Registration for Part III Seminars Lent 2016

The Lent 2016 Part III seminar series will be taking place at the end of term, from Wednesday 9th to Friday 11th March. This term's talks are slightly longer than last term's; we recommend that they're 45 minutes, and there will be 15 minutes for questions after each talk.

If you'd like to give a talk, leave a message below with

  1. Your name and CRSid (abc123)
  2. The title of your talk.
  3. The general subject area(s) of the talk. Pick the best approximation(s) from the following: Algebra, Analysis, Combinatorics, Geometry & Topology, Logic, Number Theory, Probability, Statistics, Operational Research & Mathematical Finance, Particle Physics, Quantum Fields & Strings, Relativity & Gravitation, Astrophysics, Quantum Computation/Information/Foundations, Philosophy of Physics, Applied & Computational Analysis, Continuum Mechanics.
  4. Specification/clarification of the subject area (if not one of the above).
  5. Any restrictions on when you can give a talk between Wed 9/3 afternoon and Fri 11/3 (e.g. "Can only do Thursday afternoon")
  6. Whether you'd like us to record your talk on camera.
  7. Your abstract

The deadline for signing up is Sunday 28th February, 23:59.

If you have any questions, please contact the Part III Seminar directors: James Bell (jhb43), Amalia Thomas (at682), Jonathan Michael Foonlan Tsang (jmft2).

1.) Nils Prigge (np440) 2.)

1.) Nils Prigge (np440)

2.) Topological invariance of rational Pontrjagin classes

3.) Geometry and Topology

4.) Geometric Topology

5.)

6.) Ok

7.) We usually define the Pontrjagin classes of a smooth manifold as certain characteristic classes of the tangent bundle.  Novikov proved that the rational Pontrjagin classes of homeomorphic smooth manifolds actually coincide, which may come as a surprise given that we needed the smooth structure to define them in the first place. In this talk , I will try to discuss the ideas of the original proof from Novikov that uses many useful arguments from differential and geometric topology.

Conformal symmetry in two spacetime dimensions

1. Kieran Macfarlane (ksm39)
2. Conformal symmetry in two spacetime dimensions
3. Quantum Fields & Strings
4. Related to essay topic "Affine Lie algebras and their physical applications (Dorey)"
5. Ideally on Wednesday
6. Yes
7. A field theory has conformal symmetry if looks the same at all lengthscales, ie the theory can only “see” angles. The global conformal symmetry group in d spacetime dimensions can be identified with the non-compact Lie group SO(d+1,2). Depending on your point of view, conformal symmetry can be interpreted as either a gauge redundancy or a true continuous symmetry with associated Noether currents.

So called conformal field theories (CFTs) are heavily restricted quantum field theories (QFTs), and are typically studied in a different way to more general QFTs. In particular, there can be no characteristic lengthscale, which implies that all fields in the Lagrangian must be massless. Instead of computing S-matrix elements, one studies correlation functions of operators.

In two spacetime dimensions, the theory of CFTs is much richer than the group of global conformal transformations. In the neighbourhood of any point, there is an infinite dimensional space of locally conformal transformations. This is closely related to the theory of holomorphic (analytic) functions in the complex plane.

In this talk I will give a brief introduction to some of the formalism mentioned above and develop some of the special theory of the two dimensional case. If there is time I will outline how these ideas can be applied to string theory.

TBC

Abstract TBC

mjc249

  1. mjc249
  2. Combinatrial Constructions in Ergodic Theory - the Arnoux, Ornstein Weiss Theorem.
  3. General subject areas: Analysis, Combinatorics.
  4. More analysis than combinatorics.
  5. Can't do Friday, Thursday would be best.
  6. Fine with recording.
  7. I shall discuss 'cutting and stacking transformations' of the unit interval which can be thought of as either a generalisation of the circle rotation or as an infinite sequence of 'Rokhlin towers' (don't worry if you don't now what these are, no prior knowledge will be assumed).  This will lead to the AOW theorem which says that any invertible, aperiodic MPS on a Borel probability space is isomorphic to such a construction. This can be used to realise such MPSs as flows on an open 2-manifold and is a step towards realising such transformations on compact manifolds. No prior knowledge except some basic measure theory (Lebesgue measure, sigma algebras etc) will be assumed. For those who took the Ergodic Theory course, this will be useful in gaining intuition about the cutting and stacking transformations there (at least I found it useful!).

Causal inference and invariant prediction

  1. Adam Foster (aef39)
  2. Causal inference and invariant prediction
  3. Statistics
  4. Causal inference
  5. None
  6. Yes
  7. Often statistical analysis is the precursor to an intervention such as passing legislation, bidding more for a particular commodity, or prescribing a drug to patients. Intervention can change the distribution of the underlying random variables, meaning that the predictions from the prior statistical analysis may end up being wrong. Causal inference seeks models which can make good predictions for many different distributions, including ones which have never been observed. This talk focuses on a new method for causal inference called invariant prediction. Unlike earlier methods for causal inference, invariant prediction offers confidence intervals for the conclusions it draws.

    This talk is based on the 2015 paper "Causal inference using invariance prediction: identification and confidence intervals" by Peters, Buehlmann and Meinshausen. I will include the necessary background on causal inference. This talk will be accessible to all Part III students

Francois Renaud fjr31

A gift for analysis : infinitesimals (with love, logic)

Logic and analysis

If you like analysis, if you use analysis, or if you wish to understand analysis, here is a useful tool for you : infinitesimals.

Mathematical analysis has for a long time relied on notions such as "infinitesimally small variations of time" or "infinitesimal quantities". When scientists such as Newton and Liebniz develop the first general theories of differential and integration calculus they intensively use these notions, that are fundamental to their work. However as the field was further developed, and the theories required to be better formalized, the question of a reliable foundation for analysis and the notion of infinitesimal became of central interest. The notion of limit designed by Cauchy and Weirstrass has been used to solve this problem, and the notion of infinitesimal has become merely an informal, intuitive support for the understanding of formal analysis, only used practically in more applied fields of sciences ...
However logic has been developed since then, and our understanding of how to incorporate these infinitesimals inbetween the real numbers is now as reliable as the notion of real number itself.

In this talk, I would like to give you a clear picture of how to introduce infinitesimal numbers in the real line, and how to use them to discribe analysis in an easy and intuitive way. The resulting theory is the same ! You just have new tools to describe it, tools that make life (teaching, research, modellisation, calculations) easier. As mathematics can be understood as the development of abstract technology to design models and describe phenomena, this is an excellent example of how the sophistication of logic opens new perspectives by giving acces to new formalised ideas, new effective abstract technology.

A record would be welcome.

Catherine Willis cw529

A Set Theory you can fit in your Pocket

 

Logic

 

Should be accessible to anyone with a vague knowledge of the axioms of ZFC

 

Please don't record me.

Matt Butler mb928

1. Matt Butler (mb928)

2. A Design for Artificial Microswimmers

3. Continuum Mechanics

4. Artificial Microswimmers

5. N/A

6. Ok

7. In the last few years there have been some significant developments, both experimentally and theoretically, in designing and modelling artificial microswimmers. In this talk I will try to explain why this is important, and outline some of the progress that has been made. There will then be a focus on one particular design, namely Janus particles, where we will look in some detail as to how they achieve locomotion. Some knowledge of Stokes flows would be helpful to understand the talk, but it should be accessible to all Part III students.

Covariant Loop Quantum Gravity

  1. Josh Kirklin (jjvk2)
  2. Covariant Loop Quantum Gravity
  3. Relativity & Gravitation, Quantum Fields & Strings
  4. n/a
  5. Thursday or Friday works best
  6. ok
  7. Quantum physics and general relativity are well known to be at odds. The most popular option for solving this problem is String Theory. We will consider an alternative: Covariant Loop Quantum Gravity. While String Theory takes a perturbative approach, with strings as quantum objects embedded in a background spacetime, Covariant LQG treats spacetime itself as an exact quantum entity, and is thus really a theory of quantum geometry. We will see that for quantum physics and general relativity to work together, we must treat spacetime as granular. To this end, we will investigate Regge calculus and the 2-complex discretisation of spacetime. Quantisation leads to a spectrum of lengths between any two points, and a natural expansion of spacetime states in a spin network basis. The introduction of a positive cosmological constant helpfully fixes any divergences in the action. Although we will mainly look at the 3D case, I will give a descriptive account of the difficulties that arise when moving to 4D. If there is time we will briefly look at some applications.

na382 ANALYSIS, GEOMETRY&TOPOLOGY ,RELATIVITY&GRAVITATION

1)Nikolaos Athanasiou (NA382)

2)The Cauchy problem in General relativity

3) Analysis, Geometry & Topology, Relativity & Gravitation

4)

5) Would be good if it was not on Friday, but can work around it if needed.

6)Yes

7) TBC

(TBC) Axions

1. tgp27

2. (TBC) Axions

3. Particle Physics

4. N/A

5. None

6. Yes

7. TBC

apr44

  1. Arpon Raksit (apr44)
  2. Group actions on categories
  3. Algebra
  4. Category theory & representation theory
  5. Cannot do Friday
  6. Sure
  7. This talk will be about a "categorification" of the notions of group actions and representations. I'll define what it means for a group to act on a category, and what it means to take the fixed points of such an action. More importantly, I'll try to convince you these definitions are useful and interesting, e.g. by indicating how the Fourier transform fits neatly into this language.

Rob Tovey

  1. Rob Tovey, rt446
  2. Total Variational Spectral Analysis in Image Processing
  3. Applied and Computational Analysis
  4. n/a
  5. Cannot do Wednesday morning or Thursday
  6. Ok

Abstract:

Regularisation techniques are popular in image processing due to their intuitive construction and adaptibility. This adaptibility normally presents as the ability to fine tune a few parameters which may be chosen with an apriori knowledge, such as the amount of noise in your image, or simple trial and error to find the best looking output image. The classical Total Variational method is such an example with one positive parameter although more recently it has been found that considering the parameter as a variable and looking at the set of solutions as a whole can allow you to recover much more information from the image. This talk will mainly investigate how this new technique can be used to extract fine texture details from an image and either enhance or diminish them or add a completely new texture.

For people who want a better idea of how the technique looks in practice I would recommend going to:
http://arxiv.org/pdf/1511.04687v1.pdf
Ignore the text and skip to pages 8-9 for the pictures!

Jeremy Owen (jao44)

  1. Jeremy Owen (jao44)
  2. When can preferences be aggregated?
  3. Operational Research (perhaps? see below). Could also be appropriate in Geometry + Topology.
  4. Topological social choice theory
  5. Cannot do 11am - 12pm Thursday 10/3
  6. Yes
  7. In many instances, one might wish to aggregate the preferences of many individuals, for example, in order to guide a collective decision. When is this possible? In the 1980s, G. Chichilnisky used algebraic topology to answer this and related questions, continuing the tradition of topology shedding light on economics (previously, most famously in the form of various fixed point theorems). My talk will sketch this exciting approach to social choice problems, explaining in particular: (i) how one models preferences and their aggregation in the topological framework, (ii) under what conditions fair social choice is possible, and (iii) how these ideas help unify several related results, including Arrow's impossibility theorem.

    Some results will depend on simple algebraic topology, but the thrust of the talk will be accessible to a broader audience. 

Number Theory

1. Daochen Wang (dw443)

2. Generalised Fermat Equation

3. Number Theory

4. /

5. /

6. OK

7. TBC

Lattice QCD

1. Rodrigo Lanza Munoz (rl465)

2. Lattice QCD and Hadron Spectroscopy

3. Quantum Fields & Strings

4. N/A

5. Cannot do Thursday morning or Friday afternoon

6. OK

7. Quantum chromodynamics (QCD) is a quantum field theory describing the strong interaction in particle physics. It has the property of asymptotic freedom, which means that at low energies (or long distances), the interaction becomes truly strong and perturbation theory breaks down, so we must use some non-perturbative method if we want to make quantitative predictions. 

In this talk, I will present a brief overview of the most popular non-perturbative approach, lattice QCD, where we formulate the theory on a lattice of points in spacetime, and briefly discuss some issues that arise when trying to implement fermion fields in this way. I will then discuss the application of lattice QCD to calculating the spectra of hadrons and (if there is time) glueballs. This talk should be accessible to anyone with a rudimentary knowledge of quantum field theory, but I will try to assume as little knowledge as possible. 

Sam Davenport (sjd81) Statistics

1) Sam Davenport (sjd81)

2) TBC

3) Statistics

4) n\a

5) None

6)

7) TBC

Synthetic Differential Geometry Talk

1. José Siqueira (jvp27)

2. Synthetic Differential Geometry

3. Logic

4. Category Theory

5. None

6. Yes

7. Synthetic Differential Geometry is an axiomatic form of geometry based on a sensible treatment of infinitesimals and its consequences, which should appeal to mathematicians and physicists alike. I will comment on the axiomatics, why it is interesting and its issues, but will quickly move towards foundational matters and discuss how a well-adapted model of the theory could be built - roughly, this means a concrete structure for which the axioms of Synthetic Differential Geometry hold and in which the classical manifolds are ``nicely" embedded. Central to this effort is the notion of $\mathcal{C}^{\infty}$-ring, which ``does for smooth functions what commutative rings do for polynomials" -  these are interesting objects in their own right and should be deemed worthy to algebraists, geometers and category theorists. 

mk732

1. Matt Kellett (mk732)

2. The Coleman-Mandula Theorem

3. Quantum Fields & Strings

4. Supersymmetry

5. n/a

6. Yes

7. The Coleman-Mandula Theorem (1967) asserts that, subject to reasonable assumptions, there are no ways to extend Poincaré symmetry in any non-trivial manner. That is to say, we can only add internal symmetry generators which commute with all of the Poincaré generators. This essentially ensures that supersymmetry (which circumvents this theorem) is the only non-trivial extension of the Poincaré algebra of point particles (it need not apply in the case of extended objects, for example, strings).

In this talk I will present a proof of the theorem that is essentially a streamlined and rearranged version of the proof by Coleman and Mandula. While the interest in this theorem is largely due to supersymmetry, no knowledge will be required. Pre-requisites are a facility with QFT and aspects of Lie algebras relevant to particle physics, especially the Poincaré algebra (unless you want to take my assertions on trust).

David Mehrle

 

  1. David Mehrle (dfm33)
  2. What the heck is a Hecke algebra?
  3. Algebra
  4.  
  5. None 
  6.  
  7. Hecke algebras are really popular and useful in representation theory these days, but the Wikipedia article about them is comically unhelpful. I'll try to explain what the heck these things are and why they're interesting.