Which positive integers are the area of a right angled triangle with rational sides? In this talk I will discuss this classical problem, its reformulation in terms of rational points on elliptic curves and Tunnell's theorem which gives a complete solution to this problem assuming the Birch and Swinnerton-Dyer conjecture.
The unknotting number of a knot is an incredibly difficult invariant to compute.
In fact, there are many knots which are conjectured to have unknotting number 2 but for
which no proof of this is currently available. It therefore remains an unsolved problem to find an
algorithm that determines whether a knot has unknotting number one. In my talk, I will
show that an analogous problem for links is soluble. We say that a link has splitting number
one if some crossing change turns it into a split link. I will give an algorithm that
determines whether a link has splitting number one. (In the case where the link has
two components, we must make a hypothesis on their linking number.) The proof
that the algorithm works uses sutured manifolds and normal surfaces.
abstract:In this talk, we describe how to compute the critical point for various lattice models of planar statistical physics. We will first introduce the percolation, Ising, Potts and random-cluster models on the square lattice. Then, we will discuss how critical points of these different models are related. In a final part, we will compute the critical point of these models. This last part harnesses two main ingredients that we will describe in details: duality and sharp threshold theorems. No background is necessary.
Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma
conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuous-time process favouring sites with more local time. We show that the VRJP is a mixture of time-changed Markov jump processes and calculate the mixing measure. The mixing measure is interpreted as a marginal of the supersymmetric hyperbolic sigma model introduced by Disertori, Spencer and Zirnbauer.
This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement (in fact, on graphs of bounded degree), using a localisation result of Disertori and Spencer (2010).
(Joint work with Pierre Tarrès.)
Instantons and W-bosons in 5d N=2 Yang-Mills theory arise from a circle
compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding
self-dual strings, respectively. We study an index which counts BPS
instantons with electric charges in Coulomb and symmetric phases. We first
prove the existence of unique threshold bound state of U(1) instantons for
any instanton number. By studying SU(N) self-dual strings in the Coulomb
phase, we find novel momentum-carrying degrees on the worldsheet. The total
number of these degrees equals the anomaly coefficient of SU(N) (2,0) theory.
We finally propose that our index can be used to study the symmetric phase of
this theory, and provide an interpretation as the superconformal index of the
sigma model on instanton moduli space.
The geometric Lévy model (GLM) is a natural generalisation of the geometric Brownian motion (GBM) model. The theory of such models simplifies considerably if one takes a pricing kernel approach. In one dimension, once the underlying Lévy process has been specified, the GLM has four parameters: the initial price, the interest rate, the volatility and the risk aversion. The pricing kernel is the product of a discount factor and a risk aversion martingale. For GBM, the risk aversion parameter is the market price of risk. In this talk I show that for a GLM, this interpretation is not valid: the excess rate of return above the interest rate is a nonlinear function of the volatility and the risk aversion such that it is positive, and is increasing with respect to these variables. In the case of foreign exchange, Siegel’s paradox implies that one can construct foreign exchange models for which the excess rate of return is positive for both the exchange rate and the inverse exchange rate. Examples are worked out for a range of Lévy processes. (The talk is based on a recent paper: Brody, Hughston & Mackie, Proceedings of the Royal Society London, to appear in May 2012).
Please note that this is a joint seminar with the William Dunn School of Pathology and will be held in the EPA Seminar Room
Please note the unusual start-time.
In order to run accurate electrochemical models of batteries (and other devices) it is necessary to know a priori the values of many geometric, electrical and electrochemical parameters (10-100 parameters) e.g. diffusion coefficients, electrode thicknesses etc. However a basic difficulty is that the only external measurements that can be made on cells without deconstructing and destroying them are surface temperature plus electrical measurements (voltage, current, impedance) at the terminals. An interesting research challenge therefore is the accurate, robust estimation of physically realistic model parameters based only on external measurements of complete cells. System identification techniques (from control engineering) including ‘electrochemical impedance spectroscopy’ (EIS) may be applied here – i.e. small signal frequency response measurement. However It is not clear exactly why and how impedance correlates to SOC/ SOH and temperature for each battery chemistry due to the complex interaction between impedance, degradation and temperature.
I will give a brief overview of some of the recent work in this area and try to explain some of the challenges in the hope that this will lead to a fruitful discussion about whether this problem can be solved or not and how best to tackle it.
I will explain how to define a notion of stable-independence in NIP
theories, which is an attempt to capture the "stable part" of types.
The displacement of a liquid by an air finger is a generic two-phase flow that
underpins applications as diverse as microfluidics, thin-film coating, enhanced
oil recovery, and biomechanics of the lungs. I will present two intriguing
examples of such flows where, firstly, oscillations in the shape of propagating
bubbles are induced by a simple change in tube geometry, and secondly, flexible
vessel boundaries suppress viscous fingering instability.
1) A simple change in pore geometry can radically alter the behaviour of a
fluid displacing air finger, indicating that models based on idealized pore
geometries fail to capture key features of complex practical flows. In
particular, partial occlusion of a rectangular cross-section can force a
transition from a steadily-propagating centred finger to a state that exhibits
spatial oscillations via periodic sideways motion of the interface at a fixed
location behind the finger tip. We characterize the dynamics of the
oscillations and show that they arise from a global homoclinic connection
between the stable and unstable manifolds of a steady, symmetry-broken
solution.
2) Growth of complex dendritic fingers at the interface of air and a viscous
fluid in the narrow gap between two parallel plates is an archetypical problem
of pattern formation. We find a surprisingly effective means of suppressing
this instability by replacing one of the plates with an elastic membrane. The
resulting fluid-structure interaction fundamentally alters the interfacial
patterns that develop and considerably delays the onset of fingering. We
analyse the dependence of the instability on the parameters of the system and
present scaling arguments to explain the experimentally observed behaviour.
The Arnoldi method for standard eigenvalue problems possesses several
attractive properties making it robust, reliable and efficient for
many problems. We will present here a new algorithm equivalent to the
Arnoldi method, but designed for nonlinear eigenvalue problems
corresponding to the problem associated with a matrix depending on a
parameter in a nonlinear but analytic way. As a first result we show
that the reciprocal eigenvalues of an infinite dimensional operator.
We consider the Arnoldi method for this and show that with a
particular choice of starting function and a particular choice of
scalar product, the structure of the operator can be exploited in a
very effective way. The structure of the operator is such that when
the Arnoldi method is started with a constant function, the iterates
will be polynomials. For a large class of NEPs, we show that we can
carry out the infinite dimensional Arnoldi algorithm for the operator
in arithmetic based on standard linear algebra operations on vectors
and matrices of finite size. This is achieved by representing the
polynomials by vector coefficients. The resulting algorithm is by
construction such that it is completely equivalent to the standard
Arnoldi method and also inherits many of its attractive properties,
which are illustrated with examples.
We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about some recent results on existence, regularity and geometric properties of regular reflection solutions for potential flow equation. In particular, we discuss optimal regularity of solutions near sonic curve, and stability of the normal reflection soluiton. Open problems will also
be discussed. The talk will be based on the joint work with Gui-Qiang Chen, and with Myoungjean Bae.
The mid 1980s saw a shift in the nature of the relationship between mathematics and physics. Differential equations and geometry applied in a classical setting were no longer the principal players; in the quantum world topology and algebra had come to the fore. In this talk we discuss a method of classifying 2-dim invertible Klein topological quantum field theories (KTQFTs). A key object of study will be the unoriented cobordism category $\mathscr{K}$, whose objects are closed 1-manifolds and whose morphisms are surfaces (a KTQFT is a functor $\mathscr{K}\rightarrow\operatorname{Vect}_{\mathbb{C}}$). Time permitting, the open-closed version of the category will be considered, yielding some surprising results.
Particle-based stochastic reaction-diffusion models have recently been used to study a number of problems in cell biology. These methods are of interest when both noise in the chemical reaction process and the explicit motion of molecules are important. Several different mathematical models have been used, some spatially-continuous and others lattice-based. In the former molecules usually move by Brownian Motion, and may react when approaching each other. For the latter molecules undergo continuous time random-walks, and usually react with fixed probabilities per unit time when located at the same lattice site.
As motivation, we will begin with a brief discussion of the types of biological problems we are studying and how we have used stochastic reaction-diffusion models to gain insight into these systems. We will then introduce several of the stochastic reaction-diffusion models, including the spatially continuous Smoluchowski diffusion limited reaction model and the lattice-based reaction-diffusion master equation. Our work studying the rigorous relationships between these models will be presented. Time permitting, we may also discuss some of our efforts to develop improved numerical methods for solving several of the models.
A perfect obstruction theory for a commutative ring is a morphism from a perfect complex to the cotangent complex of the ring
satisfying some further conditions. In this talk I will present work in progress on how to associate in a functorial manner commutative
differential graded algebras to such a perfect obstruction theory. The key property of the differential graded algebra is that its zeroth homology
is the ring equipped with the perfect obstruction theory. I will also indicate how the method introduced can be globalized to work on schemes
without encountering gluing issues.
We call a graph $H$ \emph{Ramsey-unsaturated} if there is an edge in the
complement of $H$ such that the Ramsey number $r(H)$ of $H$ does not
change upon adding it to $H$. This notion was introduced by Balister,
Lehel and Schelp who also showed that cycles (except for $C_4$) are
Ramsey-unsaturated, and conjectured that, moreover, one may add {\em
any} chord without changing the Ramsey number of the cycle $C_n$, unless
$n$ is even and adding the chord creates an odd cycle.
We prove this conjecture for large cycles by showing a stronger
statement: If a graph $H$ is obtained by adding a linear number of
chords to a cycle $C_n$, then $r(H)=r(C_n)$, as long as the maximum
degree of $H$ is bounded, $H$ is either bipartite (for even $n$) or
almost bipartite (for odd $n$), and $n$ is large.
This motivates us to call cycles \emph{strongly} Ramsey-unsaturated.
Our proof uses the regularity method.
Recent results (starting with Scheffer and
Shnirelman and continuing with De Lellis and Szekelhyhidi ) underline
the importance of considering solutions of the incompressible Euler
equations as limits of solutions of more physical examples like
Navier-Stokes or Boltzmann.
I intend to discuss several examples illustrating this issue.