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.