This week is a women's only week in which we are joined by the Mirzakhani society, the society for undergraduate women in maths, for lunch.
I will review some recent progress on D=4, N=2 superconformal field theories in what has come to be known as "Class-S". This is a huge class of (mostly non-Lagrangian) SCFTs, whose properties are encoded in the data of a punctured Riemann surface and a collection (one per puncture) of nilpotent orbits in an ADE Lie algebra.
How much do you know actually about the research that is going on across the department? The SIAM Student Chapter brings you a 3 minute thesis competition challenging a group of DPhil students to go head to head to explain their research in just 3 minutes with the aid of a single slide. This is the perfect opportunity to hear about a wide range of topics within applied mathematics, and to gain insight into the impact that mathematical research can have. The winner will be decided by a judging panel comprising Professors Helen Byrne, Jon Chapman, Patrick Farrell, and Christina Goldschmidt.
A matrix M of real numbers is called totally positive if every minor of M is nonnegative. This somewhat bizarre concept from linear algebra has surprising connections with analysis - notably polynomials and entire functions with real zeros, and the classical moment problem and continued fractions - as well as combinatorics. I will explain briefly some of these connections, and then introduce a generalization: a matrix M of polynomials (in some set of indeterminates) will be called coefficientwise totally positive if every minor of M is a polynomial with nonnegative coefficients. Also, a sequence (an)n≥0 of real numbers (or polynomials) will be called (coefficientwise) Hankel-totally positive if the Hankel matrix H = (ai+j)i,j ≥= 0 associated to (an) is (coefficientwise) totally positive. It turns out that many sequences of polynomials arising in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive; in some cases this can be proven using continued fractions, while in other cases it remains a conjecture.
The magma chamber - an underground vat of fluid magma that is tapped during volcanic eruptions - has been the foundation of models of volcanic eruptions for many decades and successfully explains many geological observations. However, geophysics has failed to image the postulated large magma chambers, and the chemistry and ages of crystals in erupted magmas indicate a more complicated history. New conceptual models depict subsurface magmatic systems as dominantly uneruptible crystalline networks with interstitial melt (mushes) extending deep into the Earth's crust to the mantle, containing lenses of potentially eruptible (low-crystallinity) magma. These lenses would commonly be less dense than the overlying mush and so Rayleigh Taylor instabilities should develop leading to ascent of blobs of magma unless the growth rate is sufficiently slow that other processes (e.g. solidification) dominate. The viscosity contrast between a buoyant layer and mush is typically extremely large; a consequence is that the horizontal dimension of a magma reservoir is commonly much less than the theoretical fastest growing wavelength assuming an infinite horizontal layer.
I will present laboratory experiments and linear stability analysis for low Reynolds number, laterally confined Rayleigh Taylor instabilities involving one layer that is much thinner and much less viscous than the other. I will then apply the results to magmatic systems, comparing timescales for development of the instability and the volumes of packets of rising melt generated, with the frequencies and sizes of volcanic eruptions. I will then discuss limitations of this work and outstanding fluid dynamical problems in this new paradigm of trans-crustal magma mush systems.
The Goldbach conjecture is a famous unsolved problem in mathematics. It asks whether every even number greater than or equal to 4 is the sum of two primes. I will discuss some of the history of the problem, explaining among other things why the answer is surely yes, and also why this appears to be very hard to prove.
Despite progress in understanding many aspects of malignancy, resistance to therapy is still a frequent occurrence. Recognised causes of this resistance include 1) intra-tumour heterogeneity resulting in selection of resistant clones, 2) redundancy and adaptability of gene signalling networks, and 3) a dynamic and protective microenvironment. I will discuss how these aspects influence each other, and then focus on the tumour microenvironment.
The tumour microenvironment comprises a heterogeneous, dynamic and highly interactive system of cancer and stromal cells. One of the key physiological and micro-environmental differences between tumour and normal tissues is the presence of hypoxia, which not only alters cell metabolism but also affects DNA damage repair and induces genomic instability. Moreover, emerging evidence is uncovering the potential role of multiple stroma cell types in protecting the tumour primary niche.
I will discuss our work on in silico cancer models, which is using genomic data from large clinical cohorts of individuals to provide new insights into the role of the tumour microenvironment in cancer progression and response to treatment. I will then discuss how this information can help to improve patient stratification and develop novel therapeutic strategies.
Harmonic map equations are an elliptic PDE system arising from the
minimisation problem of Dirichlet energies between two manifolds. In
this talk we present some some recent works concerning the symmetry
and stability of harmonic maps. We construct a new family of
''twisting'' examples of harmonic maps and discuss the existence,
uniqueness and regularity issues. In particular, we characterise of
singularities of minimising general axially symmetric harmonic maps,
and construct non-minimising general axially symmetric harmonic maps
with arbitrary 0- or 1-dimensional singular sets on the symmetry axis.
Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$
to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data, for
$p \geq 2$. The stability fails for $p <2$ due to Almgren--Lieb and
Mazowiecka--Strzelecki.
(Joint work with Prof. Robert M. Hardt.)
Over the past decade, and particularly over the past five years, research at the interface of topology and neuroscience has grown remarkably fast. In this talk I will briefly survey a few quite different applications of topology to neuroscience in which members of my lab have been involved over the past four years: the algebraic topology of brain structure and function, topological characterization and classification of neuron morphologies, and (if time allows) topological detection of network dynamics.
The Oxford-Princeton Workshops on Financial Mathematics & Stochastic Analysis have been held approximately every eighteen months since 2002, alternately in Princeton and Oxford. They bring together leading groups of researchers in, primarily, mathematical and computational finance from Oxford University and Princeton University to collaborate and interact. The series is organized by the Oxford Mathematical and Computational Finance Group, and at Princeton by the Department of Operations Research and Financial Engineering and the Bendheim Center for Finance.
Oxford Mathematics Public Lectures
Hooke Lecture
Michael Berry - Chasing the dragon: tidal bores in the UK and elsewhere
15 November 2018 - 5.15pm
In some of the world’s rivers, an incoming high tide can arrive as a smooth jump decorated by undulations, or as a breaking wave. The river reverses direction and flows upstream.
Understanding tidal bores involves
· analogies with tsunamis, rainbows, horizons in relativity, and ideas from quantum physics;
· the concept of a ‘minimal model’ in mathematical explanation;
· different ways in which different cultures describe the same thing;
· the first unification in fundamental physics.
Michael Berry is Emeritus Professor of Physics, H H Wills Physics Laboratory, University of Bristol
5.15pm, Mathematical Institute, Oxford
Please email @email to register.
Watch live:
https://www.facebook.com/OxfordMathematics
https://livestream.com/oxuni/Berry
Oxford Mathematics Public Lectures are generously supported by XTX Markets.
I will give an overview of some recent progress on potential automorphy results over CM fields, that is joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne. I will focus on explaining an application to the generalized Ramanujan-Petersson conjecture.
Lagrangian Floer homology has been used by Ozsvath and Szabo to define a package of three-manifold invariants known as Heegaard Floer homology. I will give an introduction to the topic.
Boundary layers control the transport of momentum, heat, solutes and other quantities between walls and the bulk of a flow. The Prandtl-Blasius boundary layer was the first quantitative example of a flow profile near a wall and could be derived by an asymptotic expansion of the Navier-Stokes equation. For higher flow speeds we have scaling arguments and models, but no derivation from the Navier-Stokes equation. The analysis of exact coherent structures in plane Couette flow reveals ingredients of such a more rigorous description of boundary layers. I will describe how exact coherent structures can be scaled to obtain self-similar structures on ever smaller scales as the Reynolds number increases.
A quasilinear approximation allows to combine the structures self-consistently to form boundary layers. Going beyond the quasilinear approximation will then open up new approaches for controlling and manipulating boundary layers.
Abstract: Akshay Venkatesh and his coauthors (Galatius, Harris, Prasanna) have recently introduced a derived Hecke algebra and a derived Galois deformation ring acting on the homology of an arithmetic group, say with p-adic coefficients. These actions account for the presence of the same system of eigenvalues simultaneously in various degrees. They have also formulated a conjecture describing a finer action of a motivic group which should preserve the rational structure $H^i(\Gamma,\Q)$. In this lecture we focus in the setting of classical modular forms of weight one, where the same systems of eigenvalues appear both in degree 0 and 1 of coherent cohomology of a modular curve, and the motivic group referred to above is generated by a Stark unit. In joint work with Darmon, Harris and Venkatesh, we exploit the Theta correspondence and higher Eisenstein elements to prove the conjecture for dihedral forms.
In many applications requiring the solution of a linear system Ax=b, the matrix A has been shown to have a low-rank property: its off-diagonal blocks have low numerical rank, i.e., they can be well approximated by matrices of small rank. Several matrix formats have been proposed to exploit this property depending on how the block partitioning of the matrix is computed.
In this talk, I will discuss the block low-rank (BLR) format, which partitions the matrix with a simple, flat 2D blocking. I will present the main characteristics of BLR matrices, in particular in terms of asymptotic complexity and parallel performance. I will then discuss some recent advances and ongoing research on BLR matrices: their multilevel extension, their use as preconditioners for iterative solvers, the error analysis of their factorization, and finally the use of fast matrix arithmetic to accelerate BLR matrix operations.
I will prove the 2d Biot-Savart law for the vorticity being an unbounded measure $\mu$, i.e. such that $\mu(\mathbb{R}^2)=\infty$, and show how can one infer some useful information concerning Kaden's spirals using it. Vorticities being unbounded measures appear naturally in the engineering literature as self-similar approximations of 2d Euler flows, see for instance Kaden's or Prandtl's spirals. Mathematicians are interested in such objects since they seem to be related to the questions of well-posedness of Delort's solutions of the 2d vortex sheet problem for the Euler equation. My talk is based on a common paper with K.Oleszkiewicz, M. Preisner and M. Szumanska.
Polycyclic groups either have polynomial growth, in which case they are virtually nilpotent, or exponential growth. I will give two interesting examples of "small" polycyclic groups which are extensions of $\mathbb{R}^2$ and the Heisenberg group by the integers, and attempt to justify the claim that they are small by sketching an argument that every exponential growth polycyclic group contains one of these.
Nets of lines are line arrangements satisfying very strict intersection conditions. We will see that nets can be defined in a very natural way in algebraic geometry, and, thanks to the strict intersection properties they satisfy, we will see that a lot can be said about classifying them over the complex numbers. Despite this, there are still basic unanswered questions about nets, which we will discuss.
A lot of physical processes are modelled by conservation laws (mass, momentum, energy, charge, ...) Because of natural symmetries, these conservation laws express often that some symmetric tensor is divergence-free, in the space-time variables. We extract from this structure a non-trivial information, whenever the tensor takes positive semi-definite values. The qualitative part is called Compensated Integrability, while the quantitative part is a generalized Gagliardo inequality.
In the first part, we shall present the theoretical analysis. The proofs of various versions involve deep results from the optimal transportation theory. Then we shall deduce new fundamental estimates for gases (Euler system, Boltzmann equation, Vlaov-Poisson equation).
One of the theorems will have been used before, during the Monday seminar (PDE Seminar 4pm Monday 12 November).
All graduate students, post-docs faculty and visitors are welcome to come to the lectures. If you aren't a member of the CDT please email @email to confirm that you will be attending.
During the talk I will present some new computational technique based on excursion theory for Markov processes. Some new results for classical processes like Bessel processes and reflected Brownian Motion will be shown. The most important point of presented applications will be the new insight into Hartman-Watson (HW) distributions. It turns out that excursion theory will enable us to deduce the simple connections of HW with a hyperbolic cosine of Brownian Motion.
I will explain how complex varieties which have asymptotically large intersections with finite grids can be seen to correspond to projective geometries, exploiting ideas of Hrushovski. I will describe how this leads to a precise characterisation of such varieties. Time permitting, I will discuss consequences for generalised sum-product estimates and connections to diophantine problems. This is joint work with Emmanuel Breuillard.
The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayley and Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.
This work determines an aim point selection strategy for players in order to improve their chances of winning at the classic darts game of 501. Although many previous studies have considered the problem of aim point selection in order to maximise the expected score a player can achieve, few have considered the more general strategical question of minimising the expected number of turns required for a player to finish. By casting the problem as a Markov decision process, a framework is derived for the identification of the optimal aim point for a player in an arbitrary game scenario.