Past

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Note: joint with Algebra seminar.

String algebras are tame - their finite-dimensional representations have been classified - and the Auslander-Reiten quiver of such an algebra shows some of the morphisms between them.  But not all.  To see the morphisms which pass between components of the Auslander-Reiten quiver, and so obtain a more complete picture of the category of representations, we should look at certain infinite-dimensional representations and use ideas and techniques from the model theory of modules.

This is joint work with Rosie Laking and Gena Puninski:
G. Puninski and M. Prest,  Ringel's conjecture for domestic string algebras, arXiv:1407.7470;
R. Laking, M. Prest and G. Puninski, Krull-Gabriel dimension of domestic string algebras, in preparation.

String algebras are tame - their finite-dimensional representations have been classified - and the Auslander-Reiten quiver of such an algebra shows some of the morphisms between them.  But not all.  To see the morphisms which pass between components of the Auslander-Reiten quiver, and so obtain a more complete picture of the category of representations, we should look at certain infinite-dimensional representations and use ideas and techniques from the model theory of modules.

This is joint work with Rosie Laking and Gena Puninski:
G. Puninski and M. Prest,  Ringel's conjecture for domestic string algebras, arXiv:1407.7470;
R. Laking, M. Prest and G. Puninski, Krull-Gabriel dimension of domestic string algebras, in preparation.

Traditionally threshold partial pivoting is used to ensure stability of sparse LDL^T factorizations of symmetric matrices. This involves comparing a candidate pivot with all entries in its row/column to ensure that growth in the size of the factors is limited by a threshold at each stage of the factorization. It is capabale of delivering a scaled backwards error on the level of machine precision for practically all real world matrices. However it has significant flaws when used in a massively parallel setting, such as on a GPU or modern supercomputer. It requires all entries of the column to be up-to-date and requires an all-to-all communication for every column. The latter requirement can be performance limiting as the factorization cannot proceed faster than k*(communication latency), where k is the length of the longest path in the sparse elimination tree.

We introduce a new family of communication-avoiding pivoting techniques that reduce the number of messages required by a constant factor allowing the communication cost to be more effectively hidden by computation. We exhibit two members of this family. The first deliver equivalent stability to threshold partial pivoting, but is more pessimistic, leading to additional fill in the factors. The second provides similar fill levels as traditional techniques and, whilst demonstrably unstable for pathological cases, is able to deliver machine accuracy on even the worst real world examples.

More than twenty years ago Erdős conjectured that a triangle-free graph $G$ of chromatic number $k$ contains cycles of at least $k^{2−o(1)}$ different lengths. In this talk we prove this conjecture in a stronger form, showing that every such $G$ contains cycles of $ck^2\log k$ consecutive lengths, which is tight. Our approach can be also used to give new bounds on the number of different cycle lengths for other monotone classes of $k$-chromatic graphs, i.e.,  clique-free graphs and graphs without odd cycles.

Joint work with A. Kostochka and J. Verstraete.

The relationship of diagonal dominance ideas to the convergence of stationary iterations is well known. There are a multitude of situations in which such considerations can be used to guarantee convergence when the splitting matrix (the preconditioner) is positive definite. In this talk we will describe and prove sufficient conditions for convergence of a stationary iteration based on a splitting with an indefinite preconditioner. Simple examples covered by this theory coming from Optimization and Economics will be described.

This is joint work with Michael Ferris and Tom Rutherford

We will discuss the relation between perturbative gauge theory and
perturbative gravity, and look at how this relation extends to some exact
classical solutions. First, we will review the double copy prescription that
takes gauge theory amplitudes into gravity amplitudes, which has been
crucial to progress in perturbative studies of supergravity. Then, we will
see how the relation between the two theories can be made manifest when we
restrict to the self-dual sector, in four dimensions. A key role is played
by a kinematic algebraic structure mirroring the colour structure, which can
be extended from the self-dual sector to the full theory, in any number of
dimensions. Finally, we will see how these ideas can be applied also to some
exact classical solutions, namely black holes and plane waves. This leads to
a relation of the type Schwarzschild as (Coulomb charge)^2.

We consider the two-phase Stefan problem with p-degenerate diffusion, p larger than two, and we prove continuity up to the boundary for weak solutions, providing a modulus of continuity which we conjecture to be optimal. Since our results are proven in the form of a priori estimates for appropriate regularized problems, as corollary we infer the existence of a globally continuous weak solution for continuous Cauchy-Dirichlet datum.

Consider a nonsingular projective variety $X$ defined by a system of $R$ forms of the same degree $d$. The circle method proves the Hasse principle and Manin's conjecture for $X$ when $\text{dim}X > C(d,R)$. I will describe how to improve the value of $C$ when $R$ is large. I use a technique for estimating mean values of exponential sums which I call a ``moat lemma". This leads to a novel and intriguing system of auxiliary inequalities.

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.

In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges.

Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. We discuss why, for a large class of rules, the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process.

Based on joint work with Oliver Riordan.

Given a 3-dimensional TQFT, the "conformal blocks" are the
values of that TQFT on closed Riemann surfaces.
The construction that we'll present (joint work with Douglas &
Bartels) takes as only input the value of the TQFT on discs. Towards
the end, I will explain to what extent the conformal blocks that we
construct agree with the conformal blocks constructed e.g. from the
theory of vertex operator algebras.

We give a necessary and sufficient condition for a map defined on a compact, quasiconvex and simply-connected space to factor through a tree. This condition can be checked using currents. In particular if the target is some Euclidean space and the map is H\"older continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over the winding number. Moreover, this shows that if the target is the Heisenberg group equipped with the Carnot-Carath\'eodory metric and the H\"older exponent of the map is bigger than 2/3, the map factors through a tree.

I will discuss the implementation of explicit stabilisation of all closed string moduli in fluxed type IIB Calabi-Yau compactifications with chiral matter.  Using toric geometry we construct Calabi-Yau manifolds with del Pezzo singularities. D-branes located at such singularities can support the Standard Model gauge group and matter content. We consider Calabi-Yau manifolds with a discrete symmetry that reduces the effective number of complex structure moduli, which allows us to calculate the corresponding periods and find explicit flux vacua. We compute the values of the flux superpotential and the string coupling at these vacua. Starting from these explicit complex structure solutions, we obtain AdS and dS minima where the Kaehler moduli are stabilised by a mixture of D-terms, non-perturbative and perturbative alpha'-corrections as in the LARGE Volume Scenario.

Struggling for ideas at the weekends? Learn how to draw with colour and discover the creation of colour from our natural environment. Have a go at making your own natural paint colours. Create your own mini planet inspired by alchemy and the Radcliffe Observatory. Paint making demonstrations throughout the day with artist Nabil Al. All materials provided. Suitable for all ages from 6 to 60. Invite your friends.

We will present results from studies of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. We work in the framework of fast singular limits that derives from the work of Bogoliubov and Mitropolsky [1961], Klainerman and Majda [1981], Shochet [1994], Embid and Ma- jda [1996] and others.

In order to understand how the flow approaches and interacts with the slow dynamics we decompose the full solution, where u is a vector of all the unknowns, as

u = u α + u ′α where α represents the Ro → 0, F r → 0 or the simultaneous limit of both (QG for

quasi-geostrophy), with

P α u α = u α    P α u ′α = 0 ,

and where Pαu represents the projection of the full solution onto the null space of the fast operator. We use this decomposition to find evolution equations for the components of the flow (and the corresponding energy) on and off the slow manifold.

Numerical simulations indicate that for the geometry considered (triply periodic) and the type of forcing applied, the fast waves act as a conduit, moving energy onto the slow manifold. This decomposition clarifies how the energy is exchanged when either the stratification or the rotation is weak. In the quasi-geostrophic limit the energetics are less clear, however it is observed that the energy off the slow manifold equilibrates to a quasi-steady value.

We will also discuss generalizations of the method of cancellations of oscillations of Schochet for two distinct fast time scales, i.e. which fast time scale is fastest? We will give an example for the quasi-geostrophic limit of the Boussinesq equations.

At the end we will briefly discuss how understanding the role of oscillations has allowed us to develop convergent algorithms for parallel-in-time methods.

Beth A. Wingate - University of Exeter

Jared Whitehead - Brigham Young University

Terry Haut - Los Alamos National Laboratory

1. A Hybrid Monte-Carlo Partial Differential Solver for Stochastic  Volatility Models (Cozma)

In finance, Monte-Carlo and Finite Difference methods are the most popular approaches for pricing options. If the underlying asset is modeled by a multidimensional system of stochastic differential equations, an analytic solution is rarely available and working under a given computational budget comes at the cost of accuracy. The mixed Monte-Carlo partial differential solver introduced by Loeper and Pironneau (2009) is one way to overcome this issue and we investigate it thoroughly for a number of stochastic volatility models. Our main concern is to provide a rigorous mathematical proof of the convergence of the hybrid method under different frameworks, which in turn justifies the use of Monte-Carlo simulations to compute the expected discounted payoff of the financial derivative. Then, we carry out a quantitative assessment based on a European call option by comparison with alternative numerical methods.

2. tbc (Brackmann)

A G\"odel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin's problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed point for the formula is obtained. Some further examples of self-referential sentences are considered, such as sentences that \anf{say of themselves} that they are $\Sigma^0_n$-true (or $\Pi^0_n$-true), and their formal properties are investigated.

In this seminar I will discuss a function field analogue of classical problems in analytic number theory, concerning the auto-correlations of divisor functions, in the limit of a large finite field.