Past

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 external-relations@maths.ox.ac.uk 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 external-relations@maths.ox.ac.uk 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.