Past

Abstract: The universal algorithm is a Turing machine program that can in principle enumerate any finite sequence of numbers, if run in the right model of PA, and furthermore, can always enumerate any desired extension of that sequence in a suitable end-extension of that model. The universal finite set is a set-theoretic analogue, a locally verifiable definition that can in principle define any finite set, in the right model of set theory, and can always define any desired finite extension of that set in a suitable top-extension of that model. Recent work has uncovered a $\Sigma_1$-definable version that works with respect to end-extensions. I shall give an account of all three results, which have a parallel form, and describe applications to the model theory of arithmetic and set theory. Post questions and commentary on my blog at http://jdh.hamkins.org/parallels-in-universality-oxford-math-logic-semi…;

Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at least $\eta d$. This says that if we choose a random vertex $x\in A$ and a random neighbour $y$ of $x$, then the probability that $y\in A$ is at least $\eta$. In recent joint work with Tim Gowers, we were aiming to obtain a qualitative description of closed subsets of the Cayley graph $\Gamma$ whose vertex set is $\mathbb{F}_2^{n_1}\otimes \dots \otimes \mathbb{F}_2^{n_d}$ with two vertices joined by an edge if their difference is of the form $u_1\otimes \cdots \otimes u_d$. For the matrix case (that is, when $d=2$), such a description was obtained by Khot, Minzer and Safra, a breakthrough that completed the proof of the 2-to-2 conjecture. We have formulated a conjecture for higher dimensions, and proved it in an important special case. In this talk, I will sketch this proof. Also, we have identified a statement about $\eta$-closed sets in Cayley graphs on arbitrary finite Abelian groups that implies the conjecture and can be considered as a "highly asymmetric Balog-Szemerédi-Gowers theorem" when it holds. I will present an example to show that this statement is not true for an arbitrary Cayley graph. It remains to decide whether the statement can be proved for the Cayley graph $\Gamma$.

The next generation emissive displays including quantum dot LED(QLED) and organic LED(OLED) could be efficiently manufactured by inkjet printing, where nano-scale droplets are injected in banked substrate and after evaporation they leave layers of thin film that forms pixels of a display. This novel manufacturing method would greatly reduce cost and improve reliability. However, it is observed in practice that the deposit becomes much thicker near the bank edge and emission is faint there. This motivated the project and in this talk, we will mathematically model the phenomeno, understand its origin and investigate ways of making more uniform deposit by means of simulation.

Abstract regular polytopes are finite quotients of Coxeter complexes
with string diagram, satisfying a natural intersection property, see
e.g. [MMS2002]. They arise in a number of geometric and group-theoretic
contexts. The first class of such objects, beyond the
well-understood examples coming from finite and affine Coxeter groups,
are locally toroidal cases, e.g.  extensions of quotients of the affine
F_4 complex [3,3,4,3].  In 1996 P.McMullen & E.Schulte constructed a
number of examples of locally toroidal abstract regular polytopes of
type [3,3,4,3,3], and conjectured completeness of their list. We
construct counterexamples to the conjecture using a Y-shaped
presentation for a subgroup of the Monster, and discuss various
related questions.
 

When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise the efficient sampling of white noise realizations can be challenging. In this talk we present a novel sampling technique that can be used to efficiently compute white noise samples in a finite element and multilevel Monte Carlo (MLMC) setting.
After discretization, the action of white noise on a test function yields a Gaussian vector with the FEM mass matrix as covariance. Sampling such a vector requires an expensive Cholesky factorization and for this reason P0 representations, for which the mass matrix is diagonal, are generally preferred in the literature. This however has other disadvantages. In this talk we introduce an alternative factorization that is naturally parallelizable and has linear cost and memory complexity (in the number of mesh elements).
Moreover, in a MLMC framework the white noise samples must be coupled between subsequent levels so as to respect the telescoping sum. We show how our technique can be used to enforce this coupling even in the case in which the hierarchy is non-nested via a supermesh construction. We conclude the talk with numerical experiments that demonstrate the efficacy of our method. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest. In a MLMC setting, a good coupling is enforced and the telescoping sum is respected.
 

Many protein interaction databases provide confidence scores based on the experimental evidence underpinning each in- teraction. The databases recommend that protein interac- tion networks (PINs) are built by thresholding on these scores. We demonstrate that varying the score threshold can re- sult in PINs with significantly different topologies. We ar- gue that if a node metric is to be useful for extracting bio- logical signal, it should induce similar node rankings across PINs obtained at different thresholds. We propose three measures—rank continuity, identifiability, and instability— to test for threshold robustness. We apply these to a set of twenty-five metrics of which we identify four: number of edges in the step-1 ego network, the leave-one-out dif- ference in average redundancy, average number of edges in the step-1 ego network, and natural connectivity, as robust across medium-high confidence thresholds. Our measures show good agreement across PINs from different species and data sources. However, analysis of synthetically gen- erated scored networks shows that robustness results are context-specific, and depend both on network topology and on how scores are placed across network edges. 

Non-abelian gauge theories underly particle physics, including collision processes at particle accelerators. Recently, quantum scattering probabilities in gauge theories have been shown to be closely related to their counterparts in gravity theories, by the so-called double copy. This suggests a deep relationship between two very different areas of physics, and may lead to new insights into quantum gravity, as well as novel computational methods. This talk will review the double copy for amplitudes, before discussing how it may be extended to describe exact classical solutions such as black holes. Finally, I will discuss hints that the double copy may extend beyond perturbation theory. 

Suppose that we want to minimise the L-infinity norm of the Laplacian of a function (or a similar quantity) under Dirichlet boundary conditions. This is a convex, but not strictly convex variational problem. Nevertheless, it turns out that it has a unique solution, which is characterised by a system of PDEs. The behaviour is thus quite different from the better-known first order problems going back to Aronsson. This is joint work with N. Katzourakis (Reading).
 

In an article published in 2009, Dave Benson described, for a finite group $G$, the mod $p$ homology of the space $\Omega(BG^\wedge_p)$ --- the loop space of the $p$-completion of $BG$ --- in purely algebraic terms. In joint work with Carles Broto and Ran Levi, we have tried to better understand Benson's result by generalizing it. We showed that when $\mathcal{C}$ is a small category, $|\mathcal{C}|$ is its geometric realization, $R$ is a commutative ring, and $|\mathcal{C}|^+_R$ is a plus construction of $|\mathcal{C}|$ with respect to homology with coefficients in $R$, then $H_*(\Omega(|\mathcal{C}|^+_R);R)$ is the homology any chain complex of projective $R\mathcal{C}$-modules that satisfies certain conditions. Benson's theorem is then the special case where $\mathcal{C}$ is the category associated to a finite group $G$ and $R=F_p$, so that $p$-completion is a special case of the plus construction.
 

We study several important fine properties for the family of fractional Brownian motions with Hurst parameter H under the (p,r)-capacity on classical Wiener space introduced by Malliavin. We regard fractional Brownian motions as Wiener functionals via the integral representation discovered by Decreusefond and \"{U}st\"{u}nel, and show non differentiability, modulus of continuity, law of iterated Logarithm(LIL) and self-avoiding properties of fractional Brownian motion sample paths using Malliavin calculus as well as the tools developed in the previous work by Fukushima, Takeda and etc. for Brownian motion case.

We develop a method to study the implied volatility for exotic options and volatility derivatives with European payoffs such as VIX options. Our approach, based on Malliavin calculus techniques, allows us to describe the properties of the at-the-money implied volatility (ATMI) in terms of the Malliavin derivatives of the underlying process. More precisely, we study the short-time behaviour of the ATMI level and skew. As an application, we describe the short-term behavior of the ATMI of VIX and realized variance options in terms of the Hurst parameter of the model, and most importantly we describe the class of volatility processes that generate a positive skew for the VIX implied volatility. In addition, we find that our ATMI asymptotic formulae perform very well even for large maturities. Several numerical examples are provided to support our theoretical results.  

This is a welcome to everyone who is interested in discussing and learning more about topics relating to life in academia and issues faced by minorities. We will tell you more about Mathematrix and the events upcoming in the term, as well as discussing ideas for future terms.

All staff, ECRs and postgrad students are invited to join. The lunches are free, relaxed and informal, and people may come and go as they please.
 

I will talk about the Witten index of supersymmetric quantum mechanics obtained from 3d gauge theories compacted on a Riemann surface. In particular, I will show that the twisted indices of 3d N=4 theories compute enumerative invariants of the moduli space, which can be identified as a space of quasi-maps to the Higgs branch. I will also discuss 3d mirror symmetry in this context which provides a non-trivial relation between a pair of generating functions of the invariants.

The focus of this talk is the regularity theory for time-harmonic Maxwell's equations with complex anisotropic parameters. By using the Helmholtz decomposition of the fields, we show how the problem can be completely reduced to a regularity question for elliptic equations, for which classical results may be applied. In particular, we prove the Hölder regularity of solutions under minimal assumptions on the coefficients.
 

For a reductive group $ G $, Steinberg established a map from the Weyl group to nilpotent $ G $-orbits using momentmaps on double flag varieties.  In particular, in the case of the general linear group, he re-interpreted the Robinson-Schensted correspondence between the permutations and pairs of standard tableaux of the same shape in terms of product of complete flags.

We generalize his theory to the case of symmetric pairs $ (G, K) $, and obtained two different maps.  In the case where $ (G, K) = (\GL_{2n}, \GL_n \times \GL_n) $, one of the maps is a generalized Steinberg map, which induces a generalization of the RS correspondence for degenerate permutations.  The other is an exotic moment map, which maps degenerate permutations to signed Young diagrams, i.e., $ K $-orbits in the Cartan space $ (\lie{g}/\lie{k})^* $.

We explain geometric background of the theory and combinatorial procedures which produces the above mentioned maps.

This is an on-going joint work with Lucas Fresse.
 

Oxford Mathematics and the Clay Mathematics Institute Public Lectures

Roger Penrose - Eschermatics
24 September 2018 - 5.30pm

Roger Penrose’s work has ranged across many aspects of mathematics and its applications from his influential work on gravitational collapse to his work on quantum gravity. However, Roger has long had an interest in and influence on the visual arts and their connections to mathematics, most notably in his collaboration with Dutch graphic artist M.C. Escher. In this lecture he will use Escher’s work to illustrate and explain important mathematical ideas.

Oxford Mathematics is hosting this special event in its Public Lecture series during the conference to celebrate the 20th Anniversary of the foundation of the Clay Mathematics Institute. After the lecture Roger will be presented with the Clay Award for the Dissemination of Mathematical Knowledge.

5.30-6.30pm, Mathematical Institute, Oxford

Please email @email to register.

Watch live:

https://www.facebook.com/OxfordMathematics
https://livestream.com/oxuni/Penrose

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

The derivation of first-order nonlinear transport PDEs via interacting particles subject only to deterministic forces is crucial in the socio-biological sciences and in the real world applications (e.g. vehicular traffic, pedestrian movements), as it provides a rigorous justification to a "continuum" description in situations more naturally described by a discrete approach. This talk will collect recent results on the derivation of entropy solutions to scalar conservation laws (arising e.g. in traffic flow) as many particle limits of "follow-the-leader"-type ODEs, including extensions to the case with Dirichlet boundary conditions and to the Hughes model for pedestrian movements (the results involve S. Fagioli, M. D. Rosini, G. Russo). I will then describe a recent extension of this approach to nonlocal transport equations with a "nonlinear mobility" modelling prevention of overcrowding for high densities (in collaboration with S. Fagioli and E. Radici). 

In this lecture Persi Diaconis will take a look at some of our most primitive images of chance - flipping a coin, rolling a roulette wheel and shuffling cards - and via a little bit of mathematics (and a smidgen of physics) show that sometimes things are not very random at all. Indeed chance is sometimes confused with frequency and this confusion caries over to a confusion between chance and evidence. All of which explains our wild misuse of probability and statistical models.

Persi Diaconis is world-renowned for his study of mathematical problems involving randomness and randomisation. He is the co-author of 'Ten Great Ideas about Chance (2017) and is the Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University. 

Please email @email to register.

Watch live:

https://www.facebook.com/OxfordMathematics
https://livestream.com/oxuni/PersiDiaconis

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

We consider the behaviors of global solutions to the initial value problems for the multi-dimensional Navier-Stokes(Euler)-Fokker-Planck equations. It is shown that due to the micro-macro coupling effects of relaxation damping type, the sound wave type propagation of this NSFP or EFP system for two-phase fluids is observed with the wave speed determined by the two-phase fluids. This phenomena can not be observed for the pure Fokker-Planck equation and the Navier-Stokes(Euler) equation with frictional damping.

Geometric Invariant Theory is a central tool in the construction of moduli spaces, and shares the property ubiquitous among such tools that certain so-called 'unstable' objects must be excluded if the moduli space is to be well behaved. However, instability in GIT is a structured phenomenon: after making a choice of a certain invariant inner product, one has the HKKN stratification of the parameter space which, morally, sorts the objects according to how unstable they are. I will explain how one can use recent results of Berczi-Doran-Hawes-Kirwan in Non-Reductive GIT to perform quotients of these unstable strata as well, extending the classifications given by classical moduli spaces. This can be carried out, at least in principle, for any moduli problem that can be posed using GIT, and I will discuss two examples in particular: unstable (i.e. singular) curves, and coherent sheaves of fixed Harder-Narasimhan type. The latter of these is joint work with Gergely Berczi, Victoria Hoskins and Frances Kirwan.
 

I will explain how the definition of Bridgeland stability condition on a triangulated category C can be generalised to allow for massless objects. This allows one to construct a partial compactification of the stability space Stab(C) in which each `boundary stratum' is related to Stab(C/N) for a thick subcategory N of C, and has a neighbourhood which fibres over (an open subset of) Stab(N). This is joint work with Nathan Broomhead, David Pauksztello, and David Ploog.
 

Enumerative  invariants since 1995 are defined as integrals of cohomology classes over a particular homology class, called the virtual fundamental class. When there is a torus action, the virtual fundamental class is localized to the fixed points and this turned out to be the most effective technique for computation of the virtual integrals so far. About 10 years ago, Jun Li and I discovered that when there is a cosection of the obstruction sheaf, the virtual fundamental class is localized to the zero locus of the cosection. This also turned out to be quite useful for computation of Gromov-Witten invariants and more. In this talk, I will discuss a generalization of the cosection localization to real classes which provides us with a purely topological theory of Fan-Jarvis-Ruan-Witten invariants (quantum singularity theory) as well as some GLSM invariants. Based on a joint work with Jun Li at arXiv:1806.00116.