Part III Seminars Michaelmas 2014

The Michaelmas 2014 Part III seminars will take place on Wednesday 3rd, Thursday 4th and Friday 5th December. Each talk lasts 30 minutes followed by 15 minutes of questions.

Wednesday afternoon

Analysis and PDEs Statistics Continuum Mechanics and Quantum Information
Statistical learning, VC-dimension and SVMs
Wenda Zhou (2:00 MR5)
Super-resolution of clustered sparse signals using convex optimization
Raf Mertens (2:45 MR9)
The Metropolis-Hastings Algorithm and its Applications
Andi Wang (2:45 MR5)
Coffee, tea and snacks — 3:30-3:45 in the core
Causation without intervention
Kweku Abraham (3:45 MR4)
Coating Flow on a Rotating Vertical Disc
Matthew Crowe (3:45 MR13)
Bayesian Networks - A marriage of Probability and Graph Theory, in the context of Statistics.
Hyun Jik Kim (4:30 MR4)
Linear Analysis of Traveling and Standing Waves in Thermosolutal Convection
Sam Turton (4:30 MR13)

Thursday morning

Analysis and PDEs Algebra and Number Theory High Energy Physics 1
Infinite dimensional oddities: differential geometry with Banach spaces
David Vasak (9:45 MR14)
Conjugacy Probability
Misja Steinmetz (9:45 MR12)
Chebyshev series and radii polynomials in solving nonlinear analytical differential equations
Qingyun Zeng (10:30 MR14)
The average number of integral points on elliptic curves is bounded
Levent Alpoge (10:30 MR12)
Geometries in Physics
Vir Bulchandani (10:30 MR11)
Coffee, tea and snacks — 11:15-11:30 in the core
Fractional Calculus and PDEs
Erik Paemurru (11:30 MR14)
Iwasawa Theory
Tibor Backhausz (11:30 MR12)
Renormalization of Quantum Spacetime
Sebastian Mizera (11:30 MR9)
On (a,B,c)-ideals in Banach Spaces
Ksenia Niglas (12:15 MR14)
Singular Moduli
Alex Best (12:15 MR12)
Assorted thoughts on Gauge theories/ What do we mean by Curvature?
Calum Ross (12:15 MR9)

Thursday afternoon

Geometry and Topology Cosmology
Symplectic embeddings
Fritz Hiesmayr (2:00 MR12)
A derivation of the FRW-metric (used in Cosmology)
Nastasha Wijers (2:00 MR5)
Clifford algebras and Spin structures on manifolds
Mauricio Benjamin Garcia Tec (2:45 MR12)
Cosmic inflation, scalar-tensor theories and cosmological attractors
Emiel Woutersen (2:45 MR5)
Coffee, tea and snacks — 3:30-3:45 in the core
Atiyah Singer index formula for Dirac operators
Yang Li (3:45 MR12)

Friday morning

Algebraic Geometry Probability and Finance High Energy Physics 2
Topological Categories - When are function spaces (nice) topological spaces?
Hector Miller-Bakewell (9:00 MR5)
Hidden Symmety of Coulomb's Potential
Mishiko Gelenava (9:45 MR11)
Solution schemes
Andrea Dotto (10:30 MR5)
Existence and uniqueness of Brownian motion by use of the Haar wavelet.
Victor Rohde (10:30 MR4)
An introduction to on-shell amplitude methods
Julio Parra Martinez (10:30 MR11)
Coffee, tea and snacks — 11:15-11:30 in the core
Moduli spaces of pointed genus zero curves and effective cones
Morgan Opie (11:30 MR5)
Quantum Random Walks
Benjamin Walter (11:30 MR4)
Intersecting branes from 11-dimensional theory
Andres Olivares del Campo (11:30 MR11)
Two cohomology theories on Algebraic Varieties
John Gowers (12:15 MR5)
From neurons to equations and back again: topics in mathematical neuroscience
Maxwell Sherman (12:15 MR4)

Friday afternoon

Geometry and Topology Probability and Finance Continuum Mechanics and Quantum Information
Stochastic Particle Methods with Applications
Mo Dick Wong (2:00 MR4)
Modelling an artificial flagellar swimmer at low Reynolds number
Archie Bott (2:00 MR13)
Persistent homology and Topological Data Analysis
Mauricio Benjamin Garcia Tec (2:45 MR5)
Tranching and Rating
Peter Vang Uttenthal (2:45 MR4)
An Introduction to Tensor Networks in Many-Body Physics
Pieter Bogaert (2:50 MR15)
Coffee, tea and snacks — 3:30-3:45 in the core
The Borsuk-Ulam Theorem
Tobias Hemmert (3:45 MR5)
The Combinatorics and Cohomology of Open Covers
Hunter Spink (4:30 MR5)
Party!!! 5:30 in the Part III room

Analysis and PDEs

Wednesday afternoon. Group leader: Natalia Kudryashova (nk375).

Super-resolution of clustered sparse signals using convex optimization — Raf Mertens (raim2) — 2:45 MR9
Super-resolution is the problem of recovering the high end of the spectrum of a signal from measurements of the low end of the spectrum. In this talk I focus on super-resolution of signals which are superpositions of point sources from low frequency Fourier series coefficients. To overcome underdermination of the problem, I introduce a minimum separation condition on the distance between the sources. In addition, I assume that the sources are clustered. I propose a simple (but intractable) convex optimisation problem assuming that the signal support is known. I then derive a tractable dual problem through a connection with trigonometric polynomials and present Matlab simulations discussing how much support information is needed. I end by suggesting some methods of obtaining the support information.


Statistics

Wednesday afternoon. Group leader: Tom Berrett (tbb26).

Statistical learning, VC-dimension and SVMs — Wenda Zhou (wz258) — 2:00 MR5
We introduce the learning problem: given a set of hypothesis, and sample data from a given (unknown) distribution, how can we select the hypothesis that gives the best performance in general? Quantifying the performance of a given hypothesis in terms of its risk, we are then lead to the problem of bounding the generalization risk by the empirical risk. We will obtain a bound based on a quantity called the Rademacher complexity of our hypothesis set, and show how we can simplify this quantity to obtain a bound in term of the VC-dimension, a purely combinatorial quantity of our hypothesis set. Given the time, we will apply this framework to the case of high-margin classifiers such as SVM (support vector machines), and show that in this case, the error bounds are in fact independent of the dimension of the space.

The Metropolis-Hastings Algorithm and its Applications — Andi Wang (aqw20) — 2:45 MR5
In recent years Markov Chain Monte Carlo methods have been a very useful tool to sample from complex high-dimensional probability distributions. The Metropolis-Hasting Algorithm is a versatile, efficient and powerful simulation technique which underpins many of these MCMC methods. In my talk I will derive it, give some intuition, and finally show how it gives rise to the famous Gibbs sampler as a special case.

Causation without intervention — Kweku Abraham (lkwa2) — 3:45 MR4
In many circumstances, randomised controlled trials – science's main method for finding causal relationships – are impractical or unethical to perform. The mathematical field of causal inference has therefore been developed to help identify causal relationships in such circumstances. I will give a brief introduction to this field, outlining some of the methods used in it, the assumptions under which these methods are valid, and highlight some of the issues that arise in practice.

Bayesian Networks - A marriage of Probability and Graph Theory, in the context of Statistics. — Hyun Jik Kim (hjk42) — 4:30 MR4
Bayesian Networks provide an intuitive and elegant representation of many real-world phenomena, and are particularly useful for modelling causal relations. In this talk, we will first motivate the idea of a Bayesian Network with an example. Then we will identify the reasons for using such a model, followed by covering the formal definition of a Bayesian Network, its properties, the problems of learning and inference, and finally its applications to real life problems.


Continuum Mechanics and Quantum Information

Wednesday afternoon. Group leader: Adrien Lefauve (aspl2).

Coating Flow on a Rotating Vertical Disc — Matthew Crowe (mc756) — 3:45 MR13
It is possible to rotate a vertical disc with a thin fluid layer on the face such that the fluid does not drip off. Using lubrication theory, we can find an equation for the height profile across the disc. In the case of no surface tension we can find curves of constant height analytically and using an asymptotic expansion in small surface tension we can numerically calculate how the height profile varies across the disc.

Linear Analysis of Traveling and Standing Waves in Thermosolutal Convection — Sam Turton (set42) — 4:30 MR13
I will discuss a research project that I carried out at PMMH-ESPCI during summer 2014, and whose results I presented at the APS-DFD conference in November this year. The work was motivated by studies of vortex shedding in wake of a cylinder, but I will discuss the thermosolutal problem: convection driven by opposing thermal and solutal gradients. In the archetypal two-dimensional geometry with free-slip vertical walls and periodic horizontal boundaries, traveling waves are created by a Hopf bifurcation. I find that the linearization about the mean fields of these traveling waves yields an eigenvalue whose real part is almost zero and whose imaginary part corresponds very closely to the nonlinear wavespeed, consistent with similar analyses performed on the cylinder wake. The Hopf bifurcation also produces a branch of standing waves. Linearization about the mean field of the standing waves yields results which are quite different. I finally demonstrate that success of the mean field linear stability analysis occurs when the temporal power spectrum is sharply peaked. During the seminar I will give a brief overview of dynamical systems, and supercritical Hopf bifurcations in particular, in the context of thermosolutal convection. I also intend to discuss some of the numerical methods I used to generate my results.


Analysis and PDEs

Thursday morning. Group leader: Natalia Kudryashova (nk375).

Infinite dimensional oddities: differential geometry with Banach spaces — David Vasak (dsv24) — 9:45 MR14
Mathematicians have long liked to see patterns across the dimensions, pairing circles and spheres, triangles and tetrahedra or squares and cubes. As geometrical foundations became more sophisticated, even higher dimensional analogues of these objects became apparent. In the last century, the foundations of geometry have become sophisticated enough to allow discussions of these objects in infinite dimensions. But it turns out that they are very different to their finite dimensional cousins, in their topological structure, algebraic invariants and differential geometry. This talk will discuss a few of the surprises that await the unwary mathematician.

Chebyshev series and radii polynomials in solving nonlinear analytical differential equations — Qingyun Zeng (zq216) — 10:30 MR14
Chebyshev series is similar to Fourier series and it has advantage in numerical computing. The Chebyshev coefficients of an analytical function will decay exponentially, which means we can evaluate nonlinear equations by estimation the discrete convolutions of nonlinear terms. It turns out the the Banach algebra structure of analytical functions on [-1,1] gives us the same decay rate in discrete convolutions. Hence we can use this in solving initial value problems (IVP) in nonlinear
differential equations. We transform the IVP problem into a IVP operator, and construct a iteration scheme. Then We will use numerical verication method and construct a so called radii polynomials to show
that if there is an r such that radii polynomials take negative values, then the real
solution x of the IVP problem is inside Br(x).

Fractional Calculus and PDEs — Erik Paemurru (ep449) — 11:30 MR14
We all know how to take a derivative from a function and how to take higher order derivatives. But can we take half a derivative? Would it even make sense? This question is answered in Fractional Calculus which I will introduce and motivate from various viewpoints. If time allows, I will also talk about an alternative method to solve first order linear PDEs.

On (a,B,c)-ideals in Banach Spaces — Ksenia Niglas (kr411) — 12:15 MR14
In this talk we focus on subspaces of Banach spaces that are (a, B, c)-ideals. We study (a, B, c)-ideals in l2∞ and present easily verifiable conditions for a subspace of l2∞ to be an (a, B, c)-ideal. Our main results concern the transitivity of (a, B, c)-ideals. We show that if X is an (a, B, c)-ideal in Y and Y is a (d, E, f)-ideal in Z, then X is a certain type of ideal in Z. Relying on this result, we show that if X is an (a, B, c)-ideal in its bidual, then X is a certain type of ideal in X(2n) for every n ∈ N.


Algebra and Number Theory

Thursday morning. Group leader: Nicolas Dupré (nd332).

Conjugacy Probability — Misja Steinmetz (mfas3) — 9:45 MR12
Suppose G is a finite group. I will talk about a very natural question: if we randomly pick two elements from this group, then what is the probability that these elements are conjugate? This 'conjugacy probability' has, somewhat surprisingly, been studied very little. In fact, there exists only one paper on this topic by Blackburn, Britnell and Wildon from 2012. In my talk I will derive and explain some initial results from this paper and look more closely at the conjugacy probability of the symmetric group on n elements, in particular the asymptotic behaviour of this probability as n goes to infinity. If there is enough time, I will explain a result by Madeleine Whybrow and myself on the asymptotic behaviour of the conjugacy probability of the alternating group.

The average number of integral points on elliptic curves is bounded — Levent Alpoge (la312) — 10:30 MR12
I will show the claim in the title. (I will also tell you what "average" means --- it's the same "average" as occurs in Bhargava-Shankar's work on the "average" rank of elliptic curves.) The methods include bounds on spherical codes, methods from Diophantine approximation, detailed study of the Weil and Neron-Tate heights on 'most' curves, and repeated use of the fact that integers and integral points on curves repel each other. But don't worry, I will keep things strictly elementary and detail the ideas rather than the details in the talk.

Iwasawa Theory — Tibor Backhausz (tab47) — 11:30 MR12
Iwasawa Theory studies the growth of number theoretic objects as the base field varies in a "tower of fields".
For different kinds of towers of fields, there are different Main Conjectures that state that this growth is quite strictly controlled by a p-adic zeta function.
After an introduction to the basic definitions, I will state the Main Conjecture of Iwasawa Theory for cyclotomic extensions.

Singular Moduli — Alex Best (ajb304) — 12:15 MR12
Singular moduli is the name given to certain special values of a very special function (the j-invariant). These values were studied by number theorists in the 19th century and many thought the story was complete, until more new properties of singular moduli were discovered in the early 80's. In this talk we'll look initially at the classical theory of singular moduli, how they relate to class field theory and how we can use them to explain some striking arithmetic equalities. After this I'll try to provide an overview of some more recent work concerning singular moduli such as Gross and Zagier's work on factoring their differences.


High Energy Physics 1

Thursday morning. Group leader: Adam Solomon (as2089).

Geometries in Physics — Vir Bulchandani (vb311) — 10:30 MR11
Vector bundles are ubiquitous in physics yet they are only discussed in Part III differential geometry. I shall summarise the basic definitions, and show how bundles offer a unified framework for thinking about classical, quantum and relativistic physics. I will then show how some important ideas in pure mathematics, such as principal bundles and spin structures on manifolds, are most naturally thought of in physical terms. If there is time, I will discuss how to construct a coordinate independent isomorphism between Lagrangian and Hamiltonian mechanics.

Renormalization of Quantum Spacetime — Sebastian Mizera (sam228) — 11:30 MR9
Quantum gravity theories provide a way of describing spacetimes with quantized geometry. Despite numerous theoretical consistency checks, they usually suffer from the lack of phenomenological predictions. Most notably, it is still not known whether they recover classical GR in the continuum limit. We will briefly discuss the motivations behind the quantum gravity programme and provide a rough overview of the state-of-the-art developments. After a short introduction to the idea of non-perturbative renormalization, we will show how the application of tensor network techniques—commonly used tools in the condensed matter research—allows for the study of the phase-space structure of lattice gauge theories; in particular the covariant loop quantum gravity models.

Assorted thoughts on Gauge theories/ What do we mean by Curvature? — Calum Ross (cdhr2) — 12:15 MR9
Non-Abelian Gauge theories, what they are why we use them and some interesting physics that comes out of the nonlinearity. This will be a varied tour through a lot of the things I find interesting about gauge theories, starting in the abelian case with a discussion of Maxwellian electrodynamics and proceeding from there. The level of rigour will change based on the topic being discussed, my familiarity with the topic, the degree to which being rigourous will help to get points across and generally my mood at the time of preparing the outline. I would hope that the talk should be accessible to most people though at some points I may assume some familiarity with basic particle physics while at others assuming an enthusiasim for a coordinate free expressions or at least a willingness to be exposed to them.


Geometry and Topology

Thursday afternoon. Group leader: Dmitry Tonkonog (dt385).

Symplectic embeddings — Fritz Hiesmayr (flh27) — 2:00 MR12
Symplectic manifolds are smooth manifolds equipped with a non-degenerate closed 2-form. The smooth structure induced by the 2-form on the manifold can be thought of as more rigid than its volume: a map preserving the symplectic structure will necessarily preserve the manifold's volume, but the converse is not true in general. Moreover, it's trickier to visualise such symplectic maps. I will discuss the basic definitions and properties of symplectic manifolds, to then state some results about existence (and non-existence) of symplectic embeddings. If time permits, I would like to talk about the construction of symplectic embeddings via the moment map.

Clifford algebras and Spin structures on manifolds — Mauricio Benjamin Garcia Tec (mbg29) — 2:45 MR12
Clifford or ‘geometric’ algebras provide a setting that helps to understand many of the basic features of Spin structures on manifolds. They generalize the real numbers, complex numbers, quaternions, etc. The Spin groups are related to a family of automorphisms of the geometric algebra that extend the usual ‘rotations’ of a vector space (this is behind their usual definition as the double cover of SO(n)). In this talk we will focus on the elementary theory of Clifford Algebras and the existence of Spin structures on manifolds and their obstructions in terms of Steifel-Whitney classes. Time permitting, I will present some classical results applied to differential geometry. Hopefully, this talk will provide some background for the talk on the Atiyah Singer index formula for Dirac operators and for Sir Atiyah's talk on spinors later this day. Although spin structures are pervasive in the theory, the algebra behind them is very rich and a topic in itself.

Atiyah Singer index formula for Dirac operators — Yang Li (yl454) — 3:45 MR12
The Atiyah Singer index formula enables us to compute the Fredholm index of the an elliptic operator over a vector bundle (a priori a question about elliptic PDEs) using topological data about the vector bundle (the characteristic classes). We will restrict our attention to the case of Dirac operators, which are first order operators on spin manifolds (and so named because of its physical analog for Minkowski space). The proof (using heat equations) is rather involved so I will not give full details but hopefully I can convince you that the problem is doable. I expect you to know about basic differential geometry, and have some exposure to PDEs or at least have the willingness to take analytical results on trust.


Cosmology

Thursday afternoon. Group leader: Davide Gerosa (dg438).

A derivation of the FRW-metric (used in Cosmology) — Nastasha Wijers (naw37) — 2:00 MR5
The FLRW (Friedmann-Lemaitre-Robertson-Walker) metric describes a space-time, of which the spatial part is uniform and without a preferred direction. In cosmology, this metric is used to describe our universe on the largest scales. Predictions of the evolution of the universe using this metric largely fit observations. In this talk, I will discuss a derivation of the FLRW metric using the high degree of symmetry this space-time possesses, focusing on showing uniqueness of this solution.

Cosmic inflation, scalar-tensor theories and cosmological attractors — Emiel Woutersen (ew459) — 2:45 MR5
Though Big Bang cosmology is succesful in predicting many features of the universe, it fails to account for the high degree of homogeneity of the Cosmic Microwave Background and the relative flatness of the universe. To solve these problems, cosmic inflation is introduced. In this talk, I will present gravitational scalar-tensor theory as a framework to investigate the scalar fields which cause inflation. In scalar-tensor theory, a scalar field is non-minimally coupled to gravity. This feature allows for a wider spectrum of potentials which allow for inflation, which becomes apparent after a conformal transformation of the Einstein action. Special attention is paid to a class of models which all produce the same observables in the strong coupling limit. Finally, a subset of this class is studied which also displays this attractor behaviour in the weak coupling limit.


Algebraic Geometry

Friday morning. Group leader: Georgios Charalambous (gc439).

Topological Categories - When are function spaces (nice) topological spaces? — Hector Miller-Bakewell (hm380) — 9:00 MR5
I hope to cover why the category of topological spaces isn't "cartesian closed". We will cover what this means and why we care about it, and enough category theory to show this result. Only basic topology should be required, and we will cover the necessary category theory in the presentation.

Solution schemes — Andrea Dotto (ad724) — 10:30 MR5
Procure yourself a field and some polynomial equations: can you tell whether they have a common solution?
Elimination theory provides an answer to this question when working over algebraically closed fields. Using the language of schemes, it is possible to associate to your equations a geometric object which parametrizes their solutions, plus some more information. Doing this one obtains a new perspective on the classical answer, together with some sense of what schemes do and how they look like. How this is done is the subject of this talk.

Moduli spaces of pointed genus zero curves and effective cones — Morgan Opie (mpo27) — 11:30 MR5
I'll informally talk about moduli spaces, focusing on the moduli space of stable rational curves with marked points. I'll then dicuss divisors, the divisor class group, and effective cones. As time allows, I'll summarize my undergraduate thesis results related to these topics.

Two cohomology theories on Algebraic Varieties — John Gowers (wjg29) — 12:15 MR5
I shall give an introduction to Čech cohomology and derived functor cohomology, first developing the theories over topological spaces for illustration and then commenting in the extension to ringed spaces. Time permitting, I shall briefly introduce several other cohomology theories, which all agree for nice enough geometric objects (particularly algebraic varieties).


Probability and Finance

Friday morning. Group leader: James Bell (jhb43).

Existence and uniqueness of Brownian motion by use of the Haar wavelet. — Victor Rohde (vur20) — 10:30 MR4
Starting from the Haar wavelet, an orthonormal basis in L^2, I will build Brownian motion in the spirit of N. Wiener’s original idea. This construction has a nice intuitive approach to proving existence of Brownian motion and is, to my mind, well worth spending time studying as an alternative to the one that was given in the Advanced Probability lectures.

Quantum Random Walks — Benjamin Walter (bw360) — 11:30 MR4
Rethinking a random walk in terms of the quantum formalism yields strikingly new behaviour. The talk will comprise also a short introduction to general quantum mechanics for non-physicists and an even shorter introduction into classical random-walks (on graphs) for non-probability theorists.

From neurons to equations and back again: topics in mathematical neuroscience — Maxwell Sherman (mas250) — 12:15 MR4
The human brain - three pounds of conductive jelly encased in bone - has created the richly detailed world of mathematics. Yet, basic functionality of the brain remains a mystery. In fortuitous reflexivity, several branches of mathematics provide powerful tools to explore these mysteries. Here we will provide an introduction to several such tools. Over the past several decades, useful technologies to non-invasively measure the activity of the human brain have emerged (e.g. EEG, MEG, fMRI). An enduring challenge of these techniques is their lack of temporal and/or spatial resolution, which prevents a direct understanding of these signals in terms of underlying brain activity. Our desire is thus to use mathematics to infer some causal link between true brain activity and these macroscopic signals. Through a combination of applied statistics, probability, and differential equations, such inference is possible. First, various signal processing techniques can be used to quantify the salient statistical features of the macroscopic neural signal. Then, using a combination of partial differential equations and stochastic processes, we can construct biophysically and anatomically accurate computational models of the human brain down to details of the individual neuron. We can then use these models to quantify precisely what neural activity is responsible for our observed signal. This talk will provide a brief introduction to relevant neuroscience; briefly discuss common time-series statistical tests for neuroimaging data; and then motivate and discussing the construction of mathematical models of the human brain. We will consistently refer to examples and applications throughout.


High Energy Physics 2

Friday morning. Group leader: Benjamin Wallisch (bw336).

Hidden Symmety of Coulomb's Potential — Mishiko Gelenava (mg707) — 9:45 MR11
Coulomb's potential is a very important potential in theoretical physics. The motion of bodies on astrophysical and atomic scales are governed by Coulomb's potential. In this talk, I show the dynamical symmetry of Coulomb's potential which makes it different from just the central symmetry field. Planets around the sun have periodic trajectories and electron's energy levels in the hydrogen atom are degenerate. The aim of this talk is to show what kind of degenerations we have and what conservation laws are connected to them in classical, non-relativistic quantum and relativistic quantum mechanics.

An introduction to on-shell amplitude methods — Julio Parra Martinez (jp667) — 10:30 MR11
In the last years many surprising properties and hidden symmetries of scattering amplitudes have been unveiled. In part, this is because we have translated our previous knowledge to a new language, the language of spinor-helicity variables, recursion relations, grassmannians and twistors, among other things. In my talk I will try to introduce this new set of variables and the BCFW recursion relations, which will allow us to forget about Feynman diagrams and "virtual nonsense" and calculate scattering amplitudes in gauge theories from the 3-particle amplitudes, which are given purely by symmetry arguments. At the end of the talk I will sketch the proof of the Parke-Taylor amplitude formula using these tools. This is the formula whose simplicity, hidden within a mess of Feynman diagrams, sparkled the work on this field.

Intersecting branes from 11-dimensional theory — Andres Olivares del Campo (ao369) — 11:30 MR11
In String Theory, branes are extended dimensional objects that can be understood as topological defects. I would like to review the intersecting branes solutions that arise from the 11-dimensional theory. Starting from the general form of the p-brane equations in D dimensions and using Kaluza-Klein dimensional reduction techniques, I will present different examples of intersecting branes descending from the 11-dimensional theory.


Geometry and Topology

Friday afternoon. Group leader: Dmitry Tonkonog (dt385).

Persistent homology and Topological Data Analysis — Mauricio Benjamin Garcia Tec (mbg29) — 2:45 MR5
Persistent homology is the most widely used application of topology to data analysis. It is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of length and are deemed more likely to represent true features of the underlying space, rather than artifacts of sampling, noise, or particular choice of parameters. The general idea of its application to data analysis is to transform a dataset into a filtered simplicial complex and compute the rank of its homology groups. I will present an application of this method using the software R. A homotopotical version of this technique is not known. Gromov’s theory of quantitative homotopy theory might shed some light on this. There is a recent paper by Blumberg and Mandell also dealing with the subject as well. Time permitting, I will discuss something about this.

The Borsuk-Ulam Theorem — Tobias Hemmert (th410) — 3:45 MR5
The Bursuk-Ulam theorem, in one of its various formulations, states that for every continuous map from the n-sphere to n-dimensional Euclidean space, there exists a point on the sphere that has the same image as its antipodal point. It has many proofs, some of them elementary, some using more sophisticated machinery. In this talk, I will illustrate the theorem and give a proof using singular homology. Moreover, the Borsuk-Ulam theorem has many interesting and nice applications in geometry and combinatorics and I intend to talk about a few of them.
The proof of the theorem that I will give requires some knowledge of homology, but the rest of the talk will be accessible to anyone interested.

The Combinatorics and Cohomology of Open Covers — Hunter Spink (hs499) — 4:30 MR5
Can R^2 be covered by 3 connected open sets which intersect but not triply? Can R^3 be covered by 4 connected open sets which triply intersect but not quadruply? One of these is possible, the other isn't! I hope to unwind a high-level proof to its most basic components, and show why it fails to generalize.


Probability and Finance

Friday afternoon. Group leader: James Bell (jhb43).

Stochastic Particle Methods with Applications — Mo Dick Wong (mdw46) — 2:00 MR4
Stochastic particle method is one of the most popular tools in solving complex problems in applied probability and statistics nowadays. The talk will commence with a multilevel splitting algorithm for the estimation of rare event probabilities in the context of computational finance, which will give us motivations for the pursuit of a more general framework of simulation. I shall then give a quick overview of the particle methodology. After that we shall see how this more sophisticated approach also fits into the filtering problem. If time permits we may also cover noncommutative models and Feynman-Kac sensitivity measure.

Tranching and Rating — Peter Vang Uttenthal (pvu20) — 2:45 MR4
Empirical evidence from the years leading up to the recent financial crisis suggests that collateralized debt obligation (CDO) tranches were often sold at yields based solely on their credit ratings. Ratings, in turn, only capture total default probabilities or expected default losses, and do not take into account the state of the world in which default losses are most likely to occur. Yet, CDOs tend to experience a high degree of correlation among default events of the underlying reference securities. This gave rise to arbitrage opportunities from securitising debt. I will show how to construct a model that can quantify such gains from tranching. The model helps explain why CDOs were sold on such a large scale, and why ratings are inadequate measures of their riskiness.


Continuum Mechanics and Quantum Information

Friday afternoon. Group leader: Adrien Lefauve (aspl2).

Modelling an artificial flagellar swimmer at low Reynolds number — Archie Bott (afab2) — 2:00 MR13
Earlier this year the first functional self-propelled, synthetic flagellar micro-swimmer at low Reynolds number was produced (by a group from the Department of Mechanical and Aerospace Engineering at Arizona State University). A typical such device is a few millimetres long, and consist of a short, rigid head and a long, thin flexible polymeric tail on which cardiomyocytes (cardiac cells) are selectively cultured. The periodic contraction and deformation of these cells causes propulsion. However, although some theoretical modelling of the swimmer was done before carrying out the fabrication process as a design aid, the approach used makes some problematic assumptions, and does not capture some experimentally observed qualitative behaviour. This talk will attempt to rectify both these issues whilst maintaining a simple model. I will also address the issue of slow swimming speeds exhibited by the device relative to natural equivalents, and how this might be improved.

An Introduction to Tensor Networks in Many-Body Physics — Pieter Bogaert (pawb2) — 2:50 MR15
Quantum many-body systems with weak correlations can be described by a mean field ansatz. However, in cases with strong correlations, such as models for high T_c superconductivity, different approaches are needed. Tensor networks are a powerful tool to approximate ground states of a large class of Hamiltonians, even in the presence of strong correlations. The talk will introduce tensor networks by looking at entanglement properties of quantum lattice systems (so-called area laws for the entanglement entropy). In the second half of the talk, we will see how to manipulate tensor networks by looking at an application in classical statistical physics.