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.
In Riemannian geometry there are several notions of rank
defined for non-positively curved manifolds and with natural extensions
for groups acting on non-positively curved spaces.
The talk shall explain how various notions of rank behave for
mapping class groups of surfaces. This is joint work with J. Behrstock.
We will give a quick overview of the semigroup perspective on splitting schemes for S(P)DEs which present a robust, "easy to implement" numerical method for calculating the expected value of a certain payoff of a stochastic process driven by a S(P)DE. Having a high numerical order of convergence enables us to replace the Monte Carlo integration technique by alternative, faster techniques. The numerical order of splitting schemes for S(P)DEs is bounded by 2. The technique of combining several splittings using linear combinations which kills some additional terms in the error expansion and thus raises the order of the numerical method is called the extrapolation. In the presentation we will focus on a special extrapolation of the Lie-Trotter splitting: the symmetrically weighted sequential splitting, and its subsequent extrapolations. Using the semigroup technique their convergence will be investigated. At the end several applications to the S(P)DEs will be given.
Based on ideas from rough path analysis and operator splitting, the Kusuoka-Lyons-Victoir scheme provides a family of higher order methods for the weak approximation of stochastic differential equations. Out of this family, the Ninomiya-Victoir method is especially simple to implement and to adjust to various different models. We give some examples of models used in financial engineering and comment on the performance of the Ninomiya-Victoir scheme and some modifications when applied to these models.
String theory on a torus requires the introduction of dual coordinates
conjugate to string winding number. This leads to physics and novel geometry in a doubled space. This will be
compared to generalized geometry, which doubles the tangent space but not the manifold.
For a d-torus, string theory can be formulated in terms of an infinite
tower of fields depending on both the d torus coordinates and the d dual
coordinates. This talk focuses on a finite subsector consisting of a metric
and B-field (both d x d matrices) and a dilaton all depending on the 2d
doubled torus coordinates.
The double field theory is constructed and found to have a novel symmetry
that reduces to diffeomorphisms and anti-symmetric tensor gauge
transformations in certain circumstances. It also has manifest T-duality
symmetry which provides a generalisation of the usual Buscher rules to
backgrounds without isometries. The theory has a real dependence on the full
doubled geometry: the dual dimensions are not auxiliary. It is concluded
that the doubled geometry is physical and dynamical.
Inverse methods are frequently used in geosciences to estimate model parameters from indirect measurements. A common inverse problem encountered when modelling the flow of large ice masses such as the Greenland and the Antarctic ice sheets is the determination of basal conditions from surface data. I will present an overview over some of the inverse methods currently used to tackle this problem and in particular discuss the use of Bayesian inverse methods in this context. Examples of the use of adjoint methods for large-scale optimisation problems that arise, for example, in flow modelling of West-Antarctica will be given.
Absence of arbitrage is a highly desirable feature in mathematical models of financial markets. In its pure form (whether as NFLVR or as the existence of a variant of an equivalent martingale measure R), it is qualitative and therefore robust towards equivalent changes of the underlying reference probability (the "real-world" measure P). But what happens if we look at more quantitative versions of absence of arbitrage, where we impose for instance some integrability on the density dR/dP? To which extent is such a property robust towards changes of P? We discuss these uestions and present some recent results.
The talk is based on joint work with Tahir Choulli (University of Alberta, Edmonton).
In this lecture I will exploit a model of asset prices where speculators overconfidence is a source of heterogeneous beliefs and arbitrage is limited. In the model, asset buyers are the most positive investors, but prices exceed their optimistic valuation because the owner of an asset has the option of reselling it in the future to an even more optimistic buyer. The value of this resale option can be identified as a bubble. I will focus on assets with a fixed terminal date, as is often the case with credit instruments. I will show that the size of a bubble satisfies a Partial Differential Equation that is similar to the equation satisfied by an American option and use the PDE to evaluate the impact of parameters such as interest rates or a “Tobin tax” on the size of the bubble and on trading volume.
Feature Selection is a ubiquitous problem in across data mining,
bioinformatics, and pattern recognition, known variously as variable
selection, dimensionality reduction, and others. Methods based on
information theory have tremendously popular over the past decade, with
dozens of 'novel' algorithms, and hundreds of applications published in
domains across the spectrum of science/engineering. In this work, we
asked the question 'what are the implicit underlying statistical
assumptions of feature selection methods based on mutual information?'
The main result I will present is a unifying probabilistic framework for
information theoretic feature selection, bringing almost two decades of
research on heuristic methods under a single theoretical interpretation.
Exponential time integrators are a powerful tool for numerical solution
of time dependent problems. The actions of the matrix functions on vectors,
necessary for exponential integrators, can be efficiently computed by
different elegant numerical techniques, such as Krylov subspaces.
Unfortunately, in some situations the additional work required by
exponential integrators per time step is not paid off because the time step
can not be increased too much due to the accuracy restrictions.
To get around this problem, we propose the so-called time-stepping-free
approach. This approach works for linear ordinary differential equation (ODE)
systems where the time dependent part forms a small-dimensional subspace.
In this case the time dependence can be projected out by block Krylov
methods onto the small, projected ODE system. Thus, there is just one
block Krylov subspace involved and there are no time steps. We refer to
this method as EBK, exponential block Krylov method. The accuracy of EBK
is determined by the Krylov subspace error and the solution accuracy in the
projected ODE system. EBK works for well for linear systems, its extension
to nonlinear problems is an open problem and we discuss possible ways for
such an extension.
Min-Max equations, also called Isaacs equations, arise from many applications, eg in game theory or mathematical finance. For their numerical solution, they are often discretised by finite difference
methods, and, in a second step, one is then faced with a non-linear discrete system. We discuss how upper and lower bounds for the solution to the discretised min-max equation can easily be computed.
In this seminar I will present two results regarding the uniqueness (and further properties) for the two-dimensional continuity equation
and the ordinary differential equation in the case when the vector field is bounded, divergence free and satisfies additional conditions on its distributional curl. Such settings appear in a very natural way in various situations, for instance when considering two-dimensional incompressible fluids. I will in particular describe the following two cases:\\
(1) The vector field is time-independent and its curl is a (locally finite) measure (without any sign condition).\\
(2) The vector field is time-dependent and its curl belongs to L^1.\\
Based on joint works with: Giovanni Alberti (Universita' di Pisa), Stefano Bianchini (SISSA Trieste), Francois Bouchut (CNRS &
Universite' Paris-Est-Marne-la-Vallee) and Camillo De Lellis (Universitaet Zuerich).
In this talk our aim is to explain why there exist hyperkähler metrics on the cotangent bundles and on coadjoint orbits of complex Lie groups. The key observation is that both the cotangent bundle of $G^\mathbb C$ and complex coadjoint orbits can be constructed as hyperkähler quotients in an infinite-dimensional setting: They may be identified with certain moduli spaces of solutions to Nahm's equations, which is a system of non-linear ODEs arising in gauge theory.
In the first half we will describe the hyperkähler quotient construction, which can be viewed as a version of the Marsden-Weinstein symplectic quotient for complex symplectic manifolds. We will then introduce Nahm's equations and explain how their moduli spaces of solutions may be related to the above Lie theoretic objects.
In recent years there has been extensive interest in word maps on groups, and various results were obtained, with emphasis on simple groups. We shall focus on some new results on word maps for more general families of finite and infinite groups.
I revisit the identification of Nekrasov's K-theoretic partition function, counting instantons on $R^4$, and the (refined) Donaldson-Thomas partition function of the associated local Calabi-Yau threefold. The main example will be the case of the resolved conifold, corresponding to the gauge group $U(1)$. I will show how recent mathematical results about refined DT theory confirm this identification, and speculate on how one could lift the equality of partition functions to a structural result about vector spaces.
The use of tissue engineered implants could facilitate unions in situations where there is loss of bone or non-union, thereby increasing healing time, reducing the risk of infections and hence reducing morbidity. Currently engineered bone tissue is not of sufficient quality to be used in widespread clinical practice. In order to improve experimental design, and thereby the quality of the tissue-constructs, the underlying biological processes involved need to be better understood. In conjunction with experimentalists, we consider the effect hydrodynamic pressure has on the development and regulation of bone, in a bioreactor designed specifically for this purpose. To answer the experimentalists’ specific questions, we have developed two separate models; in this talk I will present one of these, a multiphase partial differential equation model to describe the evolution of the cells, extracellular matrix that they deposit, the culture medium and the scaffold. The model is then solved using the finite element method using the deal.II library.
This is a report of joint work with T. Koppe, P. Majumdar, and K.
Ray.
I will define new partition functions for theories with targets on toric
singularities via
products of old partition functions on crepant resolutions. I will
present explicit examples
and show that the new partition functions turn out to be homogeneous on
MacMahon factors.