Part III Seminars Lent 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.

Wednesday afternoon

Number Theory 1 Analysis and Mathematical Finance
Serre's Conjecture and Fermat's Last Theorem
Ping Ngai Chung (1:00 MR9)
o-minimality and Diophantine geometry
Levent Alpoge (2:00 MR9)
Coffee, tea and snacks — 3:00-3:15 in the core
What the heck is the Langlands program? - cancelled
Alex Best (3:15 MR9)
Model-Free No-Arbitrage Bounds - A look at the Root/Rost Solution
Henrique Guerreiro (4:30 MR4)

Thursday morning

Statistics and Probability
Streaming and sketching
Wenda Zhou (9:00 MR4)
Nonparametric Estimation Under Shape Constraints
Andi Wang (10:00 MR4)
Coffee, tea and snacks — 11:00-11:15 in the core
Scaling Limits of Loop Erased Random Walks
Benjamin Walter (11:15 MR4)

Thursday afternoon

Geometry and Topology Analysis and Mathematical Finance
Quantitative homotopy theory and an application to data analysis
Mauricio B. García Tec (1:00 MR4)
An introduction to Hartree-Fock theory
Mi-Song Dupuy (1:00 MR5)
John Gowers (2:00 MR4)
Application of Monte Carlo Methods to Quantum Mechanics
Teodor Nikolov (2:00 MR5)
Coffee, tea and snacks — 3:00-3:15 in the core
Homotopy Theoretical Localization
Ilyas Khan (3:15 MR4)
Maximilian Moll (3:15 MR5)

Friday morning

Continuum Mechanics and Statistical Physics Quantum and Relativity
Global Modes in Shear Flow
Sam Turton (10:00 MR3)
Coffee, tea and snacks — 11:00-11:15 in the core
Directed Polymers in Random Medium
Martin Buchacek (11:15 MR3)
The Velo-Zwanziger problem.
Jesus Cruz Rojas (11:15 MR12)

Friday afternoon

Number Theory 2 Quantum and Relativity
Quantum Hamiltonian Complexity
Pieter Bogaert (1:00 MR4)
Auslander-Reiten Theory
Stacey Law (2:00 MR5)
Superconformal and dual superconformal symmetry in twistor space
Julio Parra Martinez (2:00 MR4)
Coffee, tea and snacks — 3:00-3:15 in the core
Class Field Theory (or why you think you know some number theory but don't)
David Vasak (3:15 MR5)
Holographic Superconductors
Stuart Patching (3:15 MR4)
The Class Number One Problem
Tim King (4:15 MR5)
Supersymmetry and Geometry.
Calum Ross (4:15 MR4)
Party!!! 5:30 in the Part III room

Number Theory 1

Wednesday afternoon. Group leader: Sean Moss (skm45).

Serre's Conjecture and Fermat's Last Theorem — Ping Ngai Chung (pnc29) — 1:00 MR9
Serre's conjecture provides a tight relation between two important arithmetic objects: modular forms and (two-dimensional) mod p Galois representations. A particular consequence of the conjecture (now a theorem of Khare), as observed by Serre himself, is Fermat's Last Theorem. The goal of this talk is to sketch a proof of Fermat's Last Theorem (FLT) assuming Serre's conjecture and a few other deep results.
Even building enough background to motivate and give the precise statement of the conjecture shall take a whole Part III course. Therefore this talk will be focused on motivations of the conjecture, the important ingredients in the conjecture and how it is related to FLT. I will only assume the audience to know basic Galois theory, basic complex analysis and what an elliptic curve is, though anyone with more background in the theory of elliptic curves, modular forms and Galois representations will find it easier to follow the talk.
Pre-requisite: Know what Fermat's Last Theorem is and think it is (pretty) hard to prove.

o-minimality and Diophantine geometry — Levent Alpoge (la312) — 2:00 MR9
I will discuss Pila et al.'s recent (spectacular) application of model-theoretic methods to the question of counting rational points of bounded height on varieties.

What the heck is the Langlands program? - cancelled — Alex Best (ajb304) — 3:15 MR9

Analysis and Mathematical Finance

Wednesday afternoon. Group leader: James Kilbane (jk511).

Model-Free No-Arbitrage Bounds - A look at the Root/Rost Solution — Henrique Guerreiro (hg383) — 4:30 MR4
When pricing financial options, one usually postulates a stochastic model for the underlying, then extracts from the market the prices of vanilla options, uses those to calibrate the parameters in the model, and prices contingent claims as the expectation under the risk-neutral measure. This of course exposes us to Knightian uncertainty, since we may have guessed the wrong model for the underlying. I will try to show how we can actually find Model-Free bounds for prices: by first exhibiting the relation between that problem and the Skorokhod Embedding problem, and then focusing especially on the Root/Rost solutions, which have some very desirable properties. I will also address their financial application.

Statistics and Probability

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

Streaming and sketching — Wenda Zhou (wz258) — 9:00 MR4
With the advent of big data computation, it is often no longer reasonable to assume that our algorithms may access all of the data in an out-of-order fashion. Instead, I will present an alternative paradigm, streaming algorithms, in which the input is a sequence of elements, processed one at a time, and we restrict ourselves to a logarithmic amount of memory (in the number of elements of the stream). Although it is then in general impossible to obtain exact answers to queries regarding the data, it is often possible to recover approximate information in a randomized fashion.
A large class of such algorithms are linear sketches, which can be viewed as (often random) linear projections of the data into a lower-dimensional space. I will start by describing some classical sketches (count-min, count-sketch), that allow us to query some values of a large vector, in spirit very similar to the problem of compressed sensing, and then move on to some more recent developments on sketching data with more structures, and in particular, graphs.

Nonparametric Estimation Under Shape Constraints — Andi Wang (aqw20) — 10:00 MR4
Shape constraints combine some of the best elements of both nonparametric and parametric statistics. In my seminar I will introduce this increasingly popular subject with two important examples - the Grenander estimator for a non-increasing density and the maximum likelihood estimator of a log-concave density in arbitrary dimensions. I will also present some of my own ideas on misspecification in generalised additive models with shape constraints.

Scaling Limits of Loop Erased Random Walks — Benjamin Walter (bw360) — 11:15 MR4
This talk motivates and introduces Schramm-Loewner-Evolutions as the scaling limit of Loop Erased Random Walks in the complex plane. We will briefly discuss general stochastic processes on lattices and why scaling limits are of special interest for statistical physics. After outlining key properties of Loop Erased Random Walks, we then use the Loewner evolution to describe its evolution in the limiting continuous case which will lead us to the Schramm-Loewner Evolutions. LERW just form a little fraction of the many applications of SLE, that combine probability theory, complex analysis and ODE in an intriguing way. Thus, this talk also intends to give a short impression of general theory of SLE.

Geometry and Topology

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

Quantitative homotopy theory and an application to data analysis — Mauricio B. García Tec (mbg29) — 1:00 MR4
I will present an application of Gromov's ideas on quantitative homotopy theory. Then I will discuss an application to topological data analysis based on (Blumberg & Mandell, 2013).

Stacks — John Gowers (wjg29) — 2:00 MR4
My aim is that by the end of the talk you will be fairly comfortable with the idea of a fibred category and will have seen at least one example of a stack. I shall introduce stacks by talking about a prototypical topological example, and will then move on to an interesting algebraic example. I hope to give a proof of faithfully flat descent for modules over commutative rings, and explain the consequences for geometry. Time permitting, I will talk about some of the philsophy of stacks and their most useful properties. The only prerequisites are basic category theory and familiarity with rings, modules, tensor products and topological spaces, though some background with algebraic geometry will help.

Homotopy Theoretical Localization — Ilyas Khan (imk23) — 3:15 MR4
The method of localization (of a ring, module, etc.) affords one a method of examining the behavior of some algebraic object “locally” (e.g. around some prime ideal), by analyzing a simpler object which nonetheless retains those “local” properties. We can exploit this algebraic technique in topology by first establishing an analogous concept of a local space, and subsequently determining how to “localize” a topological space X by finding a universal map from X into a local space. After motivating and describing the construction of local spaces, we will state a few basic properties, and spend the rest of the lecture demonstrating an application of topological localization to computing the homotopy groups of spheres.

Analysis and Mathematical Finance

Thursday afternoon. Group leader: James Kilbane (jk511).

L(E)QG-Theory — Maximilian Moll (mkm35) — 3:15 MR5
Dynamic programming is used in a variety of areas to solve problems efficiently. One possible application is to find optimal control variables for a process. I will show how to solve such problems in the case of linear state equations with Gaussian noise under quadratic costs. The theory behind this is quite complete and exhibits a separation principle, which states that state estimation and control optimisation can be separated. From there, I will show how to generalise this model to incorporate the risk-sensitivity of the controller.

Application of Monte Carlo Methods to Quantum Mechanics — Teodor Nikolov (tn300) — 2:00 MR5
The talk will start with a review on the path integral formulation of quantum mechanics for a 1D quantum system. Quantities of interest such as energy levels, probability distributions and averages of operators will be expressed as multidimensional integrals on a time lattice. This will be followed by a discussion of Monte Carlo methods and Markov Chains with the idea to obtain a numerical approximation of the physical quantities introduced earlier. We then proceed to discuss The Metropolis-Hasting algorithm which will provide a concrete example of an MCMC method. If time permits I will present my Java application implementing the algorithm and discuss results as applied to the exactly solvable quantum harmonic oscillator.

An introduction to Hartree-Fock theory — Mi-Song Dupuy (msd43) — 1:00 MR5
The Hartree-Fock method is a first approximation to compute the energy of a quantum many-body system in a stationary state. For both atoms and molecules, the Hartree-Fock solution is the central starting point for most methods that describe the many-electron system more accurately. The talk will be focused on the derivations of the Hartree-Fock equations and the basic properties of its solution. I will also present the main algorithms for solving these equations and discuss their efficiency.

Continuum Mechanics and Statistical Physics

Friday morning. Group leader: Matt Arran (mia31).

Directed Polymers in Random Medium — Martin Buchacek (mb863) — 11:15 MR3
Statistical physics describes polymers simply as elastic strings placed in random potential. We can map the resulting equations to problems in fluid dynamics or many-body quantum physics. I will show how understanding bound states of interacting bosons can help us to describe statistical properties of polymers. I will also review some recent results of lattice models, which can be used to explain existence of triple-stranded DNA near the melting point of the double-helix.

Global Modes in Shear Flow — Sam Turton (set42) — 10:00 MR3
Stability of simple parallel shear flows is a well understood area of hydrodynamic instability theory. However, the extension of this theory to more realistic weakly nonparallel and weakly nonlinear flows remains an active area of research. It is possible to adopt a local approach in which the basic flow is assumed to vary over sufficiently slow scales so that one can perform a local stability analysis at each streamwise position, allowing us to reintroduce the concepts of absolute and convective instability from the parallel case. Whilst it is accepted that a sufficiently large region of local absolute instability is a necessary condition for a global mode to be sustained by the system, it is not clear a priori how to determine a selection criterion to extract the global mode frequency from the local data. I will begin by motivating the study of frequency selection criteria, before reviewing some of the work in the literature on global-mode theories and discussing their success in the case of the cylinder wake problem.

Quantum and Relativity

Friday morning. Group leader: Chris Blair (cdab3).

The Velo-Zwanziger problem. — Jesus Cruz Rojas (jc874) — 11:15 MR12
Massive Higher Spin (s>1) particles exist in the form of hadronic resonances. They are composite, and their interactions are described by complicated form factors. However, when the exchanged momenta are small compared to their masses, one should be able to describe their dynamics by consistent local actions.
In this talk I will introduce the free field equations describing massive particles of any spin s > 1 found by Fierz and Pauli in 1939 but in a more modern treatment. Then I will explain what Velo and Zwanziger pointed out in 1969: that, for s > 1, the Fierz–Pauli equations generically lead to acausal propagation in backgrounds with non-zero electric and magnetic fields. Finally I will review how this is potentially a problem for string theory and how the theory lead to equations that evade this problem.

Number Theory 2

Friday afternoon. Group leader: Andreas Bode (ab921).

Auslander-Reiten Theory — Stacey Law (swcl2) — 2:00 MR5
A main question in representation theory is to classify representations of our object of interest up to isomorphism. For certain algebras, we can visualise most of the information about their representations, homomorphisms between them and short exact sequences in directed graphs known as Auslander-Reiten quivers. We'll motivate various concepts from homological algebra including almost split sequences to get to these quivers, then look at how they naturally give rise to combinatorial interpretations through specific examples for path algebras. In these cases, it turns out we can extract information out of them using surprisingly simple algorithms.

Class Field Theory (or why you think you know some number theory but don't) — David Vasak (dsv24) — 3:15 MR5
Most introductions to algebraic number theory discuss the classical question of which primes can be expressed as the sum of two squares. The original proof of this can be rephrased in terms of UFDs and field norms, giving the illusion that the problem (and generalisations to primes of the form x^2 + ny^2) has been solved by using some fairly unsophisticated machinery. The key step in the proof relies upon knowing the image of a given ideal inside the class group of the number field, e.g. Q(i). However, it is rarely considered that this still needs an effective way to compute an ideal's class inside the class group. This talk will introduce class field theory as a method to solve this problem.

The Class Number One Problem — Tim King (tk406) — 4:15 MR5
Given a number field, how likely is it to be a unique factorisation domain? Equivalently, how likely is it that its class number is one? In this talk I shall attempt to answer this in the case of quadratic number fields. It appears that real quadratic fields have class number one 'most of the time' although so far nobody has proved this. The imaginary case is very different; there are only nine distinct fields of class number one. I shall sketch a proof of this fact which involves relating fields of class number one to integral points on a certain modular curve. (Prerequisites: I will assume knowledge of basic algebraic number theory and Galois theory. I will not assume prior knowledge of algebraic geometry or modular curves.)

Quantum and Relativity

Friday afternoon. Group leader: Chris Blair (cdab3).

Quantum Hamiltonian Complexity — Pieter Bogaert (pawb2) — 1:00 MR4
I shall explore a few aspects of quantum Hamiltonian complexity theory. More specifically, after giving a short overview (/recapitulation depending on the audience) of the necessary concepts, I shall give a non-exhaustive overview of QMA-completeness results (QMA being the quantum analogon for NP) for many-body Hamiltonians defined on ever simpler geometric configurations of subsystems. Some of the proofs require 'perturbation gadgets' stemming from perturbation theory, which I shall also discuss. As the proofs of some of these results are quite lengthy, I shall limit myself to proof sketches with particular emphasis on the key concepts.

Holographic Superconductors — Stuart Patching (sp631) — 3:15 MR4
The BCS model gives an excellent description of superconductivity. However, for certain high-temperature superconductors, this model is inadequate. One proposed solution to this problem is to use the AdS/CFT correspondence, which relates General Relativistic theories in Anti-de Sitter space to conformal field theories at the boundary of that space. I aim to give a general idea of how this correspondence can be applied to superconductors, by considering Black Holes in Anti-de Sitter space and showing how phase transitions can arise at the boundary.

Supersymmetry and Geometry. — Calum Ross (cdhr2) — 4:15 MR4
Supersymmetry (Susy) is a concept that shows up a lot in physics. Often this is in the context of phenomenology beyond the standard model. However there are a range of places where susy is used more as a mathematical tool. A major example of this is the ability to localise path integrals, allowing them to be evaluated. Links can also be made between the nature of the supersymmetry and the geometry of the target space of your theory. I would hope to sketch out some basic properties of susy before moving on to discussing this link with geometry and morse theory and finally discuss some of the relationships that you can prove with this framework.

Superconformal and dual superconformal symmetry in twistor space — Julio Parra Martinez (jp667) — 2:00 MR4
After a quick review of conformal symmetry (and its algebra) and how Supersymmetry can be added to the mix to get maximal spacetime symmetry, I will introduce the on-shell superspace formalism, momentum (super)Twistors and explain why these are the most natural variables to study the kinematics of scattering processes of theories with such symmetries. I will comment on how these techniques have led to the discovery of new symmetries of the S-Matrix of some Quantum Field Theories, and, if there is time, I will explain which are the difficulties of defining the S-Matrix of a Confomal Field Theory and how to circumvent them.