Please note the unusual day of the week for this workshop (a Monday) and also the unusual location.
A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities:
(i) The optimal terminal wealth X*(T) := Xφ* (T) of the classical problem to
maximise the expected U-utility of the terminal wealth Xφ(T) generated by admissible
portfolios φ(t); 0 ≤ t ≤ T in a market with the risky asset price process modeled as a semimartingale;
(ii) The optimal scenario dQ*/dP of the dual problem to minimise the expected
V -value of dQ/dP over a family of equivalent local martingale measures Q. Here V is
the convex dual function of the concave function U.
In this talk we consider markets modeled by Itô-Lėvy processes, and we present
in a first part a new proof of the above result in this setting, based on the maximum
principle in stochastic control theory. An advantage with our approach is that it also
gives an explicit relation between the optimal portfolio φ* and the optimal scenario
Q*, in terms of backward stochastic differential equations. In a second part we present
robust (model uncertainty) versions of the optimization problems in (i) and (ii), and
we prove a relation between them. We illustrate the results with explicit examples.
The presentation is based on recent joint work with Bernt ¬Oksendal, University of
Oslo, Norway.
Unstable dynamical systems can be stabilized, and hence the solution
recovered from noisy data, provided two conditions hold. First, observe
enough of the system: the unstable modes. Second, weight the observed
data sufficiently over the model. In this talk I will illustrate this for the
3DVAR filter applied to three dissipative dynamical systems of increasing
dimension: the Lorenz 1963 model, the Lorenz 1996 model, and the 2D
Navier-Stokes equation.
In vertical annular two-phase flow, large amplitude waves ("disturbance waves") are the most significant means by which the liquid is transported by the action of the gas phase. The presentation is of certain experimental results with the intention of defining a conceptual model suitable for possible mathematical interpretation.
These large waves have been studied for over 50 years but there has been little corresponding advance in the mathematical understanding of the phenomenon.
The aim of the workshop is to discuss what analysis might be possible and how this might contribute to the understanding of the phenomena involved.
I shall give a non-technical survey of Pure Inductive Logic, a branch of Carnap's Inductive Logic which was
anticipated early on in that subject but has only recently begun to be developed as an area of Mathematical Logic. My intention
is to cover its origins and aims, and to pick out some of the key concepts which have emerged in the last decade or so.
In this talk I will present the best up-to-date bounds for the argument of the Riemann zeta-function on the critical line, assuming the Riemann hypothesis. The method applies to other objects related to the Riemann zeta-function and uses certain special families of functions of exponential type. This is a joint work with Vorrapan Chandee (Montreal) and Micah Milinovich (Mississipi).
We formulate a new theory for equilibria of 2-braids, i.e., structures
formed by two elastic rods winding around each other in continuous contact
and subject to a local interstrand interaction. Unlike in previous work no
assumption is made on the shape of the contact curve. The theory is developed
in terms of a moving frame of directors attached to one of the strands with
one of the directors pointing to the position of the other strand. The
constant-distance constraint is automatically satisfied by the introduction
of what we call braid strains. The price we pay is that the potential energy
involves arclength derivatives of these strains, thus giving rise to a
second-order variational problem. The Euler-Lagrange equations for this
problem (in Euler-Poincare form) give balance equations for the overall
braid force and moment referred to the moving frame as well as differential
equations that can be interpreted as effective constitutive relations
encoding the effect that the second strand has on the first as the braid
deforms under the action of end loads. Hard contact models are used to obtain
the normal contact pressure between strands that has to be non-negative for
a physically realisable solution without the need for external devices such
as clamps or glue to keep the strands together. The theory is first
illustrated by a few simple examples and then applied to several problems
that require the numerical solution of boundary-value problems. Both open
braids and closed braids (links and knots) are considered and current
applications to DNA supercoiling are discussed.
This talk will give a quick and dirty introduction to orbifold bordism. We will start by briefly recalling some basic properties and definitions of orbifolds and sketch (very roughly) how orbifolds can be defined in the language of $C^\infty$-stacks due to Joyce (after introducing these). We will then review classical bordism theory for manifolds (in some nonstandard way) and discuss which definitions and results generalize to the orbifold case. A word of warning: this talk is intended to be an introduction and wants to give an overview over the subject, so it is likely that we will be sloppy here and there.
Using the Borel-Schur algebra, we construct explicit characteristic-free resolutions for Weyl modules for the general linear group. These resolutions provide an answer to the problem, posed in the 80's by A. Akin and D. A. Buchsbaum, of constructing finite explicit and universal resolutions of Weyl modules by direct sums of divided powers. Next we apply the Schur functor to these resolutions and prove a conjecture of Boltje and Hartmann on resolutions of co-Specht modules. This is joint work with I. Yudin.
We consider the problem of taking a matrix A and finding diagonal matrices D and E such that the rows and columns of B = DAE satisfy some specific constraints. Examples of constraints are that
* the row and column sums of B should all equal one;
* the norms of the rows and columns of B should all be equal;
* the row and column sums of B should take values specified by vectors p and q.
Simple iterative algorithms for solving these problems have been known for nearly a century. We provide a simple framework for describing these algorithms that allow us to develop robust convergence results and describe a straightforward approach to accelerate the rate of convergence.
We describe some of the diverse applications of balancing with examples from preconditioning, clustering, network analysis and psephology.
This is joint work with Kerem Akartunali (Strathclyde), Daniel Ruiz (ENSEEIHT, Toulouse) and Bora Ucar (ENS, Lyon).
I will outline Bergeron-Wise’s proof that the Virtual Haken Conjecture follows from Wise’s Conjecture on virtual specialness of non-positively curved cube complexes. If time permits, I will sketch some highlights from the proof of Wise’s Conjecture due to Agol and based on the Weak Separation Theorem of Agol-Groves-Manning.
We'll provide some motivation for the appearance of factorization algebras in physics, before discussing the definition of a factorization monoid. We'll then review the definition of a principal G-bundle and show how a factorization monoid can help us understand the moduli stack Bun_G of principal G-bundles.
In this talk aimed at a general audience I will discuss the ways in which we have used mathematical models of the regulation of haematopoiesis (blood cell production) to understand haematological diseases, and suggest successful treatment strategies for these diseases. At the end I will talk about our current work on tailoring chemotherapy so that it has less damaging effects on the haematopoietic system and, consequently, improve the quality of life for patients being treated for a variety of tumours.
I will discuss some recent developments in Schubert calculus and a potential relation to classical combinatorics for symmetric groups and possible extensions to complex reflection groups.
We prove that if $\frac{\log^{117} n}{n} \leq p \leq 1 -
n^{-1/8}$, then asymptotically almost surely the edges of $G(n,p)$ can
be covered by $\lceil \Delta(G(n,p))/2 \rceil$ Hamilton cycles. This
is clearly best possible and improves an approximate result of Glebov,
Krivelevich and Szab\'o, which holds for $p \geq n^{-1 + \varepsilon}$.
Based on joint work with Daniela Kuhn, John Lapinskas and Deryk Osthus.
******************** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON TUESDAY ********************
Well-known iterative schemes for the solution of ill-posed linear equations are the Landweber iteration, the cg-iteration and semi-iterative algorithms like the $\nu$-methods. After introducing these methods, we show that for ill-posed problems a slight modification of the underlying three-term recurrence relation of the $\nu$-methods provides accelerated Landweber algorithms with better performance properties than the $\nu$-methods. The new semi-iterative methods are based on the family of co-dilated ultraspherical polynomials. Compared to the standard $\nu$-methods, the residual polynomials of the modified methods have a faster decay at the origin. This results in an earlier termination of the iteration if the spectrum of the involved operator is clustered around the origin. The convergence order of the modified methods turns out to be the same as for the original $\nu$-methods. The new algorithms are tested numerically and a simple adaptive scheme is developed in which an optimal dilation parameter is determined. At the end, the new semi-iterative methods are used to solve a parameter identification problem obtained from a model in elastography.
We'll present a proof of the basic Burgess bound for short character sums, following the simplified presentation of Gallagher and Montgomery.
Orthogonal calculus is a calculus of functors, inspired by Goodwillie calculus. It takes as input a functor from finite dimensional inner product spaces to topological spaces and as output gives a tower of approximations by well-behaved functors. The output captures a lot of important homotopical information and is an important tool for calculations.
In this talk I will report on joint work with Peter Oman in which we use model categories to improve the foundations of orthogonal calculus. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. The classification of n-homogeneous functors in terms of spectra with O(n)-action can then be phrased as a zig-zag of Quillen equivalences.
If one starts with a uniformly ergodic Markov chain on countable states, what sort of perturbation can one make to the transition rates and still retain uniform ergodicity? In this talk, we will consider a class of perturbations, that can be simply described, where a uniform estimate on convergence to an ergodic distribution can be obtained. We shall see how this is related to Ergodic BSDEs in this setting and outline some novel applications of this approach.