Thu, 05 Dec 2019

14:00 - 15:00
C4

Algebraic K-theory

Nadav Gropper
Abstract

In the talk we will define higher K-groups, and explain some of their relations to number theory

Thu, 05 Dec 2019

12:00 - 13:00
L2

Hölder regularity for nonlocal double phase equations

Giampiero Palatucci
(Università di Parma)
Abstract

We present some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient a = a(·, ·). The model case is driven by the following nonlocal double phase operator,

$$\int \frac{|u(x) − u(y)|^{p−2} (u(x) − u(y))} {|x − y|^{n+sp}} dy+ \int a(x, y) \frac{|u(x) − u(y)|^{ q−2} (u(x) − u(y))} {|x − y|^{n+tq}} dy$$

where $q ≥ p$ and $a(·, ·) = 0$. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are Hölder continuous, whereas similar sharp results for the classical local case do require a to be Hölder continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.

Thu, 05 Dec 2019

11:30 - 12:30
C4

Universally defining finitely generated subrings of global fields

Nicolas Daans
(Antwerpen)
Abstract

   It is a long-standing open problem whether the ring of integers Z has an existential first-order definition in Q, the field of rational numbers. A few years ago, Jochen Koenigsmann proved that Z has a universal first-order definition in Q, building on earlier work by Bjorn Poonen. This result was later generalised to number fields by Jennifer Park and to global function fields of odd characteristic by Kirsten Eisenträger and Travis Morrison, who used classical machinery from number theory and class field theory related to the behaviour of quaternion algebras over global and local fields.


   In this talk, I will sketch a variation on the techniques used to obtain the aforementioned results. It allows for a relatively short and uniform treatment of global fields of all characteristics that is significantly less dependent on class field theory. Instead, a central role is played by Hilbert's Reciprocity Law for quaternion algebras. I will conclude with an example of a non-global set-up where the existence of a reciprocity law similarly yields universal definitions of certain subrings.

Wed, 04 Dec 2019
16:00
C1

Double branched cover of knotoids, f-distance and entanglement in proteins.

Agnese Barbensi
(University of Oxford)
Abstract

Knotoids are a generalisation of knots that deals with open curves. In the past few years, they’ve been extensively used to classify entanglement in proteins. Through a double branched cover construction, we prove a 1-1 correspondence between knotoids and strongly invertible knots. We characterise forbidden moves between knotoids in terms of equivariant band attachments between strongly invertible knots, and in terms of crossing changes between theta-curves. Finally, we present some applications to the study of the topology of proteins. This is based on joint works with D.Buck, H.A.Harrington, M.Lackenby and with D. Goundaroulis.

Wed, 04 Dec 2019
11:00
N3.12

Random Groups

David Hume
(University of Oxford)
Abstract

Finitely presented groups are a natural algebraic generalisation of the collection of finite groups. Unlike the finite case there is almost no hope of any kind of classification.

The goal of random groups is therefore to understand the properties of the "typical" finitely presented group. I will present a couple of models for random groups and survey some of the main theorems and open questions in the area, demonstrating surprising correlations between these probabilistic models, geometry and analysis.

Tue, 03 Dec 2019

15:45 - 16:45
L4

Combinatorial Lefschetz theorems beyond positivity

Karim Adiprasito
(Hebrew University)
Abstract

The hard Lefschetz theorem is a fundamental statement about the symmetry of the cohomology of algebraic varieties. In nearly all cases that we systematically understand it, it comes with a geometric meaning, often in form of Hodge structures and signature data for the Hodge-Riemann bilinear form.

Nevertheless, similar to the role the standard conjectures play in number theory, several intriguing combinatorial problems can be reduced to hard Lefschetz properties, though in extreme cases without much geometric meaning, lacking any existence of, for instance,  an ample cone to do Hodge theory with.

I will present a way to prove the hard Lefschetz theorem in such a situation, by introducing biased pairing and perturbation theory for intersection rings. The price we pay is that the underlying variety, in a precise sense, has itself to be sufficiently generic. For instance, we shall see that any quasismooth, but perhaps nonprojective toric variety can be "perturbed" to a toric variety with the same equivariant cohomology, and that has the hard Lefschetz property.

Finally, I will discuss how this applies to prove some interesting theorems in geometry, topology and combinatorics. In particular, we shall see a generalization of a classical result due to Descartes and Euler: We prove that if a simplicial complex embeds into euclidean 2d-space, the number of d-simplices in it can exceed the number of (d-1)-simplices by a factor of at most d+2.

Tue, 03 Dec 2019
14:30
L1

Estimation of ODE models with discretization error quantification

Takeru Matsuda
(University of Tokyo)
Abstract

We consider estimation of ordinary differential equation (ODE) models from noisy observations. For this problem, one conventional approach is to fit numerical solutions (e.g., Euler, Runge–Kutta) of ODEs to data. However, such a method does not account for the discretization error in numerical solutions and has limited estimation accuracy. In this study, we develop an estimation method that quantifies the discretization error based on data. The key idea is to model the discretization error as random variables and estimate their variance simultaneously with the ODE parameter. The proposed method has the form of iteratively reweighted least squares, where the discretization error variance is updated with the isotonic regression algorithm and the ODE parameter is updated by solving a weighted least squares problem using the adjoint system. Experimental results demonstrate that the proposed method improves estimation accuracy by accounting for the discretization error in a data-driven manner. This is a joint work with Yuto Miyatake (Osaka University).

Tue, 03 Dec 2019
14:15
L4

Deformation of a Howe duality

Marcelo De Martino
(Oxford University)
Abstract

In this talk, I will report about a joint work with D. Ciubotaru, in which we investigate the Dunkl version of the classical Howe-duality (O(k),spo(2|2)). Similar Fischer-type decompositions were studied before in the works of Ben-Said, Brackx, De Bie, De Schepper, Eelbode, Orsted, Soucek and Somberg for other Howe-dual pairs. Our work builds on the notion of a Dirac operator for Drinfeld algebras introduced by Ciubotaru, which was inspired by the analogous theory for Lie algebras, as well as the work of Cheng and Wang on classical Howe dualities.

Tue, 03 Dec 2019

14:00 - 15:00
L6

Characterisation of quasirandom permutations by a pattern sum

Yanitsa Pehova
(University of Warwick)
Further Information

We say that a sequence $\{\Pi_i\}$ of permutations is quasirandom if, for each $k\geq 2$ and each $\sigma\in S_k$, the probability that a uniformly chosen $k$-set of entries of $\Pi_i$ induces $\sigma$ tends to $1/k!$ as $i$ tends to infinity. It is known that a much weaker condition already forces $\{\Pi_i\}$ to be quasirandom; namely, if the above property holds for all $\sigma\in S_4$. We further weaken this condition by exhibiting sets $S\subseteq S_4$, such that if a randomly chosen $k$-set of entries of $\Pi_i$ induces an element of $S$ with probability tending to $|S|/24$, then $\{\Pi_i\}$ is quasirandom. Moreover, we are able to completely characterise the sets $S$ with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight. 
This is joint work with Timothy Chan, Daniel Kráľ, Jon Noel, Maryam Sharifzadeh and Jan Volec.

Tue, 03 Dec 2019
14:00
L1

On symmetrizing the ultraspherical spectral method for self-adjoint problems

Mikael Slevinsky
(University of Manitoba)
Abstract

A mechanism is described to symmetrize the ultraspherical spectral method for self-adjoint problems. The resulting discretizations are symmetric and banded. An algorithm is presented for an adaptive spectral decomposition of self-adjoint operators. Several applications are explored to demonstrate the properties of the symmetrizer and the adaptive spectral decomposition.

 

Tue, 03 Dec 2019

12:45 - 14:00
C5

Computing multiple local minima of topology optimization problems with second-order methods

Ioannis Papadopoulos
((Oxford University))
Abstract


Topology optimisation finds the optimal material distribution of a fluid or solid in a domain, subject to PDE and volume constraints. There are many formulations and we opt for the density approach which results in a PDE, volume and inequality constrained, non-convex, infinite-dimensional optimisation problem without a priori knowledge of a good initial guess. Such problems can exhibit many local minima or even no minima. In practice, heuristics are used to obtain the global minimum, but these can fail even in the simplest of cases. In this talk, we will present an algorithm that solves such problems and systematically discovers as many of these local minima as possible along the way.

Tue, 03 Dec 2019
12:00
L4

Lie polynomials and a Penrose transform for the double copy

Lionel Mason
(Oxford)
Abstract

This talk will explain how Lie polynomials underpin the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides).  ABHY have recently shown that Lie polynomials arise naturally also in the geometry of the space K_n of momentum invariants, Mandelstams, and can be expressed in the space of n-3-forms dual to certain associahedral (n-3)-planes. They also arise in the moduli space M_{0,n} of n points on a Riemann sphere up to Mobius transformations in the n-3-dimensional homology.  The talk goes on to give a natural correspondendence between K_n and the cotangent bundle of M_{0.n} through which the relationships of some of these structures can be expressed.  This in particular gives a natural framework for expressing the CHY and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories and goes some way to expressing their double copy relations.   This is part of joint work in progress with Hadleigh Frost.

Tue, 03 Dec 2019

12:00 - 13:00
C1

Network construction methodology based on distance correlation without exogenous information

Javier Pardo Díaz
(Department of Statistics)
Abstract

We aim to generate gene coexpression networks from gene expression data. In our networks, nodes represent genes and edges depict high positive correlation in their expression across different samples. Methods based on Pearson correlation are the most commonly used to generate gene coexpression networks. We propose the use of distance correlation as an effective alternative to Pearson correlation when constructing gene expression networks. Our methodology pipeline includes a thresholding step which allows us to discriminate which pairs of genes are coexpressed. We select the value of the threshold parameter by studying the stability of the generated network, rather than relying on exogenous biological information known a priori.

Tue, 03 Dec 2019

11:00 - 12:00
L6

Babbage's mechanical notation

Adrian Johnstone
(Royal Holloway University of London)
Abstract

Charles Babbage (1791–1871) was Lucasian Professor of mathematics in Cambridge from 1828–1839. He displayed a fertile curiosity that led him to study many contemporary processes and problems in a way which emphasised an analytic, data driven view of life.

In popular culture Babbage has been celebrated as an anachronistic Victorian engineer. In reality, Babbage is best understood as a figure rooted in the enlightenment, who had substantially completed his core investigations into 'mechanisation of thought' by the mid 1830s: he is thus an anachronistic Georgian: the construction of his first difference engine design is contemporary with the earliest public railways in Britain.

A fundamental question that must strike anybody who examines Babbage's precocious designs is: how could one individual working alone have synthesised a workable computer design, designing an object whose complexity of behaviour so far exceeded that of contemporary machines that it would not be matched for over a hundred years?

We shall explore the extent to which the answer lies in the techniques Babbage developed to reason about complex systems. His Notation which shows the geometry, timing, causal chains and the abstract components of his machines, has a direct parallel in the Hardware Description Languages developed since 1975 to aid the design of large scale electronics. In this presentation, we shall provide a basic tutorial on Babbage's notation showing how his concepts of 'pieces' and 'working points' effectively build a graph in which both parts and their interactions are represented by nodes, with edges between part-nodes and interaction-nodes denoting ownership, and edges between interaction-nodes denoting the transmission of forces between individual assemblies within a machine. We shall give examples from Babbage's Difference Engine 2 for which a complete set of notations was drawn in 1849, and compare them to a design of similar complexity specified in 1987 using the Inmos HDL.

Mon, 02 Dec 2019

17:30 - 18:30
L1

Carlo Rovelli - Spin networks: the quantum structure of spacetime from Penrose's intuition to Loop Quantum Gravity

Carlo Rovelli
(Université d'Aix-Marseille)
Further Information

Oxford Mathematics Public Lectures- The Roger Penrose Lecture

Carlo Rovelli  - Spin networks: the quantum structure of spacetime from Penrose's intuition to Loop Quantum Gravity

Monday 2 December 2019

In developing the mathematical description of quantum spacetime, Loop Quantum Gravity stumbled upon a curious mathematical structure: graphs labelled by spins. This turned out to be precisely the structure of quantum space suggested by Roger Penrose two decades earlier, just on the basis of his intuition. Today these graphs with spin, called "spin networks" have become a common tool to explore the quantum properties of gravity. In this talk Carlo will tell this beautiful story and illustrate the current role of spin networks in the efforts to understand quantum gravity.

Carlo Rovelli is a Professor in the Centre de Physique Théorique de Luminy of Aix-Marseille Université where he works mainly in the field of quantum gravity and  is a founder of loop quantum gravity theory. His popular-science book 'Seven Brief Lessons on Physics' has been translated into 41 languages and has sold over a million copies worldwide.

5.30pm-6.30pm, Mathematical Institute, Oxford

Please email @email to register.

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

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

Mon, 02 Dec 2019

16:00 - 17:00
C1

What the L! The surprising world of L-functions

George Robinson
(Oxford)
Abstract

L-functions have become a vital part of modern number theory over the past century, allowing comparisons between arithmetic objects with seemingly very different properties. In the first part of this talk, I will give an overview of where they arise, their properties, and the mathematics that has developed in order to understand them. In the second part, I will give a sketch of the beautiful result of Herbrand-Ribet concerning the arithmetic interpretations of certain special values of the Riemann zeta function, the prototypical example of an L-function.

Mon, 02 Dec 2019

16:00 - 17:00
L4

Dislocation patterns at zero and finite temperature in the Ariza-Ortiz model

Florian Theil
(Warwick)
Abstract


The AO-model describes crystalline solids in the presence of defects like dislocation lines. We demonstrate that the model supports low-energy structures like grains and determine for simple geometries the grain boundary energy density. At small misorientation angles we recover the well-known Read-Shockley law. Due to the atomistic nature of the model it is possible to consider the the Boltzmann-Gibbs distribution at non-zero temperature. Using ideas by Froehlich and Spencer we prove rigorously the presence of long-range order if the temperature is sufficiently small.
 

Mon, 02 Dec 2019

15:45 - 16:45
L3

Areas-of-areas on Hall trees generate the shuffle algebra

CRIS SALVI
(University of Oxford)
Abstract

We consider the coordinate-iterated-integral as an algebraic product on the shuffle algebra, called the (right) half-shuffle product. Its anti-symmetrization defines the biproduct  area(.,.), interpretable as the signed-area between two real-valued coordinate paths. We consider specific sets of binary, rooted trees known as Hall sets. These set have a complex combinatorial structure, which can be almost entirely circumvented by introducing the equivalent notion of Lazard sets. Using analytic results from dynamical systems and algebraic results from the theory of Lie algebras, we show that shuffle-polynomials in areas-of-areas on Hall trees generate the shuffle algebra.

Mon, 02 Dec 2019
15:45
L6

A cellular decomposition of the Fulton Mac Pherson operad

Paolo Salvatore
(University of Rome `Tor Vergata')
Abstract

We construct a cellular decomposition of the
Axelrod-Singer-Fulton-MacPherson compactification of the configuration
spaces in the plane, that is compatible with the operad composition.
Cells are indexed by trees with bi-coloured edges, and vertices are labelled by 
cells of the cacti operad. This answers positively a conjecture stated in 
2000 by Kontsevich and Soibelman.

Mon, 02 Dec 2019

14:15 - 15:15
L3

Asset Prices in Segmented and Integrated Markets

PAOLO GUASONI
(University of Dublin)
Abstract

This paper evaluates the effect of market integration on prices and welfare, in a model where two Lucas trees grow in separate regions with similar investors. We find equilibrium asset price dynamics and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is zero or negative before integration, but significantly positive afterwards, explaining some effects commonly associated with financialization.

Mon, 02 Dec 2019

14:15 - 15:15
L4

Cohomology of non-reductive GIT quotients and hyperbolicity

Frances Kirwan
(Oxford)
Abstract

The aim of this talk is to describe joint work with Gergely Berczi using a recent extension to non-reductive actions of geometric invariant theory, and its links with moment maps in symplectic geometry, to study hyperbolicity of generic hypersurfaces in a projective space. Using intersection theory for non-reductive GIT quotients applied to  compactifications of bundles of invariant jet differentials over complex manifolds leads to a proof of the Green-Griffiths-Lang conjecture for a generic projective hypersurface of dimension n whose degree is greater than n^6. A recent result of Riedl and Yang then implies the Kobayashi conjecture for generic hypersurfaces of degree greater than (2n-1)^6.

Mon, 02 Dec 2019
12:45
L2

CFT and black holes

Manuela Kulaxizi
(Trinity College, Dublin)
Abstract

We consider CFTs with large gap in the spectrum of operators and a large number of degrees of freedom (large central charge). We analytically study a Heavy-Heavy-Light-Light correlation function, where Heavy, refers to an operator with conformal dimension which scales like the central charge and Light, refers to an operator whose dimension is of order unity in the large central charge limit. In certain regimes, the correlation function can be examined analytically leading to very simple and suggestive expressions.

Sun, 01 Dec 2019

17:30 - 18:30
L1

Bach, the Universe and Everything - The Creativity Code

Marcus du Sautoy and the Orchestra of the Age of Enlightenment
((Oxford University))
Further Information

The second in our fascinating collaboration with the Orchestra of the Age of Enlightenment (OAE) and Music at Oxford combines the muscial intelligence of the eighteenth century with the artificial intelligence of the twenty-first. Come along and hear the beauty of Bach's Nun komm, der Heiden Heiland (Now come, Saviour of the Gentiles) and the modern beauty of machine learning which may itself be the musical choice of audiences in 300 years' time.

The OAE provide the music (you even get to join in), Marcus delivers the sermon. Maths and Music; saying everything.

Book here