Week starting Monday 18 November 2019
Water resources management usually relies on mathematical and computer-based models to explore the likely impacts of management decisions. These models are often integrated, meaning that they combine elements from a range of different disciplines, including social, economic and ecological components. The core of the integrated model is the hydrological component, which simulates water availability given climatic conditions and water demands. In many Australian river basins, surface water and groundwater resources are highly interconnected and they must be modelled, and managed, as a single resource. Despite the undeniable importance of representing the interactions between connected surface water – groundwater (SW-GW) systems, there is a lack of consensus regarding a suitable approach, particularly for large study areas with sparse observational data. This PhD thesis presents the development of an integrated SW-GW model suitable for use in large-scale modelling applications where data and computational resources may be limited. Two alternative models are presented, both apply simple, ‘mass-balance’ approaches for representing surface water, but differing levels of complexity for the groundwater domain. We find that, while there is no single “best” solution, simple representations of groundwater processes can significantly improve the simulation of baseflows in groundwater dependent streams.
Abstract: A vertex algebra describes the symmetries of a two-dimensional conformal field theory (CFT), while a factorization algebra (introduced by Beilinson and Drinfeld) over a complex curve consists of local data in such a field theory. Roughly, the factorization structure encodes collisions between local operators. The two perspectives are equivalent, in the sense that, given a fixed open affine curve X, the category of vertex algebras over X is equivalent to the category of factorization algebras over X. In the late 1990s, Borcherds gave an alternate definition of some vertex algebras as "singular commutative rings" in a category of functors depending on some input data (A,H,S). He proved that for a certain choice of A, H, and S, the singular commutative rings he defines do indeed give examples of vertex algebras. In this talk I will explain how we can vary this input data to produce categories of chiral algebras and factorization algebras (in the sense of Beilinson--Drinfeld) over certain complex curves X. We'll also discuss the failure of these constructions to give equivalences of categories. I will not assume background in vertex algebras, factorization algebras, or chiral algebras.
We will watch the first two parts of a four part mini-course given by Agnes Beaudry. We might pause to discuss or discuss afterwards.
The video for the first talk is here: https://www.youtube.com/watch?v=ZsHtsx_A8j8
More information is here: https://sites.google.com/colorado.edu/agnesbeaudry/links/echt-may-2019
At the center of homotopy theory is the classical problem of understanding the stable homotopy groups of spheres. Despite its simple definition, this object is extremely intricate; there is no hope of computing it completely. It hides beauty and pattern behind a veil of complexity.
Chromatic homotopy theory gives us a way to see through this veil by using the algebraic geometry of formal groups to organize theory and computations. The simplest case is via K-theory: Bott periodicity gives rise to a repeating family of elements in the stable homotopy groups of spheres via the image of the J homomorphism. It is an insight of Morava that there are higher analogues of K-theory and that they should give rise to higher periodicity in the stable homotopy groups of spheres.
In the 1980s, Ravenel and Hopkins made a series of conjectures describing this connection, most of which were proved in the 1980s and 1990s by Devinatz, Hopkins, Smith and Ravenel. Two of these problems remain open: the chromatic splitting conjecture and the telescope conjecture. The ultimate goal of the course will be to motivate and state these two famous problems.
The course will include a quick reminder of spectra and a brief introduction to complex orientations and localizations. We will discuss periodicity and the chromatic filtration, leading to a statement of the two conjectures. We will also discuss the higher K-theories and the role they play in modern computations. Many topics will only be touched briefly, as my intention is to provide a roadmap of the field to non-experts.
I will assume the knowledge of an advanced course in algebraic topology, and some familiarity with the category of spectra, K-theory, the Adams operations and cobordism.
Subfactors of von Neumann factors have a rich representation theory that gives rise to interesting mathematical structures such as fusion categories, planar algebras or link invariants. They are highly noncommutative algebras of operators and in general not determined by their representations. It is open how to distinguish them. I will explain a natural notion of noncommutativity for a subfactor and illustrate it with a theorem that provides the first examples of very noncommutative, irreducible subfactors. This notion might also be of interest in quantum information theory.