Week starting Monday 21 October 2019
Satisfaction Guaranteed: The talks will be colloquium style. In particular, the first 20 minutes will be accessible to anybody who vaguely remembers the content of Algebra 2 and Analysis 2. They will run precisely on time.
Talk 1 (Monday 9:30): Ladder Categories (Mitchell Rowett)
Given two modules over a ring, we can form their tensor product over that ring. Moving up to the world of categories, our analogies for rings are given by (linear) monoidal categories. Indeed, we have a notion of a module over a monoidal category C; given two such modules we can define a tensor product over C, which we will call a ladder category (for reasons which will hopefully become obvious!).
Talk 2 (Monday 10:30): Beyond Expectations, But Within Limits — The Theory of Coherent Risk Measures (Yuxi Liu).
Expectation lies at the foundation of probability, but its centrality belies its contingency. Abstractly, the expectation of a random variable is a real number that measures some information about the variable, and one may well explore the consequences of replacing expectation by some alternative.Certain alternatives to expectation, termed "coherent risk measures", have been well-investigated in financial engineering, but they are relatively unknown in the field of machine learning. Here, we collect and prove some fundamental properties of coherent risk measures that we believe would be applicable to machine learning.
In chapter 1, we review the concept of risk measures, point out possible deficiencies of the expectation as a risk measure, then provide a historical overview of the study of risk measures in finance and other areas.
In chapter 2, we review basic probability concepts, then define the concept of coherent risk measures and study the geometric properties of their envelope representations. Armed with geometric insight, we prove a Kusuoka representation theorem when the underlying sample space is finite and uniform, and construct counterexamples when it is finite but nonuniform.
In chapter 3, we generalize some basic probability inequalities and concentration inequalities from expectation to conditional value at risk. Then we review Statistical Learning Theory and generalize its fundamental theorem by replacing expectation with spectral risk measures.
In chapter 4, we review limit theorems in probability, then give a new and intuitive proof of the Central Limit Theorem for the empirical estimator of conditional value at risk. We provide numerical evidence to support our results and generate conjectures.
Supervisor: Robert Williamson
Cosupervisor: Markus Hegland
Talk 3 (Monday 1pm): The direct method in the calculus of variations (Douglas Coulter).
Often, we'll want to find a function $u$ that minimises an integral like
$$ E(u) = \int f(x,u(x),Du(x)) \, dx.$$ But sometimes $E$ lacks a minimiser.
So what conditions on $f$ and the space of functions that $u$ can belong to are necessary and/or sufficient for a minimiser to exist?
Supervisor: John Urbas
Talk 4 (Monday 2pm): Transport Methods for study of Pdes (James King)
When one considers the dynamics of particles, certain partial differential equations naturally arise. For example, if one to consider the dynamics of an inert gas, one would arrive at the heat equation.
Traditionally, the existence of solutions to the equation above equation are proven usng analytical techniques -- such as the Fourier transform. This thesis outlines an alternative approach using concepts of transport and gradient flows in metric spaces to not only show the existence of solutions to the above equation and many others like it. Examples of partial differential equations that can be analysed using this technique are the porous medium equation, and the quantum drift diffusion equation.
Supervisor: John Urbas
Talk 5 (Monday 3pm): Christopher Hone
Talk 6 (Tuesday 9:30): Drinfeld centers and representations of the annular category (Keeley Hoek)
Supervisor: Scott Morrison.
Talk 7 (Tuesday 10:30): Modal Logic formalization in HOL (Yiming Xu)
We will start by introducing what is a theorem proving and what is a theorem prover. Then we will talk about what is modal logic and what we did to formalize it in HOL. By the end of the talk, we will sketch a proof of a theorem called 'finite model property', and talk about its formalization in HOL.
Supervisor: Michael Norrish.
Co-supervisor: Scott Morrison.
Talk 8 (Tuesday 1pm): Fourier Phase Retrieval (James Martini)
Supervisor: Qinian Jin
Talk 9 (Tuesday 2pm): Contact Surgery (Joshua Tomlin)
Surgery is a fundamental tool in 3-manifold topology for studying and and creating new 3-manifolds. Contact geometry is also a useful and more recently developed tool for studying 3-manifolds. We will explore how to apply surgery techniques to contact 3-manifolds, particularly focusing on their effect on tight 3-manifolds.
Supervisor: Joan Licata
We will watch a talk given by Kathryn Hess, this talk follws the one we watched on October 11th. We might pause to discuss or discuss afterwards.
Abstract: I will define operads and modules over operads, in the context of both chain complexes and topological spaces. I will then describe a number of important examples of these structures arising in algebraic topology and explain their significance and utility.
The video and some notes are available here: http://www.msri.org/workshops/684/schedules/17867