Part III Seminars Michaelmas 2015

The Michaelmas 2015 Part III seminars will take place on Wednesday 2nd and Friday 4th March. Each talk lasts 30 minutes followed by 10 minutes of questions. Note that there are Analysis and Geometry seminars on both days, and that the Geometry seminars on Friday are out of sync with the rest of the seminars to accommodate the Geometry tea talk, which you're of course welcome to attend if you'd like.

Wednesday 2pm-4pm

Combinatorics and Algebra Geometry 1
Splitting Digraphs
Matija Bucic (2:40 MR5)
Homotopy groups of a pushout of cell complexes
Nils Prigge (2:40 MR11)
Schur-Weyl Duality for the Orthogonal Group
David Mehrle (3:20 MR5)
Topological quantum field theory
Arpon Raksit (3:20 MR11)

Wednesday 4pm-6pm

Particle Physics Analysis
Noether's Theorem in Fundamental Physics
Matt Kellett (4:00 MR9)
Compact Operators and the Lomonosov Hyperinvariant Subspace Theorem
Francis White (4:00 MR11)
Symmetries and Degeneracies in the Spectrum of a Hamiltonian - The Hydrogen Atom
David Tennyson (4:40 MR9)
Abstract Cauchy-Kowalewski Theorem, Scale of Banach Spaces and a Moser Type Iteration Scheme
Matthew Colbrook (4:40 MR11)
The monopole problem, and the interactions of topological defects
Micah Brush (5:20 MR9)
The Poincaré Inequality and Heat Flow: Intuition for the Spectral Gap
Cole Graham (5:20 MR11)

Friday 1pm-4pm

Analysis, Geometry and Optimisation Geometry 2 Fluid Dynamics Number Theory Quantum Fields and Strings
When is a Vector Space like a Module?
Jasper Bird (1:40 MR4)
An analytical model of a hurricane
Adrian van Kan (1:20 MR5)
Phase Unwrapping via Total Variation — Formulating and Solving a Corresponding Non-Convex Problem
Sam Thomas (2:00 MR3)
What is a Spectral Sequence?
Robin Elliott (2:20 MR4)
Vortices in Superfluids
Alex Chamolly (2:00 MR5)
Symmetries of Minkowski space and their representations
Kieran Macfarlane (2:00 MR14)
The Ricci Flow as a Heat Equation
Tristan Giron (2:40 MR3)
(Geometry Tea)
(3:00-4:00 MR13)
Magical graphs and lifts
Alex Makelov (2:40 MR5)
Finding your place in the universe: Cosmological GPS for the interstellar vagabond
Josh Kirklin (2:40 MR14)
The Borwein integral sequence and why it's not quite as weird as it looks
Daan van de Weem (3:20 MR3)
A one-sentence proof that every prime congruent to 1 mod 4 is a sum of two squares
Yanitsa Pehova (3:20 MR5)
Casimir effect, Vacuum energy and the Cosmological constant
Sergi Navarro-Albalat (3:20 MR14)

Friday 4pm-6pm

General Relativity and Gravitation Geometry 2 Logic Theory Number Theory Quantum Fields and Strings
The vacuum energy of a scalar quantum field in curved spacetime and Hawking radiation
Frank Schindler (4:00 MR9)
Floer theory, to A-infinity and beyond
Tim Large (4:00 MR4)
Mathematics as an enquiry: a riddle by S Mac Lane, an explanation using monoidal categories again !
François Renaud (4:00 MR13)
The p-adic numbers and Ostrowski’s Theorem
Alex Kubiesa (4:00 MR5)
PT-symmetry and non-Hermitian Hamiltonians
Baptiste Ravina (4:00 MR14)
The effects of neutrinos on the linear matter power spectrum
Horng Sheng Chia (4:40 MR9)
Intersection Cohomology, or the study of perversities
Josh Lam (4:40 MR4)
How all of us can solve maths and physics problems visually with monoidal categories
Ed Ayers (4:40 MR13)
The Birch and Swinnerton-Dyer conjecture
Tomer Reiter (4:40 MR5)
The Quantisation Cookbook
Thomas Parton (4:40 MR14)
Wonderful Abstract Nonsense and Why You Should Care
José Siqueira (5:20 MR13)
Cubic rings
Evan O'Dorney (5:20 MR5)
Moonshines in String Theory
Victor Godet (5:20 MR14)
Party!!! 6pm in the Part III room

Combinatorics and Algebra

Wednesday 2pm-4pm. Group leader: Richard Freeland (rf348).

Schur-Weyl Duality for the Orthogonal Group — David Mehrle (dfm33) — 3:20 MR5
Schur-Weyl duality relates representations of GL(n) (a Lie group) to representations of S_n (which is nice and finite). What happens when we replace GL(n) by the orthogonal group O(n)? Proofs may be omitted in favor of examples and pictures.

Splitting Digraphs — Matija Bucic (mb927) — 2:40 MR5
I will talk about splitting directed graphs into smaller ones while preserving the condition of having "large" out-degree. I will give a (sketch) proof of the beautiful and complete result for undirected graphs and present currently best results towards settling the question in directed graphs as well as present a couple of open problems which do not look like they should be open.


Geometry 1

Wednesday 2pm-4pm. Group leader: ().

Homotopy groups of a pushout of cell complexes — Nils Prigge (np440) — 2:40 MR11
The computation of the homotopy groups of a space is far more subtle than of its homology groups. However, in certain situations one can calculate the first non-trivial homotopy group by computing the corresponding homology group. We will briefly discuss this link and explain how we can use it for a pushout of cell complexes by studying the universal cover. So even though the title is about homotopy theory, this will be a talk about singular homology and should be accessible for anyone with some knowledge of singular homology groups (Mayer-Vietoris) and some covering space theory.

Topological quantum field theory — Arpon Raksit (apr44) — 3:20 MR11
The notion of 'topological quantum field theory' (TQFT) arose out of people's attempts at rigorously understanding the mathematics of quantum field theory. But you know how it goes: math arising out of physics turns out to be important for all sorts of interesting reasons. Just the definition of TQFT reveals fundamental links among quantum field theory, manifold topology, and category theory. I'll give the definition and try to give some intuition for it via entertainingly/offensively bad drawings of manifolds in low dimensions, as well as an example related to the good-old representation theory of finite groups. You don't need to know any quantum field theory (well I hope so, because I don't), and hand-wavy intuition will be provided for any category theory, so you don't need to know that either.


Particle Physics

Wednesday 4pm-6pm. Group leader: Alex Arvanitakis ().

Noether's Theorem in Fundamental Physics — Matt Kellett (mk732) — 4:00 MR9
Noether's theorem is often stated as being the most beautiful theorem of physics. While I agree to an extent, statements like these often hide what a theorem is telling us. In this case, it is the fact that conserved quantities are not a fundamental concept, but are rather derived from symmetry. Here we will see a proof of Noether's theorem in the context of particle motion and field evolution, and discuss the conserved currents that arise, which range from momentum, energy and angular momentum to Newton's First Law, electric charge and baryon/lepton number. This talk should be accessible to anyone who's familiar with the Euler-Lagrange equations, but later parts may use some tensor formulation of special relativity and field theory.

Symmetries and Degeneracies in the Spectrum of a Hamiltonian - The Hydrogen Atom — David Tennyson (dt389) — 4:40 MR9
It is a well known fact that symmetries are intrinsic to physics and it is startling how much we can learn about a system just by knowing its symmetry group, G. In the case of quantum mechanics, a symmetry transformation is just given by the action of an element of G on a Hilbert space that leaves certain properties invariant, i.e. a representation of G. If we can determine the properties of representations of G then we have gone a long way to determining some of the physical properties of the quantum system. Here we will introduce the representation theory of groups with particular focus on SU(2), SO(3) and SO(4). We will apply these ideas to understand why degeneracy occurs in energy levels of a Hamiltonian and why this is broken when the Hamiltonian is perturbed. We will then consider the hydrogen atom and see why the degeneracies are much higher than one might predict with this theory. I will assume no prior knowledge of representation theory, so the talk should be accessible to anyone who is familiar with some basic quantum mechanics.

The monopole problem, and the interactions of topological defects — Micah Brush (mjb280) — 5:20 MR9
All Grand Unified theories in particle physics predict the existence of magnetic monopoles, but despite our best efforts they have yet to be observed. This is known as the monopole problem, which was the original motivation for the theory of inflation. The monopoles are created in the spontaneous symmetry breaking of the early universe, which can also create other topological defects. I will give a brief introduction to the topics above, and then discuss the interaction between two different types of topological defects: magnetic monopoles and domain walls. This talk should be very accessible to all, and there will be plenty of pictures since the interactions were studied numerically.


Analysis

Wednesday 4pm-6pm. Group leader: James Kilbane (jk511).

Compact Operators and the Lomonosov Hyperinvariant Subspace Theorem — Francis White (fw318) — 4:00 MR11
It is well-known that every linear operator T on a complex finite-dimensional vector space X possesses an eigenvalue and hence a nontrivial invariant subspace. When X is an infinite-dimensional complex topological vector space and T is assumed to be a continuous linear transformation from X to itself, then the problem of whether T has a nontrivial closed invariant subspace is far more difficult and is, for the most part, unsolved. In this talk, we will introduce and prove the Lomonosov hyperinvariant subspace theorem, which states that if X is a Banach space and T commutes with some compact operator K on X, then T possesses an invariant subspace. The talk will begin with a discussion of compact operators and their properties (briefly mentioned in the Analysis of PDE's course). We will also introduce the Schauder fixed point theorem, which is a generalization of the classical Brouwer fixed point theorem to infinite-dimensional spaces. I will assume only very basic notions from functional analysis.

Abstract Cauchy-Kowalewski Theorem, Scale of Banach Spaces and a Moser Type Iteration Scheme — Matthew Colbrook (mjc249) — 4:40 MR11
The Cauchy-Kovalevskaya Theorem guarantees the local existence and uniqueness for analytic partial differential equations associated with Cauchy initial value problems. Generalising such theorems from the ODE to the PDE setting is usually quite involved. A classical proof involves the method of majorants with lots of power series! I shall present a more abstract form (with no power series) due to L. Nirenberg which uses a scale of Banach spaces and a Moser type iteration scheme. The main idea is to exploit that, whilst the derivative is an unbounded operator, it becomes bounded when we consider holomorphic functions on a strictly smaller subdomain. This motivates considering a family of nested Banach spaces. I will present Nirenberg's proof and state some of the applications of the general theory. The talk should be accessible to anyone with a basic knowlegde of Banach spaces, contraction mapping theorem etc. For those attending the Part III PDEs course, some ideas are similar to the proof of the C.K. theorem given in lectures but in a more abstract setting with a different iteration scheme.

The Poincaré Inequality and Heat Flow: Intuition for the Spectral Gap — Cole Graham (cag69) — 5:20 MR11
The Poincaré inequality addresses a simple question: how much can a fuction deviate from average if its derivative isn't too large? We'll consider the L2 version on compact Euclidean domains, and explore its intimate connection with heat flow. I'll use this connection to (circularly) justify the inequality and find its optimal constant. This constant is the inverse of the spectral gap---the smallest nonzero eigenvalue of the Laplacian on the domain. I'll offer some geometric intuition for the spectral gap, and will advertise a beautiful gap bound for convex domains due to Payne and Weinberger. Prerequisites: high tolerance for integration by parts, sketchy variational arguments, and muttered appeals to the spectral theorem.


Analysis, Geometry and Optimisation

Friday 1pm-4pm. Group leader: James Bell (jhb43).

The Borwein integral sequence and why it's not quite as weird as it looks — Daan van de Weem (dhlv2) — 3:20 MR3
The Borwein integrals are a sequence of integrals of products of sinc functions. They exactly equal pi for the first 7 terms and then suddenly decide to be of order 10^{-12} less than that. In fact, with a small modification the integrals are exactly equal to pi again up until the 56th term after which it breaks down again. Worse even, the sums of the same functions equal the integrals, break away from pi at the same points, and then diverge from the integrals a couple hundred terms later. I will try to explain that this behaviour is not quite as bizarre as it sounds by saying the word Fourier a lot and drawing some pictures, and hopefully show that there is some geometric reason for this.

The Ricci Flow as a Heat Equation — Tristan Giron (tpg34) — 2:40 MR3
This talk will focus on the Ricci Flow, a geometric evolution equation on manifolds that has fascinating properties. First defined and explored by Richard Hamilton in 1982, the Ricci Flow has since then proven to be a powerful tool towards the understanding of the geometrisation conjecture of Thurston and the Poincaré conjecture, allowing geometers to rely on the precision of analytic tools and eventually leading to the proof of the Poincaré Conjecture by Grigori Perelman in 2002. Remarkably it turns out that this geometric equation is in fact essentially a non-linear heat equation, and thus admits very interesting analytic properties common with parabolic equations such as local existence, regularity etc. After some examples of the behaviour of the Ricci Flow on some (simple) closed manifolds, we'll show how to view the Ricci Flow as a heat equation on manifolds, and we'll prove the local existence of solutions. Time permitting we'll try to give an overview of one of the most remarkable features of the Ricci Flow, the Maximum principle. Although I'll try not to go too deep in the details to keep a more intuitive approach, I would recommend to have some sort of understanding of what the words 'metric' and 'curvature' of a manifold mean, and a vague notion of ellipticity. The chain rule for differentiation could be an asset but is not mandatory.

Phase Unwrapping via Total Variation — Formulating and Solving a Corresponding Non-Convex Problem — Sam Thomas (smt52) — 2:00 MR3
In this seminar, we look at the problem of two-dimensional phase unwrapping. This is the situation where we have data, eg for a height map, which has values `wrapped' in modulus 2*pi. For example, this may be a height map obtained from an aeroplane flying above terrain, bouncing a microwave off the ground and measuring the change in phase from leaving the plane to when it returns. This gives a measure of the height at each point, but `wrapped modulo 2*pi'; our problem is to `unwrap' this data to recover the original height map as accurately as possible. Such data is also sometimes obtained from MRI scans. We shall use a form of total variation penalisation to do this. With regards to pre-requisits, there aren't any, really. If you know a bit about optimisation, eg existence of duals and feasibility, then that would help, but it should be completely accessible to anyone.


Geometry 2

Friday 1pm-4pm. Group leader: Claudius Zibrowius (cbz20).

When is a Vector Space like a Module? — Jasper Bird (jcb96) — 1:40 MR4
From a purely algebraic point of view, vector spaces are rather trivial objects. Indeed (assuming the axiom of choice), any vector space has a basis, and the cardinality of said basis determines the space up to isomorphism. When we remove the multiplicative inverse axiom from our scalar field, however, we obtain much richer collection -- modules. I will begin the talk by discussing some pathological modules whose behaviour is nothing like vector spaces. I will then discuss the length, depth, and dimension of a module. When we place restrictions on these, we obtain much simpler objects to which vector space results can often be generalised. Only a first course in algebra (equivalent to the Cambridge IB course `Groups, Rings and Modules') is required.

What is a Spectral Sequence? — Robin Elliott (rare2) — 2:20 MR4
Spectral sequences are an important algebraic gadget for computing homology groups "by taking successive approximations". Their use pervades homological algebra, algebraic topology and algebraic geometry. At first they can look like unwieldy beasts, but in this talk I’ll motivate the definitions as a generalisation from the long exact sequence for pairs. Technical details (and proofs) aside, we’ll arrive at the big results which put spectral sequences to work. We’ll then have a play with our new toy, computing the homology of the space of based maps from S^1 to S^n. Only a cursory knowledge of homology is required.

(Geometry Tea) — () — 3:00-4:00 MR13
This isn't a Part III Seminar! This is the regular geometry tea talk; for more information see https://www.dpmms.cam.ac.uk/~col24/geometrytea.html . This week, Ruadhai Dervan will speak about "Symplectic reduction and stability of points". The abstract is as follows:
Somewhat surprisingly, there is a close link between the process of taking quotients in symplectic and algebraic geometry. This talk will discuss the Kempf-Ness Theorem, which relates the two constructions. I will not assume any previous knowledge of symplectic or algebraic geometry.


Fluid Dynamics

Friday 1pm-4pm. Group leader: Matt Arran (mia31).

Vortices in Superfluids — Alex Chamolly (ajc297) — 2:00 MR5
Superfluids are supercool. Liquid Helium is famous for defying gravity, squeezing through tiniest holes and perpetual motion. How is that possible? Similar to Bose-Einstein condensation the fluid falls into a macroscopic quantum state giving rise to its strange properties. I am going to outline the equations used for modelling a superfluid and sketch the derivation of some of these results. In particular I am going to talk about vortices: vorticity is quantised in a superfluid which gives rise to some very interesting behaviour. While the subject is quantum, my treatment will be fairly fluidsy in flavour, so the talk should appeal to people interested in either subject. I will keep it accessible for people that have not done any Statistical Physics or advanced fluids, so you have no excuse not to come. Also, there shall be jokes.

An analytical model of a hurricane — Adrian van Kan (av446) — 1:20 MR5
Among the most enigmatic features of the tropical atmosphere are the violently rotating storms known as hurricanes or typhoons (jargon: tropical cyclone). In this seminar talk, I will give a general introduction to the theory of tropical cylones in general and present a simple analytic model of a mature tropical cyclone. I will show that the model compares well with highly resolved simulations despite being highly idealised. As an example for the usefulness of the model we will deduce a relationship between the size and the strength of a hurricane.


Number Theory

Friday 1pm-4pm. Group leader: Marius Leonhardt (ml670).

Magical graphs and lifts — Alex Makelov (aam65) — 2:40 MR5
(Note: this talk is part of the Combinatorics and Algebra session, but has been rescheduled to Friday) Magical graphs, a.k.a. expanders, have been around for about half a century, spurring much research in telegraphic networks (!), theoretical computer science and, more recently, some seemingly distant areas of pure maths. In this talk, we'll give a brief overview of the concept, describe at least one application in at least some detail, and then talk a little bit about how one goes about constructing such things explicitly while imposing nice additional structure.

A one-sentence proof that every prime congruent to 1 mod 4 is a sum of two squares — Yanitsa Pehova (yyp24) — 3:20 MR5
There are various proofs of the fact that every prime p = 1 (mod 4) can be written as p = x^2+ y^2 for some integers x and y. However, sometime in the 90s an idea linking the size of a set with the number of fixed points of an involution (function whose square is the identity) arose in the mathematical community, and was believed to be capable of producing a short and sleek proof of this fact. In 1990 Don Zagier came up with a working example and published an article that was literally one sentence long! This talk will attempt to make sense of his sentence and the ideas behind it. Accessible to IAs.


Quantum Fields and Strings

Friday 1pm-4pm. Group leader: James Munro (jpm82).

Symmetries of Minkowski space and their representations — Kieran Macfarlane (ksm39) — 2:00 MR14
Lorentz boosts and rotations together form a symmetry group of Minkowski space: the Lorentz group. We can define particle states as elements of some representation space of these symmetries. I will elucidate some of the representation theory of the Lorentz group using infinitesimal transformations, briefly exploring some links with Dirac's equations. I will then extend some of these ideas to the Poincaré group (Lorentz group with translations added in).

Finding your place in the universe: Cosmological GPS for the interstellar vagabond — Josh Kirklin (jjvk2) — 2:40 MR14
The problem of finding your location on the Earth is essentially a solved one. The presence of GPS satellites allows anyone with a smartphone to easily triangulate their position, anywhere on the planet. For humankind in the short term, this is as good as anyone will need, but suppose that you are an incompetent astronaut in the far future, lost in deep space. There are no man-made satellites to assist you now, but you do have a database of every star in the universe. The only other information you can obtain is what you see in the sky around you. If you know your location, it is easy to render an image of the sky, but we will see that the inverse task is not so trivial. I will present some original tricks and methods to make this problem more tractable, and give an algorithm that works in small universes. We will mainly explore the problem in an idealised flat space, but if there is time will consider the impacts of an expanding universe.

Casimir effect, Vacuum energy and the Cosmological constant — Sergi Navarro-Albalat (sn483) — 3:20 MR14
In typical laboratory experiments the absolute value of the vacuum energy in unobservable, and only energy differences matter. One may assign zero energy density to the vacuum as a convention. This viewpoint is fine as far as the gravitational field is neglected. Since the energy density is part of the stress-energy tensor and it acts, according to General Relativity, as a source of the gravitational field, the absolute value of the vacuum energy matters. In the talk we will analyse the fundamental ingredients of the Casimir effect as a prototype to study the problem of vacuum energy in Quantum Field Theory and the cosmological constant in General Relativity. We will see different regularization methods to perform the renormalization of the vacuum stress-energy tensor. The role of Lorentz invariance will be analysed in detail. The dynamical Casimir effect will be briefly discussed.


General Relativity and Gravitation

Friday 4pm-6pm. Group leader: ().

The vacuum energy of a scalar quantum field in curved spacetime and Hawking radiation — Frank Schindler (fas41) — 4:00 MR9
Consider a massless scalar quantum field in Minkowski spacetime: The vacuum expectation value of its energy momentum tensor diverges, but this divergence can be straightforwardly redefined away. In curved spacetime, further divergences appear, and more elaborate renormalization techniques are required. I will present a nice, renormalization-avoiding argument (EM conservation!) to fully determine the expectation value in a generally curved two-dimensional spacetime by making use of the conformal anomaly (that is, the EMT is not traceless anymore). I will motivate this anomaly, and explicitly consider the case of a two-dimensional metric that corresponds to the Schwarzschild spacetime of a black hole. Requiring the EMT to be well behaved at the event horizon naturally leads to outgoing radiation at the Hawking temperature of the black hole.

The effects of neutrinos on the linear matter power spectrum — Horng Sheng Chia (hsc29) — 4:40 MR9
Cosmology provides stringent constraints on various neutrino parameters, including its effective number and total mass. These parameters play significant roles in the formation of the large scale structure of the universe, whose statistical nature is quantified in terms of the matter power spectrum. This talk focuses on some observable effects of neutrinos on the power spectrum. Furthermore, we will briefly discuss Weinberg's mode decomposition method (mentioned in Cosmology lectures) to justify the negligible radiation perturbation in Poisson's equation when one calculates the evolution of dark matter perturbation. This result is useful in our discussion of one particular effect of neutrinos on the power spectrum.


Geometry 2

Friday 4pm-6pm. Group leader: Claudius Zibrowius (cbz20).

Floer theory, to A-infinity and beyond — Tim Large (tmjl3) — 4:00 MR4
Floer theory is a powerful tool now ubiquitous in symplectic and low-dimensional topology. In its incarnation as Lagrangian Floer cohomology, it plays a central role in various mathematical formulations of string theory, and modern formulations of mirror symmetry link it to the study of coherent sheaves on algebraic varieties. We will set up Floer cohomology in its simplest setting, on oriented (Riemann) surfaces, where all the hard analysis disappears and the theory becomes wonderfully visual and essentially combinatorial. In particular, we will discover the surprising algebraic structure of A-infinity categories, with ramifications all over mathematics. Absolutely no prerequisites are needed, other than a desire to see lots of pictures!

Intersection Cohomology, or the study of perversities — Josh Lam (yhjl4) — 4:40 MR4
The suspension of a manifold is almost never a manifold: it has 'singularities' at the top and bottom, and if we take its ordinary homology, we certainly do not have Poincare duality on our side. I will introduce a generalisation of ordinary homology, known as intersection homology, which was introduced precisely for the study of such singular spaces. Since its introduction in the 70's, intersection (co)homology has proved to be a powerful tool in representation theory and algebraic geometry, in the guise of perverse sheaves. I will discuss mostly the topological aspects and compute several examples of this cohomology and I will only assume some basic understanding of homology theory.


Logic Theory

Friday 4pm-6pm. Group leader: James Bell (jhb43).

How all of us can solve maths and physics problems visually with monoidal categories — Ed Ayers (ewa21) — 4:40 MR13
When we want to understand some maths, we usually try and find a way of visualising the problem. So we draw some pictures, play around with them and then convert the pictures back into a respectable series of equations. Amazingly, a large class of these diagrams we draw are actually terms in the formal graphical language of monoidal categories, making the 'convert to respectable equations' step redundant! Come along to see how your quantum mechanics, knot theory, feynman diagrams, tensor calculus and computer programming problems can all be solved rigourously with silly boxes and arrows. No knowledge of category theory will be assumed. [Disclaimer: theory might be broken, actual thought still necessary to solve problems, some pure mathmos may find the contents of this talk offensive]

Mathematics as an enquiry: a riddle by S Mac Lane, an explanation using monoidal categories again ! — François Renaud (fjr31) — 4:00 MR13
This is a talk for detective stories lovers ... no need to be a categorist for that ! Once upon a time, S Mac Lane did a mysterious remark : X isomorphic to 1 + X^2 induces an isomorphism between X and X^7. Then mathematician-detectives tried to make sense of that statement, progressively disinterring the underlying phenomenon. A Blass gave it the flavour of trees and universal algebra. Subsequently, T Leinster and M Fiore used monoidal categories as a particularly appropriated framework for a simple and neat explanation of the riddle. As a result : facing the challenges of arithmetic in monoidal categories. Or as they say in their article Objects of categories as complexe numbers: how to derive proofs using only + and *, from proofs with complex numbers.

Wonderful Abstract Nonsense and Why You Should Care — José Siqueira (jvp27) — 5:20 MR13
Category Theory has earned quite a reputation since it's inception, being dubbed "general abstract nonsense" and regarded in some circles as essentially linguistic and devoid of any value besides as an organising tool and a nice notation to talk about Algebraic Topology and Geometry. While algebraists, topologists, geometers and computer scientists (probably!) don't need to be convinced of its merits, it is still valid to wonder why those in other fields who have always done perfectly good mathematics while ignoring it should consider going through an extra layer of abstraction and learn the basics - this mostly nontechnical talk is especially aimed at those who've had little to no contact whatsoever with Category Theory, and those concerned with Analysis, Number Theory, Physics and Applied Mathematics are most welcome.


Number Theory

Friday 4pm-6pm. Group leader: Marius Leonhardt (ml670).

The p-adic numbers and Ostrowski’s Theorem — Alex Kubiesa (ak812) — 4:00 MR5
The usual metric on the rationals allows us to form a set (field) we all know and love: the reals. But there are other interesting fields arising in such a way, namely the weird and wonderful p-adic numbers. Ostrowski’s Theorem tells us exactly what we can create from the rationals using such metrics. I will introduce absolute values, which will allow us to define the p-adic numbers analytically and give a statement of Ostrowski’s Theorem. If time allows, I may discuss some analytic or topological properties of the p-adic numbers. Some familiarity with completeness in metric spaces is required, but no knowledge from Part II or III.

The Birch and Swinnerton-Dyer conjecture — Tomer Reiter (tr398) — 4:40 MR5
The Birch and Swinnerton-Dyer conjecture is one of the open Millenium Problems. In this talk, I will explain some motivation for why it should be true as well as the content of the statement itself. The talk should be accessible to anyone who has taken a course in algebra.

Cubic rings — Evan O'Dorney (emo34) — 5:20 MR5
In 2004, Manjul Bhargava stunned the mathematical community by showing that composition of quadratic forms, a peculiar number-theoretic tool dating back to Gauss in 1801, is but one of a number of composition laws that can be described along similar lines. One of the fruits of his method is an explicit way to describe all extension rings of Z of finite rank up to 5. In my undergraduate senior thesis, I extended this parametrization to rings of small finite rank over a Dedekind domain. In this talk, I will focus on a fundamental case, the Delone-Faddeev-Gan-Gross-Savin-Deligne description of cubic rings, which is remarkably simple and adaptable, as well as revealing the pattern of the higher cases. Prerequisites should not exceed basic abstract algebra.


Quantum Fields and Strings

Friday 4pm-6pm. Group leader: James Munro (jpm82).

PT-symmetry and non-Hermitian Hamiltonians — Baptiste Ravina (br370) — 4:00 MR14

The Quantisation Cookbook — Thomas Parton (tgp27) — 4:40 MR14
Given the Lagrangian for a classical field, how do we go about quantising it? The conventional recipe is to derive a classical Hamiltonian and find its quantised energy spectrum by swapping our classical field variables for creation and annihilation operators. However, this transition from classical to quantum field theory is not straightforward when the system exhibits gauge symmetry. I will discuss Dirac’s approach to finding the most general Hamiltonian for a constrained system and the subtleties involved in successfully quantising it. To demonstrate this approach in practice I will apply it to the canonical quantisation of electromagnetism. This seminar is suitable for anyone with a background in the Lagrangian and Hamiltonian formulations of classical mechanics. Background material and notes available at https: //www.dropbox.com/sh/pyq8p65504px5xc/AACex4M_3eK5fQn3tBDEDGtNa?dl=0

Moonshines in String Theory — Victor Godet (vg305) — 5:20 MR14
Moonshines are unexpected connections between modular functions, sporadic groups and string theory. In 1978, John McKay found that the simplest modular function is connected to the largest sporadic group, the Monster. This completely mysterious relationship was shown to be very deep and culminated with Conway and Norton’s moonshine conjecture. In 1992, Borcherds finally proved Monstrous moonshine by using bosonic string theory compactified on the Leech lattice, a theory which has Monster symmetry. In 2010, another moonshine phenomenon was observed from compactification of superstrings on the K3 surface. This so-called Mathieu moonshine involves Jacobi forms and the Mathieu group M24. Although very similar to the first moonshine, Mathieu moonshine is more mysterious and not yet understood.