Abstract: A hyperkähler manifold is a Riemannian manifold $(M,g)$ with three complex structures $I,J,K$ satisfying the quaternion relations, i.e. $I^2=J^2=K^2=IJK=-1$, and such that $(M,g)$ is Kähler with respect to each of them. I will describe a construction due to Kronheimer which gives such a structure on the cotangent bundle of any complex reductive group.
Let $L/K$ be an extension of number fields and let $J_L$ and $J_K$ be the associated groups of ideles. Using the diagonal embedding, we view $L^*$ and $K^*$ as subgroups of $J_L$ and $J_K$ respectively. The norm map $N: J_L\to J_K$ restricts to the usual field norm $N: L^*\to K^*$ on $L^*$. Thus, if an element of $K^*$ is a norm from $L^*$, then it is a norm from $J_L$. We say that the Hasse norm principle holds for $L/K$ if the converse holds, i.e. if every element of $K^*$ which is a norm from $J_L$ is in fact a norm from $L^*$.
The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of $K$ fail the Hasse norm principle? More generally, for an abelian group $G$, what proportion of extensions of $K$ with Galois group $G$ fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis.
This is joint work with Christopher Frei and Daniel Loughran.
Surfactants are chemicals that adsorb onto the air-liquid interface and lower the surface tension there. Non-uniformities in surfactant concentration result in surface tension gradients leading to a surface shear stress, known as a Marangoni stress. This stress, if sufficiently large, can influence the flow at the interface.
Surfactants are ubiquitous in many aspects of technology and industry to control the wetting properties of liquids due to their ability to modify surface tension. They are used in detergents, crop spraying, coating processes and oil recovery. Surfactants also occur naturally, for example in the mammalian lung. They reduce the surface tension within the liquid lining the airways, which assists in preventing the collapse of the smaller airways. In the lungs of premature infants, the quantity of surfactant produced is insufficient as the lungs are under- developed. This leads to a respiratory distress syndrome which is treated by Surfactant Replacement Therapy.
Motivated by this medical application, we theoretically investigate a model problem involving the spreading of a drop laden with an insoluble surfactant down an inclined and pre-wetted substrate. Our focus is in understanding the mechanisms behind a “fingering” instability observed experimentally during the spreading process. High-resolution numerics reveal a multi-region asymptotic wave-like structure of the spreading droplet. Approximate solutions for each region is then derived using asymptotic analysis. In particular, a quasi-steady similarity solution is obtained for the leading edge of the droplet. A linear stability analysis of this region shows that the base state is linearly unstable to long-wavelength perturbations. The Marangoni effect is shown to be the dominant driving mechanism behind this instability at small wavenumbers. A small wavenumber stability criterion is derived and it's implication on the onset of the fingering instability will be discussed.
I will describe a method to find negatively curved structures on some groups, by manipulating metrics on piecewise hyperbolic complexes. As an example, I will prove that hyperbolic limit groups are CAT(-1).
Torsion in homology are invariants that have received increasing attention over the last twenty years, by the work of Lück, Bergeron, Venkatesh and others. While there are various vanishing results, no one has found a finitely presented group where the torsion in the first homology is exponential over a normal chain with trivial intersection. On the other hand, conjecturally, every 3-manifold group should be an example.
A group is right angled if it can be generated by a list of infinite order elements, such that every element commutes with its neighbors. Many lattices in higher rank Lie groups (like SL(n,Z), n>2) are right angled. We prove that for a right angled group, the torsion in the first homology has subexponential growth for any Farber sequence of subgroups, in particular, any chain of normal subgroups with trivial intersection. We also exhibit right angled cocompact lattices in SL(n,R) (n>2), for which the Congruence Subgroup Property is not known. This is joint work with Nik Nikolov and Tsachik Gelander.
Efficient factorization or efficient computation of class
numbers would both suffice to break RSA. However the talk lies more in
computational number theory rather than in cryptography proper. We will
address two questions: (1) How quickly can one construct a factor table
for the numbers up to x?, and (2) How quickly can one do the same for the
class numbers (of imaginary quadratic fields)? Somewhat surprisingly, the
approach we describe for the second problem is motivated by the classical
Hardy-Littlewood method.
What are the national maths societies for? What can they do for us? What can we do for them?
Featuring representatives from the Institute of Mathematics and its Applications, the London Mathematical Society, the OR Society, and the Royal Statistical Society.
This talk will describe the basic properties of the de Rham algebra, which is a generalisation of the de Rham algebra over smooth schemes, which was introduced by L. Illusie in his monograph 'Complexe cotangent et déformations'.
We aim to determine how cells faithfully complete genome replication. Accurate and complete genome replication is essential for all life. A single DNA replication error in a single cell division can give rise to a genomic disorder. However, almost all experimental data are ensemble; collected from millions of cells. We used a combination of high-resolution, genomic-wide DNA replication data, mathematical modelling and single cell experiments to demonstrate that ensemble data mask the significant heterogeneity present within a cell population; see [1-4]. Therefore, the pattern of replication origin usage and dynamics of genome replication in individual cells remains largely unknown. We are now developing cutting-edge single molecule methods and allied mathematical models to determine the dynamics of genome replication at the DNA sequence level in normal and perturbed human cells.
[1] de Moura et al., 2010, Nucleic Acids Research, 38: 5623-5633
[2] Retkute et al, 2011, PRL, 107:068103
[3] Retkute et al, 2012, PRE, 86:031916
[4] Hawkins et al., 2013, Cell Reports, 5:1132-41
I will consider automorphism groups of countable structures acting continuously on compact spaces: the viewpoint of topological dynamics. A beautiful paper of Kechris, Pestov and Todorcevic makes a connection between this and the ‘structural Ramsey theory’ of Nesetril, Rodl and others in finite combinatorics. I will describe some results and questions in the area and say how the Hrushovski predimension constructions provide answers to some of these questions (but then raise more questions). This is joint work with Hubicka and Nesetril.
Abstract: Ricci solitons are genralizations of Einstein metrics which have become subject of much interest over the last decade. In this talk I will give a basic introduction to these metrics and discuss how to reformulate the Ricci soliton equation as a Hamiltonian system assuming some symmetry conditions. Using this approach we will construct explicit solutions to the soliton equation for manifolds of dimension 5.
The sub-convexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this sub-convexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.
In the context of surplus models of insurance risk theory,
some rather surprising and simple identities are presented. This
includes an
identity relating level crossing probabilities of continuous-time models
under (randomized) discrete and continuous observations, as well as
reflection identities relating dividend payments and capital injections.
Applications as well as extensions to more general underlying processes are
discussed.
A Simple Generative Model of Collective Online Behavior (Mason Porter)
Human activities increasingly take place in online environments, providing novel opportunities for relating individual behaviors to population-level outcomes. In this paper, we introduce a simple generative model for the collective behavior of millions of social networking site users who are deciding between different software applications. Our model incorporates two distinct mechanisms: one is associated with recent decisions of users, and the other reflects the cumulative popularity of each application. Importantly, although various combinations of the two mechanisms yield long-time behav- ior that is consistent with data, the only models that reproduce the observed temporal dynamics are those that strongly emphasize the recent popularity of applications over their cumulative popularity.
This demonstrates --- even when using purely observational data with- out experimental design --- that temporal data-driven modeling can effectively distinguish between competing microscopic mechanisms, allowing us to uncover previously unidentified aspects of collective online behavior.
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Bubbles, Turing machines, and possible routes to Navier-Stokes blow-up (Robert van Gorder)
Navier-Stokes existence and regularity in three spatial dimensions for an incompressible fluid... is hard. Indeed, while the original equations date back to the 1840's, existence and regularity remains an open problem and is one of the six remaining Millennium Prize Problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Despite the difficulty, a resolution to this problem may say little about real-world fluids, as many real fluid problems do not seem to blow-up, anyway.
In this talk, we shall briefly outline the mathematical problem, although our focus shall be on the negative direction; in particular, we focus on the possibility of blow-up solutions. We show that many existing blow-up solutions require infinite energy initially, which is unreasonable. Therefore, obtaining a blow-up solution that starts out with nice properties such as bounded energy on three dimensional Euclidean space is rather challenging. However, if we modify the problem, there are some results. We survey recent results on averaged Navier-Stokes equations and compressible Navier-Stokes equations, and this will take us anywhere from bubbles to fluid Turing machines. We discuss how such results might give insight into the loss of regularity in the incompressible case (or, insight into how hard it might be to loose regularity of solutions when starting with finite energy in the incompressible case), before philosophizing about whether mathematical blow-up solutions could ever be physically relevant.
Let U be the quantized enveloping algebra coming from a semi-simple finite dimensional complex Lie algebra. Lusztig has defined a canonical basis B for the minus part of U- of U. It has the remarkable property that one gets a basis of each highest-weight irreducible U-module V, with highest weight vector v, as the set of all bv which are not 0, as b varies in B. It is not known how to give the elements b explicitly, although there are algorithms.
For each reduced expression of the longest word in the Weyl group, Lusztig has defined a PBW basis P of U-, and for each b in B there is a unique p(b) in P such that b = p(b) + a linear combination of p' in P where the coefficients are in qZ[q]. This is much easier in the simply laced case. I show that the set of p(b)v, where b varies in B and bv is not 0, is a basis of V, and I can explicitly exhibit this basis in type A, and to some extent in types B, C, D.
It is known that B and P are crystal bases in the sense of Kashiwara. I will define Kashiwara operators, and briefly describe Kashiwara's approach to canonical bases, which he calls global bases. I show how one can calculate the Kashiwara operators acting on P, in types A, B, C, D, using tableaux of Kashiwara-Nakashima.
The simplicial boundary is another way to study the boundary of CAT(0) cube complexes. I will define this boundary introducing the relevant terminology from CAT(0) cube complexes along the way. There will be many examples and many pictures, hopefully to help understanding but also to improve my (not so great) drawing skills.
Due to Shor's algorithm, quantum computers are a severe threat for public key cryptography. This motivated the cryptographic community to search for quantum-safe solutions. On the other hand, the impact of quantum computing on secret key cryptography is much less understood. In this paper, we consider attacks where an adversary can query an oracle implementing a cryptographic primitive in a quantum superposition of different states. This model gives a lot of power to the adversary, but recent results show that it is nonetheless possible to build secure cryptosystems in it.
We study applications of a quantum procedure called Simon's algorithm (the simplest quantum period finding algorithm) in order to attack symmetric cryptosystems in this model. Following previous works in this direction, we show that several classical attacks based on finding collisions can be dramatically sped up using Simon's algorithm: finding a collision requires Ω(2n/2) queries in the classical setting, but when collisions happen with some hidden periodicity, they can be found with only O(n) queries in the quantum model.
We obtain attacks with very strong implications. First, we show that the most widely used modes of operation for authentication and authenticated encryption (e.g. CBC-MAC, PMAC, GMAC, GCM, and OCB) are completely broken in this security model. Our attacks are also applicable to many CAESAR candidates: CLOC, AEZ, COPA, OTR, POET, OMD, and Minalpher. This is quite surprising compared to the situation with encryption modes: Anand et al. show that standard modes are secure when using a quantum-secure PRF.
Second, we show that slide attacks can also be sped up using Simon's algorithm. This is the first exponential speed up of a classical symmetric cryptanalysis technique in the quantum model.
We will introduce the Thurston norm in the setting of 3-manifold groups, and show how the techniques coming from L2-homology allow us to extend its definition to the setting of free-by-cyclic groups.
We will also look at the relationship between this Thurston norm and the Alexander norm, and the BNS invariants, in particular focusing on the case of ascending HNN extensions of the 2-generated free group.
This talk will start with a brief historical review of the classification of solids by their symmetries, and the more recent K-theoretic periodic table of Kitaev. It will then consider some mathematical questions this raises, in particular about the behaviour of electrons on the boundary of materials and in the bulk. Two rather different models will be described, which turn out to be related by T-duality. Relevant ideas from noncommutative geometry will be explained where needed.
The CHY formulae are a set of remarkable formulae describing the scattering amplitudes of a variety of massless theories, as certain worldsheet integrals, localized on the solutions to certain polynomial equations (scattering equations). These formulae arise from a new class of holomorphic strings called Ambitwistor strings that encode exactly the dynamics of the supergravity (Yang-Mills) modes of string theory. Despite some recent progress by W. Siegel and collaborators, it remains as an open question as to what extent this theory was connected to the full string theory. The most mysterious point being certainly that the localization equations of the ambitwistor string also appear in the zero tension limit of string theory (alpha’ to infinity), which is the opposite limit than the supergravity one (alpha’ to zero). In this talk, I’ll report on some work in progress with E. Casali (Math. Inst. Oxford) and argue that the ambitwistor string is actually a tensionless string. Using some forgotten results on the quantization of these objects, we explain that the quantization of tensionless strings is ambiguous, and can lead either to a higher spin theory, or to the ambitwistor string, hence solving the previously mentioned paradox. In passing, we see that the degenerations of the tensile worldsheet that lead to tensionless strings make connection with Galilean Conformal Algebras and the (3d) BMS algebra.
Part of the series 'What do historians of mathematics do?'
Ada Lovelace (1815-1852) is famous as "the first programmer" for her prescient writings about Charles Babbage's unbuilt mechanical computer, the Analytical Engine. Biographers have focused on her tragically short life and her supposed poetic approach – one even dismissed her mathematics as "hieroglyphics". This talk will focus on how she learned the mathematics she needed to write the paper – a correspondence course she took with Augustus De Morgan – which is available in the Bodleian Library. I'll also reflect more broadly on things I’ve learned as a newcomer to the history of mathematics.
It is known that if the boundary of a 1-ended
hyperbolic group G has a local cut point then G splits over a 2-ended group. We prove a similar theorem for CAT(0)
groups, namely that if a finite set of points separates the boundary of a 1-ended CAT(0) group G
then G splits over a 2-ended group. Along the way we prove two results of independent interest: we show that continua separated
by finite sets of points admit a tree-like decomposition and we show a splitting theorem for nesting actions on R-trees.
This is joint work with Eric Swenson.
The planar Ising model is one of the simplest and most studied models in Statistical Mechanics. On one hand, it has a rich and interesting phase transition behaviour. On the other hand, it is "solvable" enough to allow for many rigorous and exact results. This, in particular, makes it one of the prime examples in Conformal Field Theory (CFT). In this talk, I will review my joint work with C. Hongler and D. Chelkak on the scaling limits of correlations in the planar Ising model at criticality. We prove that these limits exist, are conformally covariant and given by explicit formulae consistent with the CFT predictions. This may be viewed as a step towards a rigorous understanding of CFT in the case of the Ising model.TBC
In 1995 N. Hitchin constructed explicit algebraic solutions to the Painlevé VI (1/8,-1/8,1/8,3/8) equation starting with any Poncelet trajectory, that is a closed billiard trajectory inscribed in a conic and circumscribed about another conic. In this talk I will show that Hitchin's construction is the Okamoto transformation between Picard's solution and the general solution of the Painlevé VI (1/8,-1/8,1/8,3/8) equation. Moreover, this Okamoto transformation can be written in terms of an Abelian differential of the third kind on the associated elliptic curve, which allows to write down solutions to the corresponding Schlesinger system in terms of this differential as well. This is a joint work with V. Dragovic.
We consider the random conductance model: random walk among iid, uniformly elliptic conductnace on the d-dimensional lattice. We state,and explain, the Einstein relation for this model:It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle.These invariant measures are often called steady states.
The Einstein relation follows, at least for dimensions three and larger, from an expansion of the steady states as a function of the bias.
The talk is gase on joint work with Jan Nagel and Xiaoqin Guo
Generalised Geometry is a very powerful tool to study gravity duals of strongly coupled gauge theories. In this talk I will discuss how Exceptional Geometry can be used to study marginal deformations of N=1 SCFT's in 4 and 3 dimensions.