market. We use the semimartingale optimal transport methodology to formulate this calibration problem into a constrained optimisation problem, with our model calibrated using a finite number of European options observed in the market as constraints. Given such a PDE formulation, we are able to then derive a dual formulation involving an HJB equation which we can numerically solve. We focus on two cases: (1) The stochastic interest rate is known and perfectly matches the observed term structure in the market, however the asset local volatility and correlation are not known and must be calibrated; (2) The dynamics of both the stochastic interest rate and the underlying asset are unknown, and we must jointly calibrate both to European options on the interest rate and on the asset.

ife forms have devised impressive and subtle ways to exploit fluid flows. I’ll talk about birds as flying machines whose behaviors can give surprising insights into flow physics. One story explains how flocking interactions can help to bring flapping flyers into orderly formations. A second story involves the more subtle role of aerodynamics in the highly efficient breathing of birds, which is thought to be critical to their ability to fly.

Tensors, also known as multiway arrays, have become ubiquitous as representations for operators or as convenient schemes for storing data. Yet, when it comes to compressing these objects or analyzing the data stored in them, the tendency is to ``flatten” or ``matricize” the data and employ traditional linear algebraic tools, ignoring higher dimensional correlations/structure that could have been exploited. Impediments to the development of equivalent tensor-based approaches stem from the fact that familiar concepts, such as rank and orthogonal decomposition, have no straightforward analogues and/or lead to intractable computational problems for tensors of order three and higher.

In this talk, we will review some of the common tensor decompositions and discuss their theoretical and practical limitations. We then discuss a family of tensor algebras based on a new definition of tensor-tensor products. Unlike other tensor approaches, the framework we derive based around this tensor-tensor product allows us to generalize in a very elegant way all classical algorithms from linear algebra. Furthermore, under our framework, tensors can be decomposed in a natural (e.g. ‘matrix-mimetic’) way with provable approximation properties and with provable benefits over traditional matrix approximation. In addition to several examples from recent literature illustrating the advantages of our tensor-tensor product framework in practice, we highlight interesting open questions and directions for future research.

In this seminar, we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporated the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model, which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.

Condensed Mathematics is a tool recently developed by Clausen and Scholze and it is proving fruitful in many areas of algebra and geometry. In this talk, we will cover the definition of condensed sets, the analogues of topological spaces in the condensed setting. We will also talk about condensed modules over a ring and some of their nice properties like forming an abelian category. Finally, we'll discuss some recent results that have been obtained through the application of Condensed Mathematics.

We will be joined by Professor Kobi Kremnizer, who is a trained mental health first-aider, to discuss ways to protect your mental health this season.

In view of Takesaki-Takai duality, we can go back and forth between C*-dynamical systems of an abelian group and ones of its Pontryagin dual by taking crossed products. In this talk, I present a similar duality between actions on C*-algebras of two constructions of locally compact quantum groups: one is the bicrossed product due to Vaes-Vainerman, and the other is the double crossed product due to Baaj-Vaes. I will explain the situation by illustrating the example coming from groups. If time permits, I will also discuss its consequences in the case of quantum doubles.

A metric is said to be (globally) median,  if any three points have a unique “median” which  lies  between  any  two  points  from  the  triple.  
Such  spaces  arise  naturally  in  many different contexts.  The property of being locally median can be viewed as a kind of
non-positive curvature condition.  We show that a complete uniformly locally median space is
globally median if and only if it is simply connected.  This is an analogue of the well known Cartan-Hadamard Theorem for non-positively curved manifolds, or more generally CAT(0) spaces.  However it leaves open a number of interesting questions.

We consider a bond percolation on the hypercube in the supercritical regime. We derive vertex-expansion properties of the giant component. As a consequence we obtain upper bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component. This talk is based on joint work with Joshua Erde and Michael Krivelevich.

We discuss the Local Langlands correspondence in connection with the Bernstein center and the Stable Bernstein center. We also give an example of stable Bernstein center as a stable essentially compact invariant distribution.

I will present a model for an optimal portfolio allocation and consumption problem for a portfolio composed of a risk-free bond and two illiquid assets. Two forms of illiquidity are presented, both illiquidities based on Lévy processes. The goal of the investor is to maximise a certain utility function, and the optimal utility is found as a solution of a nonlinear PIDE of the Hamilton-Jacobi-Bellman kind.