Past

Knot theory investigates the many ways of embedding a circle into the three-dimensional sphere. The study of these embeddings is not only important for understanding three-dimensional manifolds, but is also intimately related to many new and surprising phenomena appearing in dimension four. I will discuss how four-dimensional interpretations of some invariants can help us understand surfaces that bound a given link (embedding of several disjoint circles).

The Ring Learning With Errors problem (RLWE) comes in various forms. Vanilla RLWE is the decision dual-RLWE variant, consisting in distinguishing from uniform a distribution depending on a secret belonging to the dual OK^vee of the ring of integers OK of a specified number field K. In primal-RLWE, the secret instead belongs to OK. Both decision dual-RLWE and primal-RLWE enjoy search counterparts. Also widely used is (search/decision) Polynomial Learning With Errors (PLWE), which is not defined using a ring of integers OK of a number field K but a polynomial ring Z[x]/f for a monic irreducible f in Z[x]. We show that there exist reductions between all of these six problems that incur limited parameter losses. More precisely: we prove that the (decision/search) dual to primal reduction from Lyubashevsky et al. [EUROCRYPT 2010] and Peikert [SCN 2016] can be implemented with a small error rate growth for all rings (the resulting reduction is nonuniform polynomial time); we extend it to polynomial-time reductions between (decision/search) primal RLWE and PLWE that work for a family of polynomials f that is exponentially large as a function of deg f (the resulting reduction is also non-uniform polynomial time); and we exploit the recent technique from Peikert et al. [STOC 2017] to obtain a search to decision reduction for RLWE. The reductions incur error rate increases that depend on intrinsic quantities related to K and f.

Based on joint work with Miruna Roșca and Damien Stehlé.

With Gianluca Paolini (in preparation), we constructed, using a variant on the Hrushovski dimension function, for every k ≥ 3, 2^µ families of strongly minimal Steiner k systems. We study the mathematical properties of these counterexamples to Zilber’s trichotomy conjecture rather than thinking of them as merely exotic examples. In particular the long study of finite Steiner systems in reflected in results that depend on the block size k. A quasigroup is a structure with a binary operation such that for each equation xy = z the values of two of the variables determines a unique value for the third. The new Steiner 3-systems are bi-interpretable with strongly minimal Steiner quasigroups. For k > 3, we show the pure k-Steiner systems have ‘essentially unary definable closure’ and do not interpret a quasigroup. But we show that for q a prime power the Steiner q systems can be interpreted into specific sorts of quasigroups, block algebras. We extend the notion of an (a, b)-cycle graph arising in the study of finite and infinite Stein triple systems (e.g Cameron-Webb) by introducing what we call the (a, b)-path graph of a block algebra. We exhibit theories of strongly minimal block algebras where all (a, b)-paths are infinite and others in which all are finite only in the prime model. We show how to obtain combinatorial properties (e.g. 2-transitivity) by the either varying the basic collection of finite partial Steiner systems or modifying the µ function which ensures strong minimality

The Erdos-Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs.

1. A theorem of R\"odl asserts that if an n-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to 0 or 1) of size \Omega(n). We prove that if a 3-uniform hypergraph is non-universal then it contains an almost homogeneous set of size \Omega(log n). An example of R\"odl from 1986 shows that this bound is tight.

2. Let R_r(t) denote the size of the largest non-universal r-graph G so that neither G nor its complement contain a complete r-partite subgraph with parts of size t. We prove an Erd\H{o}s--Hajnal-type stepping-up lemma, showing how to transform a lower bound for R_r(t) into a lower bound for R_{r+1}(t). As an application of this lemma, we improve a bound of Conlon-Fox-Sudakov by showing that R_3(t) \geq t^{ct).

Joint work with M. Amir and A. Shapira

Analytic continuation is ill-posed, but becomes merely ill-conditioned (though with an infinite condition number) if it is known that the function in question is bounded in a given region of the complex plane.
This classical, seemingly theoretical subject has many connections with numerical practice.  One argument indicates that if one tracks an analytic function from z=1 around a branch point at z=0 and back to z=1 again by a Weierstrass chain of disks, the number of accurate digits is divided by about exp(2 pi e) ~= 26,000,000.

The problem of authorship attribution (AA) involves matching a text of unknown authorship with its creator, found among a pool of candidate authors. In this work, we examine in detail authorship attribution methods that rely on networks of function words to detect an “authorial fingerprint” of literary works. Previous studies interpreted these word adjacency networks (WANs) as Markov chains, giving transition rates between function words, and they compared them using information-theoretic measures. Here, we apply a variety of network flow-based tools, such as role-based similarity and community detection, to perform a direct comparison of the WANs. These tools reveal an interesting relation between communities of function words and grammatical categories. Moreover, we propose two new criteria for attribution based on the comparison of connectivity patterns and the similarity of network partitions. The results are positive, but importantly, we observe that the attribution context is an important limiting factor that is often overlooked in the field's literature. Furthermore, we give important new directions that deserve further consideration.

We study solutions to stationary Navier-Stokes system in two dimensional exterior domains, namely, existence of these solutions and their asymptotical behavior. The talk is based on the recent joint papers with K. Pileckas and R. Russo where the uniform boundedness and uniform convergence at infinity for arbitrary solution with finite Dirichlet integral were established. Here  no restrictions on smallness of fluxes are assumed, etc.  In the proofs we develop the ideas of the classical papers of Gilbarg & H.F. Weinberger (Ann. Scuola Norm.Pisa 1978) and Amick (Acta Math. 1988).

We address the problem of pricing and hedging general exotic derivatives. We study this problem in the scenario when one has access to limited price data of other exotic derivatives. In this presentation I explore a nonparametric approach to pricing exotic payoffs using market prices of other exotic derivatives using signatures.

The lower central series of a group $G$ is defined by $\gamma_1=G$ and $\gamma_n = [G,\gamma_{n-1}]$. The "dimension series", introduced by Magnus, is defined using the group algebra over the integers: $\delta_n = \{g: g-1\text{ belongs to the $n$-th power of the augmentation ideal}\}$.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has $\delta_n\ge\gamma_n$, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with $\delta_4/\gamma_4$ cyclic of order 2. On the positive side, Sjogren showed that $\delta_n/\gamma_n$ is always a torsion group, of exponent bounded by a function of $n$. Furthermore, it was believed (and falsely proven by Gupta) that only $2$-torsion may occur.
In joint work with Roman Mikhailov, we prove however that for every prime $p$ there is a group with $p$-torsion in some quotient $\delta_n/\gamma_n$.
Even more interestingly, I will show that the dimension quotient $\delta_n/gamma_n$ is related to the difference between homotopy and homology: our construction is fundamentally based on the order-$p$ element in the homotopy group $\pi_{2p}(S^2)$ due to Serre.
 

Abstract: Consider a gaussian field f on R^2 and a level l. One can define a random coloring of the plane by coloring a point x in black if f(x)>-l and in white otherwise. The topology of this coloring is interesting in many respects. One can study the "small scale" topology by counting connected components with fixed topology, or study the "large scale" topology by considering black crossings of large rectangles. I will present results involving these quantities.

Einstein metrics are fixed points (up to scaling) of Hamilton's Ricci flow. A natural question to ask is whether a given metric is stable in the sense that the flow returns to the Einstein metric under a small perturbation. I'll give a brief survey of this area focussing on the case when the Einstein constant is positive. An interesting class of metrics where this question is not completely resolved are the compact symmetric spaces. I'll report on some recent progress with Tommy Murphy and James Waldron where we have been able to use a criterion due to Kroencke to show the Kaehler-Einstein metric on some Grassmannians and the bi-invariant metric on the Lie group G_2 are unstable.

Get to know the Mathematrix events of this term!

We were a bit too late with ordering food, so the usual sandwich lunch will only start from week 2. However, there may be some small snacks.

The zeta-function of a manifold varies with the parameters and may be evaluated in terms of the periods. For a one parameter family of CY manifolds, the periods satisfy a single 4th order differential equation. Thus there is a straight and, it turns out, readily computable path that leads from a differential operator to a zeta-function. Especially interesting are the specialisations to singular manifolds, for which the zeta-function manifests modular behaviour. We are also able to find, from the zeta function, attractor points. These correspond to special values of the parameter for which there exists a 10D spacetime for which the 6D corresponds to a CY manifold and the 4D spacetime corresponds to an extremal supersymmetric black hole. These attractor CY manifolds are believed to have special number theoretic properties. This is joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.

The 23rd Annual Meeting of the SIAM UKIE Section will take place on Friday 11th January 2019 at the Mathematical Institute at the University of Oxford.

The meeting will feature five invited speakers covering a broad range of industrial and applied mathematics: 

- Lisa Fauci, Tulane University, Incoming SIAM President
- Des Higham, Strathclyde University 
- Carola-Bibiane Schoenlieb (IMA sponsored speaker), University of Cambridge 
- Kirk Soodhalter, Trinity College Dublin 
- Konstantinos Zygalakis, University of Edinburgh 

There will also be a poster session, open to PhD students and postdocs. Travel support will be available for PhD students with an accepted poster presentation, and Best Poster prizes will be awarded. 

All talks will take place in room L3 in the Andrew Wiles Building (Mathematical Institute, University of Oxford). 

Programme 
09:30 - 10:00 Registration, tea/coffee 
10:00 - 10:15 Welcome 
10:15 - 11:00 Des Higham: Our Friends are Cooler than Us 
11:00 - 11:45 Lisa Fauci: Complex dynamics of fibers in flow at the microscale 
11:45 - 12:15 Poster Blitz 
12:15 - 13:30 Lunch and Poster session 
13:30 - 14:00 SIAM UKIE Business Meeting, open to all 
14:00 - 14:45 Kirk Soodhalter: Augmented Arnoldi-Tikhonov Methods for Ill-posed Problems 
14:45 - 15:30 Konstantinos Zygalakis: Explicit stabilised Runge-Kutta methods and their application to Bayesian inverse problems 
15:30 - 16:00 Tea/coffee 
16:00 - 16:45 Carola-Bibiane Schoenlieb (IMA sponsored speaker): Variational models and partial differential equations for mathematical imaging 
16:45 - 17:00 Poster prize announcement

With topics ranging from prime numbers to the lottery, from lemmings to bending balls like Beckham, Professor Marcus du Sautoy will provide an entertaining and, perhaps, unexpected approach to explain how mathematics can be used to predict the future. 

We are delighted to announce our first Oxford Mathematics Midlands Public Lecture to take place at Solihull School on 9th January 2019. 

Please email @email to register

Watch live:
https://facebook.com/OxfordMathematics
https://livestream.com/oxuni/du-Sautoy

We are very grateful to Solihull School for hosting this lecture.

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Hannah Fry takes us on a tour of the good, the bad and the downright ugly of the algorithms that surround us. Are they really an improvement on the humans they are replacing?

Hannah Fry is a lecturer in the Mathematics of Cities at the Centre for Advanced Spatial Analysis at UCL. She is also a well-respected broadcaster and the author of several books including the recently published 'Hello World: How to be Human in the Age of the Machine.'

5.00pm-6.00pm, Mathematical Institute, Oxford

Please email @email to register

Watch live:
https://facebook.com/OxfordMathematics
https://livestream.com/oxuni/ChristmasLecture2018

The Oxford Mathematics Public Lectures are generously supported by XTX Markets

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.

Following recent papers by Nima Arkani-Hamed.

Authors:

Anne Balter and Antoon Pelsser

Models can be wrong and recognising their limitations is important in financial and economic decision making under uncertainty. Robust strategies, which are least sensitive to perturbations of the underlying model, take uncertainty into account. Interpreting

the explicit set of alternative models surrounding the baseline model has been difficult so far. We specify alternative models by a stochastic change of probability measure and derive a quantitative bound on the uncertainty set. We find an explicit ex ante relation

between the choice parameter k, which is the radius of the uncertainty set, and the Type I and II error probabilities on the statistical test that is hypothetically performed to investigate whether the model specification could be rejected at the future test horizon.

The hypothetical test is constructed to obtain all alternative models that cannot be distinguished from the baseline model with sufficient power. Moreover, we also link the ambiguity bound, which is now a function of interpretable variables, to numerical

values on several divergence measures. Finally, we illustrate the methodology on a robust investment problem and identify how the robustness multiplier can be numerically interpreted by ascribing meaning to the amount of ambiguity.

The Oberwolfach Research Institute for Mathematics (Mathematisches Forschungsinstitut Oberwolfach/MFO) was founded in late 1944 by the Freiburg mathematician Wilhelm Süss (1895-1958) as the „National Institute for Mathematics“. In the 1950s and 1960s the MFO developed into an increasingly international conference centre.

The aim of my project is to analyse the history of the MFO as it institutionally changed from the National Institute for Mathematics with a wide, but standard range of responsibilities, to an international social infrastructure for research completely new in the framework of German academia. The project focusses on the evolvement of the institutional identity of the MFO between 1944 and the early 1960s, namely the development and importance of the MFO’s scientific programme (workshops, team work, Bourbaki) and the instruments of research employed (library, workshops) as well as the corresponding strategies to safeguard the MFO’s existence (for instance under the wings of the Max-Planck-Society). In particular, three aspects are key to the project, namely the analyses of the historical processes of (1) the development and shaping of the MFO’s workshop activities, (2) the (complex) institutional safeguarding of the MFO, and (3) the role the MFO played for the re-internationalisation of mathematics in Germany. Thus the project opens a window on topics of more general relevance in the history of science such as the complexity of science funding and the re-internationalisation of the sciences in the early years of the Federal Republic of Germany.

Joint work with Zsolt Vizi (Bolyai Institute, University of Szeged, Hungary), Istvan Kiss (Department
of Mathematics, University of Sussex, United Kingdom)

Pairwise models have been proven to be a flexible framework for analytical approximations
of stochastic epidemic processes on networks that are in many situations much more accurate
than mean field compartmental models. The non-Markovian aspects of disease transmission
are undoubtedly important, but very challenging to incorporate them into both numerical
stochastic simulations and analytical investigations. Here we present a generalization of
pairwise models to non-Markovian epidemics on networks. For the case of infectious periods
of fixed length, the resulting pairwise model is a system of delay differential equations, which
shows excellent agreement with results based on the explicit stochastic simulations. For more
general distribution classes (uniform, gamma, lognormal etc.) the resulting models are PDEs
that can be transformed into systems of integro-differential equations. We derive pairwise
reproduction numbers and relations for the final epidemic size, and initiate a systematic
study of the impact of the shape of the particular distributions of recovery times on how
the time evolution of the disease dynamics play out.

We discuss shock reflection problem for compressible gas dynamics, von Neumann conjectures on transition between regular and Mach reflections, and existence of regular reflection solutions for potential flow equation. Then we will talk about recent results on uniqueness and stability of regular reflection solutions for potential flow equation in a natural class of self-similar solutions. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, and prove uniqueness by a version of method of continuity. A property of solutions important for the proof of uniqueness is convexity of the free boundary. 

This talk is based on joint works with G.-Q. Chen and W. Xiang.

The geometry of the moduli space of 4d  N=2  moduli spaces, and in particular of their Coulomb branches (CBs), is very constrained. In this talk I will show that through its careful study, we can learn general and somewhat surprising lessons about the properties of N=2  super conformal field theories (SCFTs). Specifically I will show that we can prove that the scaling dimension of CB coordinates, and thus of the corresponding operator at the SCFT fixed point, has to be rational and it has a rank-dependent maximum value and that in general the moduli spaces of N=2 SCFTs can have metric singularities as well as complex structure singularities. 

Finally I will outline how we can explicitly perform a classification of geometries of N>=3 SCFTs and carry out the program up to rank-2. The results are surprising and exciting in many ways.

Massless Quantum Field Theories can be described perturbatively by chiral worldsheet models - the so-called Ambitwistor Strings. In contrast to conventional string theory, where loop amplitudes are calculated from higher genus Riemann surfaces, loop amplitudes in the ambitwistor string localise on the non-separating boundary of the moduli space. I will describe the resulting framework for QFT amplitudes from (nodal) Riemann spheres, building up from tree-level to two-loop amplitudes.
 

Robert Timms

Title: Multiscale modelling of lithium-ion batteries

Lithium-ion batteries are one of the most widely used technologies for energy storage, with applications ranging from portable electronics to electric vehicles. Due to their popularity, there is a continued interest in the development of mathematical models of lithium-ion batteries. These models encompass various levels of complexity, which may be suitable to aid with design, or for real-time monitoring of performance. After a brief introduction to lithium-ion batteries, I will discuss some of the modelling efforts undertaken here at Oxford and within the wider battery modelling community.
 

Jan Vonk

Title: Singular moduli for real quadratic fields

At the 1900 ICM, David Hilbert posed a series of problems, of which the 12th remains completely open today. I will discuss how to solve this problem in the simplest open case, by considering certain exotic (so called p-adic) metrics on the set of numbers, and using its concomitant theories of analysis and geometry.
 

The accurate modelling of geophysical flows often requires information which is difficult to measure and therefore poorly quantified. Such information may relate to the fluid properties or an unknown boundary condition, for example. The premise of this talk is that when the flow is bounded by a free surface, the deformation of this free surface contains useful information which can be used to infer such unknown quantities. The increasing availability of free surface data through remote sensing using drones and satellites provides the impetus to develop new mathematical methods and numerical tools to interpret the signature embedded in the free surface deformation. This talk will explore two recent examples drawn from glaciology and inspired from volcanology for which free surface data was successfully used to reconstruct an unknown field.

Would you like to meet some of your fellow students, and some graduate students and postdocs, in an informal and relaxed atmosphere, while building your communication skills?  In this Friday@2 session, you'll be able to play a selection of board games, meet new people, and practise working together.  What better way to spend the final Friday afternoon of term?!  We'll play the games in the south Mezzanine area of the Andrew Wiles Building, outside L3.

Switch-like and oscillatory dynamical systems are widely observed in biology. We investigate the simplest biological switch that is composed of a single molecule that can be autocatalytically converted between two opposing activity forms. We test how this simple network can keep its switching behaviour under perturbations in the system. We show that this molecule can work as a robust bistable system, even for alterations in the reactions that drive the switching between various conformations. We propose that this single molecule system could work as a primitive biological sensor and show by steady state analysis of a mathematical model of the system that it could switch between possible states for changes in environmental signals. Particularly, we show that a single molecule phosphorylation-dephosphorylation switch could work as a nucleotide or energy sensor. We also notice that a given set of reductions in the reaction network can lead to the emergence of oscillatory behaviour. We propose that evolution could have converted this switch into a single molecule oscillator, which could have been used as a primitive timekeeper. I will discuss how the structure of the simplest known circadian clock regulatory system, found in cyanobacteria, resembles the proposed single molecule oscillator. Besides, we speculate if such minimal systems could have existed in an RNA world. I will also present how the regulatory network of the cell cycle could have emerged from this system and what are the consequences of this possible evolution from a single antagonistic kinase-phosphatase network.

The social sciences are undergoing a profound shift as new data and methods emerge to study human behaviour. These data offer tremendous opportunity but also mathematical and statistical challenges that the field has yet to fully understand. This talk will give an overview of social data science research faculty are undertaking at the Oxford Internet Institute, a multidisciplinary department of the University. Projects include studying the flow of information across languages, the role of political bots, and volatility in public attention.

The celebrated Lang-Vojta Conjecture predicts degeneracy of S-integral points on varieties of log general type defined over number fields. It admits a geometric analogue over function fields, where stronger results have been obtained applying a method developed by Corvaja and Zannier. In this talk, we present a recent result for non-isotrivial surfaces over function fields dominating a two-dimensional torus. This extends Corvaja and Zannier’s result in the isotrivial case and it is based on a refinement of gcd estimates for polynomials evaluated at S-units. This is a joint work with A. Turchet.

Motion at the microscale is a fascinating field visualizing non-equilibrium behaviour of matter. Evolution has optimized the ability of microscale swimming on different length scales from tedpoles, to sperm and bacteria. A constant metabolic energy input is required to achieve active propulsion which means these systems are obeying the laws imposed by a low Reynolds number. Several strategies - including topography1 , chemotaxis or rheotaxis2 - have been used to reliably determine the path of active particles. Curiously, many of these strategies can be recognized as analogues of approaches employed by nature and are found in biological microswimmers.

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Bacteria attached to the metal caps of Janus particles3

However, natural microswimmers are not limited to being exemplary systems on the way to artificial micromotion. They certainly enable us to observe how nature overcame problems such as a lack of inertia, but natural microswimmers also offer the possibility to couple them to artificial microobjects to create biohybrid systems. Our group currently explores different coupling strategies and to create so called ‘BacteriaBots’.4

1 J. Simmchen, J. Katuri, W. E. Uspal, M. N. Popescu, M. Tasinkevych, and S. Sánchez, Nat. Commun., 2016, 7, 10598.

2 J. Katuri, W. E. Uspal, J. Simmchen, A. Miguel-López and S. Sánchez, Sci. Adv., , DOI:10.1126/sciadv.aao1755.

3 M. M. Stanton, J. Simmchen, X. Ma, A. Miguel-Lopez, S. Sánchez, Adv. Mater. Interfaces, DOI:10.1002/admi.201500505.

4 J. Bastos-Arrieta, A. Revilla-Guarinos, W. E. Uspal and J. Simmchen, Front. Robot. AI, 2018, 5, 97.

The usual finite dimensional Grassmannians are well known to be classifying spaces for vector bundles. It is maybe a less known fact that one has certain natural connections on the Stiefel bundles over them, which also have a universality property. I will show how these connections are constructed and explain how this viewpoint can be used to rediscover Chern-Weil theory. Finally, we will see how a certain stabilized version of this, called the restricted Grassmannian, admits a similar construction, which can be used to show that it is a smooth classifying space for differential K-theory.

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.