Past

Standard quadratic programs have numerous applications and play an important role in copositivity detection. We consider reformulating a standard quadratic program as a mixed integer linear programming (MILP) problem. We propose alternative MILP reformulations that exploit the specific structure of standard quadratic programs. We report extensive computational results on various classes of instances. Our experiments reveal that our MILP reformulations significantly outperform other global solution approaches. 
This is joint work with Jacek Gondzio.

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton-Jacobi-Bellman equation usually calls for additional strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then apply a level-set approach to describe the reachable set. With this approach, the state constraints can be managed through an exact penalization technique. However, a new set of state and control variables enters the definition of this stochastic target problem. In particular, those controls are unbounded. A “compactification” of the problem is then performed. (joint work with Athena Picarelli)
 

This talk will be an introduction to the use of conformal methods in asymptotic analysis in general relativity. We shall consider the explicit example of flat spacetime (Minkowski spacetime). The full conformal compactification will be constructed. For a simple example of a conformally invariant equation (we'll take the wave equation), we shall see how the compactification allows to infer precise informations on the asymptotic behaviour of the solution in all directions, for a certain class of data at any rate. Then, depending on time and questions, I will either describe how a scattering theory can be constructed using the same method or, explain how conformal methods can be used on other asymptotically flat geometries.

In 1982, Gromov introduced bounded cohomology to give estimates on the minimal volume of manifolds. Since then, bounded cohomology has become an independent and active research field. In this talk I will give an introduction to bounded cohomology, state many open problems and relate it to other fields. 

Pseudorandom functions (PRFs) are one of the fundamental building blocks in cryptography. Traditionally, there have been two main approaches for PRF design: the ``practitioner's approach'' of building concretely-efficient constructions based on known heuristics and prior experience, and the ``theoretician's approach'' of proposing constructions and reducing their security to a previously-studied hardness assumption. While both approaches have their merits, the resulting PRF candidates vary greatly in terms of concrete efficiency and design complexity. In this work, we depart from these traditional approaches by exploring a new space of plausible PRF candidates. Our guiding principle is to maximize simplicity while optimizing complexity measures that are relevant to cryptographic applications. Our primary focus is on weak PRFs computable by very simple circuits (depth-2 ACC^0 circuits). Concretely, our main weak PRF candidate is a ``piecewise-linear'' function that first applies a secret mod-2 linear mapping to the input, and then a public mod-3 linear mapping to the result. We also put forward a similar depth-3 strong PRF candidate.  
The advantage of our approach is twofold. On the theoretical side, the simplicity of our candidates enables us to draw many natural connections between their hardness and questions in complexity theory or learning theory (e.g., learnability of depth-2 ACC^0 circuits and width-3 branching programs, interpolation and property testing for sparse polynomials, and natural proof barriers for showing super-linear circuit lower bounds). On the applied side, the piecewise-linear structure of our candidates lends itself nicely to applications in secure multiparty computation (MPC). Using our PRF candidates, we construct protocols for distributed PRF evaluation that achieve better round complexity and/or communication complexity (often both) compared to protocols obtained by combining standard MPC protocols with PRFs like AES, LowMC, or Rasta (the latter two are specialized MPC-friendly PRFs).
Finally, we introduce a new primitive we call an encoded-input PRF, which can be viewed as an interpolation between weak PRFs and standard (strong) PRFs. As we demonstrate, an encoded-input PRF can often be used as a drop-in replacement for a strong PRF, combining the efficiency benefits of weak PRFs and the security benefits of strong PRFs. We conclude by showing that our main weak PRF candidate can plausibly be boosted to an encoded-input PRF by leveraging standard error-correcting codes.
Joint work with Dan Boneh, Yuval Ishai, Amit Sahai, and David J. Wu.

We will talk about set theory, and, more specifically, forcing. Forcing is powerful. It is the go-to method for proving the independence of the continuum hypothesis or for understanding the (lack of) fine structure of the real numbers. However, forcing is hard. Keen to export their theorems to more mainstream areas of mathematics, set theorists have tackled this issue by inventing forcing axioms, (relatively) simple mathematical statements which describe sophisticated forcing extensions. In my talk, I will present the basics of forcing, I will introduce some interesting forcing axioms and I will show how these might be used to obtain surprising independence results.

When approximating PDEs with the finite element method, large sparse linear systems must be solved. The ideal preconditioner yields convergence that is  algorithmically optimal and parameter robust, i.e. the number of Krylov iterations required to solve the linear system to a given accuracy does not grow substantially as the mesh or problem parameters are changed.

Achieving this for the stationary Navier-Stokes has proven challenging: LU factorisation is Reynolds-robust but scales poorly with degree of freedom count, while Schur complement approximations such as PCD and LSC degrade as the Reynolds number is increased.

Building on the work of Schöberl, Olshanskii and Benzi, in this talk we present the first preconditioner for the Newton linearisation of the stationary Navier--Stokes equations in three dimensions that achieves both optimal complexity and Reynolds-robustness. The scheme combines a novel tailored finite element discretisation, discrete augmented Lagrangian stabilisation, a custom prolongation operator involving local solves on coarse cells, and an additive patchwise relaxation on each
level. We present 3D simulations with over one billion degrees of freedom with robust performance from Reynolds number 10 to 5000.

Employing the usual multilevel Monte Carlo estimator, we introduce a framework for estimating the solutions of SDEs by an Euler-Maruyama scheme. By considering the expected value of such solutions, we produce simulations using approximately normal random variables, and recover the estimate from the exact normal distribution by use of a multilevel correction, leading to faster simulations without loss of accuracy. We will also highlight this concept in the framework of reduced precision and vectorised computations.

Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. It has been studied extensively for many different systems but a full description is still lacking. Here, we provide a systematic analysis of the wrinkling of a thin hyperelastic film over a substrate in plane strain using stream functions. For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire system. We perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects.

Crime is a major risk to society’s well-being, particularly in cities, and yet the scientific literature lacks a comprehensive statistical characterization of crime that could uncover some of the mechanisms behind such pervasive social phenomenon. Evidence of nonlinear scaling of urban indicators in cities, such as wages and serious crime, has motivated the understanding of cities as complex systems—a perspective that offers insights into resources limits and sustainability, but usually without examining the details of indicators. Notably, since the nineteenth century, criminal activities have been known not to occur uniformly within a city. Crime concentrates in such way that most of the offenses take place in few regions of the city. However, though this concentration is confirmed by different studies, the absence of broad examinations of the characteristics of crime concentration hinders not only the comprehension of crime dynamics but also the proposal of sounding counter-measures. Here, we developed a framework to characterize crime concentration which splits cities into regions with the same population size. We used disaggregated criminal data from 25 locations in the U.S. and the U.K. which include offenses in places spanning from 2 to 15 years of data. Our results confirmed that crime concentrates regardless of city and revealed that the level of concentration does not scale with city size. We found that distribution of crime in a city can be approximated by a power-law distribution with exponent α that depends on the type of crime. In particular, our results showed that thefts tend to concentrate more than robberies, and robberies more than burglaries. Though criminal activities present regularities of concentration, we found that criminal ranks have the tendency to change continuously over time. Such features support the perspective of crime as a complex system which demands analyses and evolving urban policies covering the city as a whole. 

 I will introduce two obstructions for a rational homology 3-sphere to smoothly bound a rational homology 4-ball- one coming from Donaldson's theorem on intersection forms of definite 4-manifolds, and the other coming from correction terms in Heegaard Floer homology. If L is a nonunimodular definite lattice, then using a theorem of Elkies we will show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives a rephrasing of the obstruction coming from Donaldson's theorem. Furthermore, from this perspective it is easy to see that if the obstruction to bounding a rational homology ball coming from Heegaard Floer correction terms vanishes, then (under some mild hypotheses) the obstruction from Donaldson's theorem vanishes too.

The Prandtl equation was derived in 1904 by Ludwig Prandtl in order to describe the behavior of fluids with small viscosity around a solid obstacle. Over the past decades, several results of ill-posedness in Sobolev spaces have been proved for this equation. As a consequence, it is natural to look for more sophisticated boundary layer models, that describe the coupling with the outer Euler flow at a higher order. Unfortunately, these models do not always display better mathematical properties, as I will explain in this talk. This is a joint work with Helge Dietert, David Gérard-Varet and Frédéric Marbach.

In this talk I will articulate and contextualize the following sequence of results.

The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group.
Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories.  
In this Morita category, this algebra acts on the category of n-categories -- this action is given by adjoining adjoints to n-categories. 

This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.  

In semimartingale optimal transport problem, the functional to be minimized can be considered as a “stochastic action”, which is the expectationof a “stochastic Lagrangian” in terms of differential semimartingale characteristics. Therefore it would be natural to apply variational calculus approach to characterize the minimizers. R. Lassalle and A.B. Cruzeiro have used this approach to establish a stochastic Euler-Lagrangian condition for semimartingale optimal transport by perturbing the drift terms. Motivated by their work, we want to perform the same type of calculus for martingale optimal transport problem. In particular, instead of only considering perturbations in the drift terms, we try to find a nice variational family for volatility,and then obtain the stochastic Euler-Lagrangian condition for martingale laws. In the first part of this talk we will mention some basic results regarding the existence of minimizers in semimartingale optimal transport problem. In the second part, we will introduce Lassalle and Cruzeiro’s  work, and give a simple example related to this topic, where the variational family is induced by time-changes; and then we will introduce some potential problems that are needed to be solved.

We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger,  $L^{2+\epsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.  

Amplituhedra are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed as a geometric construction encoding scattering amplitudes in the four-dimensional maximally supersymmetric Yang-Mills theory, they are mathematically interesting objects on their own. In my talk I strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. I focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional
conjugates. I show how the Jeffrey-Kirwan residue prescription allows to extract the correct amplituhedron canonical differential form in all these cases. Notably, this also naturally exposes the rich combinatorial structures of amplituhedra, such as their regular triangulations

This will be the final mathematrix meeting for the term and we will be discussing Implicit Bias. In short, Implicit Bias is to do with perceptions and judgements we unconsciously make about people based on preconceptions we have about certain appearances, background or other characteristics. Even if we are not aware of making these judgements, they can affect our actions and decisions none-the-less. For a slightly longer introduction about this topic and how it can relate to academia, we suggest reading the following article: http://science.sciencemag.org/content/352/6289/1067.full

In this session we hope to explain more about what implicit bias is, how it might affect us, and discuss ways to avoid implicit bias and make ourselves and others more aware of it.

Everyone is welcome! Monday, 1300-1400, Quillen Room (N3.12), with lunch provided.

 I will describe recent progress in the study of scattering amplitudes in gauge theory and gravity at loop level, using the formalism of the scattering equations. The scattering equations relate the kinematics of the scattering of massless particles to the moduli space of the sphere. Underpinned by ambitwistor string theory, this formalism provides new insights into the relation between tree-level and loop-level contributions to scattering amplitudes. In this talk, I will describe results up to two loops on how loop integrands can be constructed as forward-limits of trees. One application is the loop-level understanding of the colour-kinematics duality, a symmetry of perturbative gauge theory which relates it to perturbative gravity.

Are you teaching intercollegiate classes or tutorials this term? Would you like to explore inclusive teaching strategies that could help all students make the most of your sessions? In this interactive workshop, we'll explore strategies that have been found effective. This will be a self-contained session, but will also be a good introduction to the "Developing Learning and Teaching" course offered by MPLS for graduate students and early career researchers. The session will be led by Vicky Neale (Mathematics) and Delia O'Rourke (Oxford Learning Institute). 
 

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. The talk will be given by Dr Richard Earl who chairs Projects Committee. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the  workload, presenting mathematics, structuring the dissertation and creating a narrative, providing references and avoiding plagiarism.

In 1973, Serre observed that the Hecke eigenvalues of Eisenstein series can be p-adically interpolated. In other words, Eisenstein series can be viewed as specializations of a p-adic family parametrized by the weight. The notion of p-adic variations of modular forms was later generalized by Hida to include families of ordinary cuspforms. In 1998, Coleman and Mazur defined the eigencurve, a rigid analytic space classifying much more general p-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is well-understood at points corresponding to cuspforms of weight k ≥ 2, while the weight one case is far more intricate.

In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations. Via this Galois-theoretic method, we obtain the q-expansion of some non-classical overconvergent forms in terms of p-adic logarithms of p-units in certain number field. Finally, we will explain how these calculations suggest a different approach to the Gross-Stark conjecture.

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation. Starting with a discussion on the intrinsic structure of images and their mathematical representation, in this talk we will learn about variational models for image analysis and their connection to partial differential equations, and go all the way to the challenges of their mathematical analysis as well as the hurdles for solving these - typically non-smooth - models computationally. The talk is furnished with applications of the introduced models to image de-noising, motion estimation and segmentation, as well as their use in biomedical image reconstruction such as it appears in magnetic resonance imaging.

In this talk I will present two recent findings concerning the preconditioning of elliptic problems. The first result concerns preconditioning of elliptic problems with variable coefficient K by an inverse Laplacian. Here we show that there is a close relationship between the eigenvalues of the preconditioned system and K. 
The second results concern the problem on mixed form where K approaches zero. Here, we show a uniform inf-sup condition and corresponding robust preconditioning. 

An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

I will discuss the basics of normal surface theory, and how they were used to give an algorithm for deciding whether a given diagram represents the unknot. This version is primarily based on Haken's work, with simplifications from Schubert and Jaco-Oertel.
 

What does abstract nonsense (category theory) have to do with the apparently opposite proverbial concreteness of physics? In this talk I will try to convey the importance of understanding physical theories from a compositional and structural perspective, where the fundamental logic of interaction between systems becomes the real protagonist. Firstly, we will see how different classes of symmetric monoidal categories can be used to model physical processes in a very natural and intuitive way. We will then explore the claim that category theory is not only useful in providing a unified framework, but it can be also used to perfect and modify preexistent models. In this direction, I will show how the introduction of a trace in the symmetric monoidal category describing QIT can be used to talk about quantum interactions induced by cyclic causal relationships.

I will discuss joint work with Gismatullin, Halupczok, and Simonetta on the following problem: given a henselian valued field of characteristic 0, possibly equipped with analytic structure (in the sense stemming originally from Denef and van den Dries), describe the possibilities for a definable group G in the valued field sort which is definably almost simple, that is, has no proper infinite definable normal subgroups. We also have results for an algebraically closed valued field K in characteristic p, but assuming also that the group is a definable subgroup of GL(n, K).

The (total) Steenrod square is a ring homomorphism from the cohomology of a topological space to the Z/2-equivariant cohomology of this space, with the trivial Z/2-action. Given a closed monotone symplectic manifold, one can define a deformed notion of the Steenrod square for quantum cohomology, which will not in general be a ring homomorphism, and prove some properties of this operation that are analogous to properties of the classical Steenrod square. We will then link this, in a more general setting, to a definition by Seidel of a similar operation on Floer cohomology.
 

In this talk I will introduce a new mathematical model for the computational modelling of the active contraction of cardiac tissue using stress-assisted conductivity as the main mechanism for mechanoelectrical feedback. The coupling variable is the Kirchhoff stress and so the equations of hyperelasticity are written in mixed form and a suitable finite element formulation is proposed. Next I will introduce a simplified version of the coupled system, focusing on its analysis in terms of solvability and stability of continuous and discrete mixed-primal formulations, and the convergence of these methods will be illustrated through two numerical tests.

The extremal number ${\rm ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [0,2]$ is realisable if there exists a graph $F$ with ${\rm ex}(n , F) = \Theta(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers $1$ and $2$.

We discuss some progress on the conjecture of Erdős and Simonovits. First, we show that $2 - \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b \equiv \pm 1 ~({\rm mod}\:a)$. This includes all previously known ones, and gives infinitely many limit points $2-\frac{1}{m}$ in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

This is joint work with Jaehoon Kim and Hong Liu.

The celebrated localisation theorem of Beilinson-Bernstein asserts that there is an equivalence between representations of a Lie algebra and modules over the sheaf of differential operators on the corresponding flag variety. In this talk we discuss certain analogues of this result in various contexts. Namely, there is a localisation theorem for quantum groups due to Backelin and Kremnizer and, more recently, Ardakov and Wadsley also proved a localisation theorem working with certain completed enveloping algebras of p-adic Lie algebras. We then explain how to combine the ideas involved in these results to construct
a p-adic analytic quantum flag variety and a category of D-modules on it, and we show that the global section functor on these D-modules yields an equivalence of categories.

In petroleum reservoir simulation, the standard preconditioner is a two-stage process which involves solving a restricted pressure system with AMG. Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not incorporate heat diffusion or the effects of temperature changes on fluid flow through viscosity and density. We seek to develop preconditioners which consider this cross-coupling between pressure and temperature. In order to study the effects of both pressure and temperature on fluid and heat flow, we first consider a model of non-isothermal single phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Then, we extend this method for multiphase flow as a two-stage preconditioner.

We present a self-contained construction of the Euclidean $\Phi^4$ quantum
field theory on $\mathbb{R}^3$ based on PDE arguments. More precisely, we
consider an approximation of the stochastic quantization equation on
$\mathbb{R}^3$ defined on a periodic lattice of mesh size $\varepsilon$ and
side length $M$. We introduce an energy method and prove tightness of the
corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow
\infty$. We show that every limit point satisfies reflection positivity,
translation invariance and nontriviality (i.e. non-Gaussianity). Our
argument applies to arbitrary positive coupling constant and also to
multicomponent models with $O(N)$ symmetry. Joint work with Massimiliano
Gubinelli.

Disease transmission and rumour spreading are ubiquitous in social and technological networks. In this talk, we will present our last results on the modelling of rumour and disease spreading in multilayer networks.  We will derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks in a multiplex network. Moreover, we will introduce a model of epidemic spreading with awareness, where the disease and information are propagated in different layers with different time scales. We will show that the time scale determines whether the information awareness is beneficial or not to the disease spreading. 

In recent years it has been discovered that also non-linear, degenerate equations like the $p$-Poisson equation $$ -\mathrm{div}(A(\nabla u))= - \mathrm{div} (|\nabla u|^{{p-2}}\nabla u)= -{\rm div} F$$ allow for optimal regularity. This equation has similarities to the one of power-law fluids. In particular, the non-linear mapping $F \mapsto A(\nabla u)$ satisfies surprisingly the linear, optimal estimate $\|A(\nabla u)\|_X \le c\, \|F\|_X$ for several choices of spaces $X$. In particular, this estimate holds for Lebesgue spaces $L^q$ (with $q \geq p'$), spaces of bounded mean oscillations and Holder spaces$C^{0,\alpha}$ (for some $\alpha>0$).

In this talk we show that we can extend this theory to Sobolev and Besov spaces of (almost) one derivative. Our result are restricted to the case of the plane, since we use complex analysis in our proof. Moreover, we are restricted to the super-linear case $p \geq 2$, since the result fails $p < 2$. Joint work with Anna Kh. Balci, Markus Weimar.

I will present a construction of large families of singularity-free stationary solutions of Einstein equations, for a large class of matter models including vacuum, with a negative cosmological constant. The solutions, which are of course real-valued Lorentzian metrics, are determined by a set of free data at conformal infinity, and the construction proceeds through elliptic equations for complex-valued tensor fields. One thus obtains infinite dimensional families of both strictly stationary spacetimes and black hole spacetimes.

In this talk I will discuss recent joint work with Dominik Gruber where 
we find a reasonable model for random (infinite) Burnside groups, 
building on earlier tools developed by Coulon and Coulon-Gruber.

The free Burnside group with rank r and exponent n is defined to be the 
quotient of a free group of rank r by the normal subgroup generated by 
all elements of the form g^n; quotients of such groups are called 
Burnside groups.  In 1902, Burnside asked whether any such groups could 
be infinite, but it wasn't until the 1960s that Novikov and Adian showed 
that indeed this was the case for all large enough odd n, with later 
important developments by Ol'shanski, Ivanov, Lysenok and others.

In a different direction, when Gromov developed the theory of hyperbolic 
groups in the 1980s and 90s, he observed that random quotients of free 
groups have interesting properties: depending on exactly how one chooses 
the number and length of relations one can typically gets hyperbolic 
groups, and these groups are infinite as long as not too many relations 
are chosen, and exhibit other interesting behaviour.  But one could 
equally well consider what happens if one takes random quotients of 
other free objects, such as free Burnside groups, and that is what we 
will discuss.
 

I will talk about R^n valued random processes driven by a "noise", which is generated by a deterministic dynamical system, randomness coming from the choice of the initial condition.

Such processes were considered by D.Kelly and I.Melbourne.I will present our joint work with I.Chevyrev, P.Friz, I.Melbourne and H.Zhang, where we consider the noise with long term memory. We prove convergence to solution of a stochastic differential equation which is, depending on the noise, driven by either a Brownian motion (optimizing the assumptions of Kelly-Melbourne) or a Lévy process.Our work is made possible by recent progress in rough path theory for càdlàg paths in p-variation topology.

The goal of the talk will be to introduce a class of functions that answer a question in topology, can be computed via analytic methods more common in the theory of dynamical systems, and in the end turn out to enjoy beautiful modular properties of the type first observed by Ramanujan. If time permits, we will discuss connections with vertex algebras and physics of BPS states which play an important role, but will be hidden "under the hood" in much of the talk.

We consider the problem of optimally hedging a portfolio of derivatives in a scenario based discrete-time market with transaction costs. Risk-preferences are specified in terms of a convex risk-measure. Such a framework has suffered from numerical intractability up until recently, but this has changed thanks to technological advances: using hedging strategies built from neural networks and machine learning optimization techniques, optimal hedging strategies can be approximated efficiently, as shown by the numerical study and some theoretical results presented in this talk (based on joint work with Hans Bühler, Ben Wood and Josef Teichmann).