Topological field theories give a connection between
topology and algebra. This connection can be exploited in both
directions: using algebra to construct topological invariants, or
using topology to prove algebraic theorems. In this talk, I will
explain an interesting example of the latter phenomena. Radford's
theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a
finite tensor category the quadruple dual functor is easy to
understand. It's somewhat mysterious that the double dual is hard to
understand but the quadruple dual is easy. Using topological field
theory, we show that Radford's theorem is exactly the consequence of
the Dirac belt trick in topology. That is, the double dual
corresponds to the generator of $\pi_1(\mathrm{SO}(3))$ and so the
quadruple dual is trivial in an appropriate sense exactly because
$\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2$. This is part of a large
project, joint with Chris Douglas and Chris Schommer-Pries, to
understand local field theories with values in the 3-category of
tensor categories via the cobordism hypothesis.
Given a flat torus, we consider certain discrete graph approximations of
it and determine the asymptotics of the number of spanning trees
("complexity") of these graphs as the mesh gets finer. The constants in the
asymptotics involve various notions of determinants such as the
determinant of the Laplacian ("height") of the torus. The analogy between
the complexity of graphs and the height of manifolds was previously
commented on by Sarnak and Kenyon. In dimension two, similar asymptotics
were established earlier by Barber and Duplantier-David in the context of
statistical physics.
Our proofs rely on heat kernel analysis involving Bessel functions, which
in the torus case leads into modular forms and Epstein zeta functions. In
view of a folklore conjecture it also suggests that tori corresponding to
densest regular sphere packings should have approximating graphs with the
largest number of spanning trees, a desirable property in network theory.
Joint work with G. Chinta and J. Jorgenson.
We construct explicitly a bridge process whose distribution, in its own filtration, is the same as the difference of two independent Poisson processes with the same intensity and its time 1 value satisfies a specific constraint. This construction allows us to show the existence of Glosten-Milgrom equilibrium and its associated optimal trading strategy for the insider. In the equilibrium the insider employs a mixed strategy to randomly submit two types of orders: one type trades in the same direction as noise trades while the other cancels some of the noise trades by submitting opposite orders when noise trades arrive. The construction also allows us to prove that Glosten-Milgrom equilibria converge weakly to Kyle-Back equilibrium, without the additional assumptions imposed in \textit{K. Back and S. Baruch, Econometrica, 72 (2004), pp. 433-465}, when the common intensity of the Poisson processes tends to infinity. This is a joint work with Umut Cetin.
Ice-dammed lakes form next to, on the surface of, and beneath glaciers
and ice sheets. Some lakes are known to drain catastrophically,
creating hazards, wasting water resources and modulating the flow of
the adjacent ice. My work aims to increase our understanding of such
drainage. Here I will focus on lakes that form next to glaciers and
drain subglacially (between ice and bedrock) through a channel. I will
describe how such a system can be modelled and present results from
model simulations of a lake that fills due to an input of meltwater
and drains through a channel that receives a supply of meltwater along
its length. Simulations yield repeating cycles of lake filling and
drainage and reveal how increasing meltwater input to the system
affects these cycles: enlarging or attenuating them depending on how
the meltwater is apportioned between the lake and the channel. When
inputs are varied with time, simulating seasonal meteorological
cycles, the model simulates either regularly repeating cycles or
irregular cycles that never repeat. Irregular cycles demonstrate
sensitivity to initial conditions, a high density of periodic orbits
and topological mixing. I will discuss how these results enhance our
understanding of the mechanisms behind observed variability in these
systems.
This is the session for our industrial sponsors to propose project ideas. Academic staff are requested to attend to help shape the problem statements and to suggest suitable internal supervisors for the projects.
Recent years have seen increasing attention to the subtle effects on
intracellular transport caused when multiple molecular motors bind to
a common cargo. We develop and examine a coarse-grained model which
resolves the spatial configuration as well as the thermal fluctuations
of the molecular motors and the cargo. This intermediate model can
accept as inputs either common experimental quantities or the
effective single-motor transport characterizations obtained through
systematic analysis of detailed molecular motor models. Through
stochastic asymptotic reductions, we derive the effective transport
properties of the multiple-motor-cargo complex, and provide analytical
explanations for why a cargo bound to two molecular motors moves more
slowly at low applied forces but more rapidly at high applied forces
than a cargo bound to a single molecular motor. We also discuss how
our theoretical framework can help connect in vitro data with in vivo
behavior.
The Discontinuous Galerkin (DG) method has been used to solve a wide range of partial differential equations. Especially for advection dominated problems it has proven very reliable and accurate. But even for elliptic problems it has advantages over continuous finite element methods, especially when parallelization and local adaptivity are considered.
In this talk we will first present a variation of the compact DG method for elliptic problems with varying coefficients. For this method we can prove stability on general grids providing a computable bound for all free parameters. We developed this method to solve the compressible Navier-Stokes equations and demonstrated its efficiency in the case of meteorological problems using our implementation within the DUNE software framework, comparing it to the operational code COSMO used by the German weather service.
After introducing the notation and analysis for DG methods in Euclidean spaces, we will present a-priori error estimates for the DG method on surfaces. The surface finite-element method with continuous ansatz functions was analysed a few years ago by Dzuik/Elliot; we extend their results to the interior penalty DG method where the non-smooth approximation of the surface introduces some additional challenges.
In 1977, Cline Parshall, Scott and van der Kallen wrote a seminal paper `Rational and generic cohomology' which exhibited a connection between the cohomology for algebraic groups and the cohomology for finite groups of Lie type, showing that in many cases one can conclude that there is an isomorphism of cohomology through restriction from the algebraic to the finite group.
One unfortunate problem with their result is that there remain infinitely many modules for which their theory---for good reason---tells us nothing. The main result of this talk (recent work with Parshall and Scott) is to show that almost all the time, one can manipulate the simple modules for finite groups of Lie type in such a way as to recover an isomorphism of its cohomology with that of the algebraic group.
H. Johnston and J.G. Liu proposed in 2004 a numerical scheme for approximating numerically solutions of the incompressible Navier-Stokes system. The scheme worked very well in practice but its analytic properties remained elusive.\newline
In order to understand these analytical aspects they considered together with R. Pego a continuous version of it that appears as an extension of the incompressible Navier-Stokes to vector-fields that are not necessarily divergence-free. For divergence-free initial data one has precisely the incompressible Navier-Stokes, while for non-divergence free initial data, the divergence is damped exponentially.\newline
We present analytical results concerning this extended system and discuss numerical implications. This is joint work with R. Pego, G. Iyer (Carnegie Mellon) and J. Kelliher, M. Ignatova (UC Riverside).
Saturated fusion systems are a next generation approach to the theory of finite groups- one major motivation being the opportunity to borrow techniques from homotopy theory. Extending work of Broto, Levi and Oliver, we introduce a new object - a 'tree of fusion systems' and give conditions (in terms of the orbit graph) for the completion to be saturated. We also demonstrate that these conditions are 'best possible' by producing appropriate counterexamples. Finally, we explain why these constructions provide a powerful way of building infinite families of fusion systems which are exotic (i.e. not realisable as the fusion system of a finite group) and give some concrete examples.
Although the importance of hydrologic uncertainty is widely recognized it is rarely considered in control problems, especially real-time control. One of the reasons is that stochastic control is computationally expensive, especially when control decisions are derived from spatially distributed models. This talk reviews relevant control concepts and describes how reduced order models can make stochastic control feasible for computationally demanding applications. The ideas are illustrated with a classic problem -- hydraulic control of a moving contaminant plume.
Let $X$ be a variety and $Z$ be a sub-variety. Can one glue vector bundles on $X-Z$ with vector bundles on some small neighborhood of $Z$? We survey two recent results on the process of gluing a vector bundle on the complement of a sub-variety with a vector bundle on some 'small' neighborhood of the sub-variety. This is joint work. The first with M. Temkin and is about gluing categories of coherent sheaves over the category of coherent sheaves on a Berkovich analytic space. The second with J. Block and is about gluing dg enhancements of the derived category of coherent sheaves.
In an Erdős--R\'enyi random graph above the phase transition, i.e.,
where there is a giant component, the size of (number of vertices in)
this giant component is asymptotically normally distributed, in that
its centred and scaled size converges to a normal distribution. This
statement does not tell us much about the probability of the giant
component having exactly a certain size. In joint work with B\'ela
Bollob\'as we prove a `local limit theorem' answering this question
for hypergraphs; the graph case was settled by Luczak and Łuczak.
The proof is based on a `smoothing' technique, deducing the local
limit result from the (much easier) `global' central limit theorem.
High-pressure freezing processes are a novel emerging technology in food processing,
offering significant improvements to the quality of frozen foods. To be able to simulate
plateau times and thermal history under different conditions, a generalized enthalpy
model of the high-pressure shift freezing process is presented. The model includes
the effects of pressure on conservation of enthalpy and incorporates the freezing point
depression of non-dilute food samples. In addition, the significant heat-transfer effects of
convection in the pressurizing medium are accounted for by solving the two-dimensional
Navier–Stokes equations.
The next question is: is high-pressure shift freezing good also in the long run?
A growth and coarsening model for ice crystals in a very simple food system will be discussed.
Topological quantum error correcting codes, such as the Toric code, are
ideal candidates for protecting a logical quantum bit against local noise.
How are we to get the best performance from these codes when an unknown
local perturbation is applied? This talk will discuss how knowledge, or lack
thereof, about the error affects the error correcting threshold, and how
thresholds can be improved by introducing randomness to the system. These
studies are directed at trying to understand how quantum information can be
encoded and passively protected in order to maximise the span of time between successive rounds of error correction, and what properties are
required of a topological system to induce a survival time that grows
sufficiently rapidly with system size. The talk is based on the following
papers: arXiv:1208.4924 and Phys. Rev. Lett. 107, 270502 (2011).