# Pure Mathematics Postgraduate Seminar 2014-15

## Schedule (Spring Term)

• Thu 29th January, 2015 - 4:00 PM, MALL 1
Anton Freund - "Computational content of non-constructive proofs: A case study in the proof assistant Minlog"
Abstract: We present an example (due to Ulrich Berger) for the extraction of computational content from a classical existence proof. More precisely we give a classical proof of the hsh-Theorem stating that the composition of functions $h \circ s \circ h$ cannot be bijective if s is not surjective. Using Harvey Friedman's A-translation we extract a quite unexpected algorithm which finds an x such that $(h \circ s \circ h)(x)$ is different from $x$. We do this in the proof assistant Minlog, which we introduce along the way. The style of the talk will be exploratory and example-based, so that it is hopefully interesting both without and with prior knowledge of proof theory.
• Thu 5th February, 2015 - 4:00 PM, MALL 1
Richard Whyman - "An Introduction to the Mathematics of Quantum Mechanics and Quantum Computation."
Abstract: In this talk, I will firstly give the necessary background in quantum mechanics and demonstrate how quantum systems differ from classical ones.

I shall then attempt to explain what quantum computation is and how quantum computers could be potentially faster than classical computers at solving certain problems, giving an algorithm for factorising numbers in polynomial time as an example.

I intend to finish by showing how quantum computers might even be capable of computing uncomputable numbers.
• Thu 12th February, 2015 - 4:00 PM, MALL 1
Daniel Wood - "Ultrafilters and Ultraproducts"
Abstract: I will give an introductory talk to the fun world of ultrafilters and ultraproducts, covering basic definitions, results and examples. I will do my best to make the talk accessible, although I should warn non-logicians that the section on ultraproducts may be difficult to follow without knowledge of first-order logic.
• Thu 19th February, 2015 - 4:00 PM, MALL 1
Erick Garcia Ramirez - "Semi-Algebraic Geometry"
Abstract: In Algebraic Geometry one studies the zero-sets of polynomials and most of the time one assumes that the background field is algebraically closed. In Semi-algebraic Geometry one drops such assumption by taking the real numbers (or any real closed field) as the background field. By doing so one enables the study of semi-algebraic sets: sets defined by equalities and inequalities between polynomials. I will describe some of the structure of these sets.
• Thu 26th February, 2015 - 4:00 PM, MALL 1
Joshua Cork - "Path Integrals, Quantum Mechanics and Multiply-Connected Spaces"
Abstract: Path integrals are somewhat more intuitive as an approach to quantum mechanics than the usual Hilbert spaces method, but they are not very well understood mathematically. In this talk I aim to first highlight the disadvantages of this formulation, but to then discuss the advantages of this approach in the context of studying quantum mechanics on spaces with non-trivial fundamental group, and the interesting physical results that can be derived from their topology. No prior knowledge of quantum mechanics will be necessary, but some classical intuition will certainly not go unwanted!
• Thu 5th March, 2015 - 4:00 PM, MALL 1
Adam Dent - "Racks and Quandles"
Abstract: There aren't many places in algebra where you can find a binary operation that distributes over itself, but an interesting structure with just such a multiplication is a rack. A quandle is a special type of rack. It can easily be shown that the notion of a quandle generalises that of a group, in a sense. Racks, especially quandles, became particularly interesting/important to knot theorists when it was found that they provide a total knot invariant. I will explain how to obtain a quandle from a knot diagram so that the quandle axioms directly correspond to the Reidemeister moves, and if there is enough time I will talk about an even more recently discovered application of racks in Lie theory as the solution to the so-called 'coquecigrue problem'.
• Thu 12th March, 2015 - 4:00 PM, MALL 1
Raphael Bennett-Tennenhaus - "Yoneda's lemma"
Abstract: Yoneda's lemma has been used to prove many results in category theory, and helps motivate the study of functor categories.
Due to its importance as an idea, one main focus of this talk will be to give a correct statement and proof of the result.
The proof will let us find some easy corollaries, and I will use them to motivate some ideas used in representation theory.
I will assume the audience is comfortable with the notions of category, functor, natural transformation, ring, and module. I will also assume they have dealt with examples. Under this assumption such language will be defined again, but no examples will be considered.
• Thu 19th March, 2015 - 4:00 PM, MALL 1
Rodrigo Pires dos Santos - "Hyperkahler manifolds and Nahm's equations"
Abstract: I'll give a brief introduction to Hyperkahler geometry and will discuss its twistor theory. The goal of the talk is to expose the concept of Hyperkahler quotient, a method from symplectic geometry, which is used to construct those geometries; I'll conclude by saying how Nahm's equations can give examples of such quotients.
• Thu 23rd April, 2015 - 4:00 PM, MALL 1
Michael Toppel - "What is Proof-Theory about and why does it look ugly?"
Abstract: I will explain the aims of proof-theory in a way so that also non-logician should understand. Hence my talk lacks precise definitions and proper proofs. I will rather focus on "explanatory examples" and black-boxing. Also my talk tries to explain why there is no such thing as a nice theory in proof-theory, that could be taught in a beginners course, and therefore might look a bit overwhelming for newcomers.
• Thu 30th April, 2015 - 4:00 PM, MALL 1
Lovkush Agarwal - "Godel's Incompleteness Theorems"
Abstract: Heisenberg, Godel, Turing, Chomsky, a knight, a knave, Cameron, Miliband and Farage walk into a bar. The barman asks them if they think this joke is funny. Heisenberg is uncertain. Godel says its impossible to know since they're inside the joke. Turing couldn't decide. Chomsky says that it is of course funny, but it has just been told wrong. The knight says yes. The knave says no. Cameron says the joke has been getting funnier over the past 5 years. Miliband says Cameron is wrong. Farage says it would be funnier without Heisenberg, Godel and Chomsky. Pauli was upset because of his exclusion from the joke. By the end of the talk, you should understand at least one tenth of this joke.

## Schedule (Autumn Term)

• Thu 2nd October, 2014 - 4:00 PM, MALL 1
Alec Thomson - "Paul Cohen's Consistency Proof for Peano Arithmetic"
Abstract: In a posthumous manuscript, Paul Cohen presented a new version of the consistency proof for Peano Arithmetic (PA). This proof utilizes the method of tableau proofs, allowing a set of sentences to be decomposed into a tree structure. If every branch of the tree contains a contradiction, then the original set of sentences must be inconsistent. Cohen's proof shows that no such tree exists for the Axioms of PA.
• Tue 7th October, 2014 - 3:00 PM, MALL 1
Ricardo Bello Aguirre - "Chinese remainder theorem."
Abstract: In this talk we will present the Chinese remainder theorem both for $\mathbb{Z}$ and for the general commutative ring case.
• Tue 14th October, 2014 - 3:00 PM, MALL 1
Jacob Hilton - The topological pigeonhole principle for ordinals
Abstract: The ordinary pigeonhole principle states that if you put more than $n$ items into $n$ containers, then one container must contain $2$ or more items. The infinite pigeonhole principle states that if you put infinitely many items into finitely many containers, then one container must contain infinitely many items.

What happens if you divide up a countable ordinal into finitely many containers? Is there some container that contains a set of order type $\omega+1$? We will study this question and see how the answer changes when we topologize the ordinals.

Familiarity with ordinals and basic topology will be helpful but not essential.

• Tue 21th October, 2014 - 2:30 PM, Mall 1
Daoud Siniora - Free Groups & Free Products
Abstract: Free groups are fundamental objects in the branch of combinatorial group theory, where the term combinatorial refers to the frequent use of combinatorial methods in this discipline. In this talk I will introduce free groups, presentation of groups via generators and defining relations, free products, and list Dehn's fundamental problems.
• Tue 28th October, 2014 - 2:30 PM, Mall 1
Jakob Vidmar - A Small Subset of "Everything You Wanted to Know About Higher Category Theory and More"
Abstract: Higher category theory is a generalisation of category theory, where we allow for existence of "higher dimensional'' morphisms. In this talk I will present some basic notions of higher category theory (along with some necessary theory of enriched categories). I will mainly focus on the so-called $(\infty, 1)$-categories and one of their geometric models, the quasi-categories.
• Tue 4th November, 2014 - 2:30 PM, Mall 1
Cesare Gallozzi - Homotopy type theory, weak universes and constructive set theory
Abstract: The aim of this talk is to provide a gentle introduction to dependent type theory and to homotopy type theory.
We will discuss in particular the univalence axiom and how it can suggest considering weakenings of type theory.
Time permitting, I will present constructive set theory and show that its type-theoretic interpretation can be used to justify these variants.
• Tue 11th November, 2014 - 2:30 PM, Mall 1
Andrzej Kucik - Multipliers of spaces of analytic functions
Abstract: A brief description of multipliers, algebras of multipliers and multiplication operators on spaces of analytic functions (mainly on the complex right half-plane). I will present some well-known basic facts and prove some elementary properties of them. If time allows I will also prove an application of those, giving a necessary and sufficient condition for weighted spaces of Lebesgue square integrable (over positive reals)) functions to be Banach algebras (with multiplication given by convolution). It is a recent result of my work, and I believe it has not been proved before.
• Tue 18th November, 2014 - 2:30 PM, Mall 1
Alexander Bullivant - Topological entanglement entropy
Abstract: Quantum computing is a field of research concerned with harnessing and exploiting the laws of quantum physics in order to process information. Theoretical work has already shown a remarkable speed up of quantum algorithms compared to their classical counterparts for applications such as searching unordered databases and factorising numbers.

Due to the fragile nature of quantum systems, no practical realisation of a quantum computer which is immune to errors is yet to be built. One proposal for a fault tolerant realisation is to use topologically ordered states of matter which possess global degrees of freedom which are immune to to local errors. The aim of my talk is to give a brief introduction to quantum computation before discussing entanglement entropy and how it may be used to determine whether a physical system is topologically ordered or not.
• Tue 2nd December, 2014 - Â3:00PM, Mall 1
Josh Cork - A Crach Course in Gauge Theory
Abstract: In classical physics, one tends to tackle mechanical problems by assigning coordinates to the system. These coordinates are dynamical variables which represent the possible configurations (degrees of freedom) of the system, and the dynamics can be realised via a particular function called the Lagrangian. This description of physics often leads to many redundant degrees of freedom (or removable coordinates); the physics of the system is equivalent in certain symmetric configurations. In other words, the system is invariant under some symmetry group of global transformations. A gauge theory is a particular type of physical theory where the system obeys what is known as a gauge principle, that is, the system is invariant under local transformations called gauge transformations. The common examples of such theories are electromagnetism, which is invariant under the action of the group of U(1) valued functions, and the non-Abelian gauge theory, or Yang-Mills theory, which has SU(n) symmetry group. Indeed, it turns out that for most of these sorts of physical systems, the group structure that describes the transformations is a finite dimensional Lie group, and so one can form a mathematical definition of a gauge transformation in the language of Lie groups. Furthermore, in order to precisely describe these physical systems, one should use the language of fiber bundles to define what we really mean by the configurations. The aim of this presentation will be to discuss in more detail the physical motivation of gauge theory and to provide a dictionary between these intuitive ideas and the concrete formalism found in differential geometry.
• Tue 2nd December, 2014 - 3:00 PM, Mall 1
David Toth - Definability of Truth in Probabilistic Logic
Abstract: Tarski's undefinability of truth theorem says that in a theory with Godel numbering gn:L->N on formulas, the set of Godel numbers of true sentences is not definable - there is no formula True such that for every L-sentence phi: phi iff True(gn(phi)). The undefinability of the formula True is a consequence of the certainity required to reason about the truth of a sentence within a theory. However, if one relaxes the requirement for the absoluteness of certainty, one can reason about the probability of a sentence to be true with an infinitesimal error. In this talk I shall present the paper Definability of Truth in Probabilistic Logic [http://intelligence.org/files/DefinabilityTruthDraft.pdf] which proves the existence of the probability distribution mu over the models of an L-theory T where L is the language containing the predicate P such that for any sentence phi of the language L, P(phi) approximates mu({M (models T) : M models phi}).
• Tue 9th December, 2014 - 2:30 PM, Mall 1
Joshua Tattersall - Tauberian Theory and the prime number theorem
Abstract: Tauberian theories relate the asymptotic behaviour of a function or series to averages of said function or series. In this talk, I shall introduce Ingham's Tauberian Theorem, and how it can be used to prove the prime number conjecture- that $(\pi(x)\log x)/x$ converges to $1$ as $x$ tends to infinity, where $\pi(x)$ is the number of primes less than $x$.