## Part III Seminars Lent 2016

The Lent 2016 Part III seminars will take place from Wednesday 9th until Friday 11th March. Each talk lasts 45 minutes followed by 15 minutes of questions.

### Wednesday afternoon

Statistics Logic
Causal inference and invariant prediction
Synthetic Differential Geometry
Jose Siqueria (3:00 MR15)
Coffee, tea and snacks — 4:00-4:30 in the core
Changepoint Detection in fMRI data
Sam Davenport (4:30 MR2)
Group actions on categories
Arpon Raksit (4:30 MR15)

### Thursday afternoon

Quantum Fields and Strings Geometry and Topology Analysis
Conformal symmetry in two spacetime dimensions
Kieran Macfarlane (1:30 MR2)
Topological invariance of rational Pontrjagin classes
Nils Prigge (1:30 MR9)
Covariant Loop Quantum Gravity
Josh Kirklin (2:30 MR2)
When can preferences be aggregated?
Jeremy Owen (2:30 MR9)
Combinatrial Constructions in Ergodic Theory - the Arnoux, Ornstein Weiss Theorem.
Matthew Colbrook (2:30 MR11)
Coffee, tea and snacks — 3:30-4:00 in the core
Rodrigo Lanza Munoz (4:00 MR2)
The Cauchy problem in General relativity
Nikolaos Athanasiou (4:00 MR9)
A gift for analysis : infinitesimals (with love, logic)
Francois Renauld (4:00 MR11)
The Coleman-Mandula Theorem
Matthew Kellett (5:00 MR2)

### Friday afternoon

Algebra and Logic Continuum Mechanics and Computational Analysis
A Set Theory you can fit in your Pocket
Catherine Willis (2:00 MR11)
What the heck is a Hecke algebra?
David Mehrle (3:00 MR11)
A Design for Artificial Microswimmers
Matthew Butler (3:00 MR3)
Generalised Fermat Equation
Daochen Wang (4:00 MR11)
Total Variational Spectral Analysis in Image Processing
Robert Tovey (4:00 MR13)
Party!!! 5:00 in the Part III room

### Statistics

Wednesday afternoon. Group leader: Alberto Coca-Cabrero (ac776).

Causal inference and invariant prediction — Adam Foster (aef39) — 3:00 MR3
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.

Changepoint Detection in fMRI data — Sam Davenport (sjd81) — 4:30 MR2
I will introduce the Bootstrap and Stationary Bootstrap, Multiple Hypothesis testing and discuss change-point problems in general. Then I will look specifically at the hypothesis testing that results from considering binary-segmentation methods for Change-point detection. I will discuss an existing means of controlling the False Discovery Rate for sequential hypothesis tests and discuss and demonstrate a new method to analyse sequential hypothesis tests and then consider applications of this to binary-segmentation methods. Time permitting I will then explain how this can be applied to identify changes in the community structure of the brain over time and consider more general applications.

### Logic

Wednesday afternoon. Group leader: Eric Faber (eef25).

Synthetic Differential Geometry — Jose Siqueria (jvp27) — 3:00 MR15
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.

Group actions on categories — Arpon Raksit (apr44) — 4:30 MR15
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.

### Quantum Fields and Strings

Thursday afternoon. Group leader: Toby Crisford (tc393).

Conformal symmetry in two spacetime dimensions — Kieran Macfarlane (ksm39) — 1:30 MR2
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.

Covariant Loop Quantum Gravity — Josh Kirklin (jjvk2) — 2:30 MR2
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.

Lattice QCD and Hadron Spectroscopy — Rodrigo Lanza Munoz (rl465) — 4:00 MR2
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.

The Coleman-Mandula Theorem — Matthew Kellett (mk732) — 5:00 MR2
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).

### Geometry and Topology

Thursday afternoon. Group leader: Claudis Zibrowius (cbz20).

Topological invariance of rational Pontrjagin classes — Nils Prigge (np440) — 1:30 MR9
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.

When can preferences be aggregated? — Jeremy Owen (jao44) — 2:30 MR9
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.

The Cauchy problem in General relativity — Nikolaos Athanasiou (na382) — 4:00 MR9
On November the 25th, 1915, Einstein presents to the prussian academy of sciences his famous Einstein equations, relating the energy and matter content of the universe to the curvature of spacetime. Naturally, much of the research that was carried out revolved around trying to find explicit solutions to those tensorial equations. However, it was not until 1952 when it was first understood that there is hope for viewing those equations as an initial value problem. This is partly because of the diffeomorphism invariance property, which makes both a suitable surface and suitable initial data hard to find. In this talk I aim to present the fundamentals geometric aspects behind formalising the Einstein equations as an IVP. For example, I will discuss among others, the constraint equations of General relativity as well as the choice of gauge that allows us to transform the equations into a system of quasilinear wave equations on a suitable manifold.

### Analysis

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

Combinatrial Constructions in Ergodic Theory - the Arnoux, Ornstein Weiss Theorem. — Matthew Colbrook (mjc249) — 2:30 MR11
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!).

A gift for analysis : infinitesimals (with love, logic) — Francois Renauld (fjr31) — 4:00 MR11
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 access to new formalised ideas, new effective abstract technology.

### Algebra and Logic

Friday afternoon. Group leader: Stacey Law (swcl2).

A Set Theory you can fit in your Pocket — Catherine Willis (cw529) — 2:00 MR11
A huge proportion of mathematics occurs at cardinalities less than or equal
to that of the continuum. This means that traditional set theory is somewhat
‘overkill’ in providing a foundation. We will instead construct a simpler set of
axioms, to create a ‘pocket-sized’ set theory.
This talk will be accessible to anyone who's seen the axioms of ZFC, so y'all should come along!

What the heck is a Hecke algebra? — David Mehrle (dfm33) — 3:00 MR11
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.

Generalised Fermat Equation — Daochen Wang (dw443) — 4:00 MR11
Do you know how to solve X^2+Y^2=Z^2? How about X^2+Y^3=Z^5? It is perhaps surprising that these equations, not dissimilar to that of Fermat, belong to a large class for which we can solve concretely and completely. In this talk, I'll try to address this using the Platonic solids. (Nothing essentially prerequisite in mind at the moment but may decide to include some material related to reduction of binary quadratic forms and the geometry of numbers as seen in Part II Number Theory and Number Fields).