Part III Seminars Lent 2014

The Lent 2014 Part III seminars will take place on Wednesday 12th, Thursday 13th and Friday 14th March. Each talk lasts 45 minutes followed by 15 minutes of questions.

Wednesday afternoon

Combinatorics and Graph Theory Number Theory 1 Quantum computing
Random processes on isoradial graphs
Will Rafey (2:00 MR9)
BSD conjecture for CM elliptic curves
Kwok Wing Tsoi (2:00 MR3)
Cops and Robbers on diameter two graphs
Zsolt Adam Wagner (3:00 MR9)
Towards Perfectoid Spaces
Alessandro Maria Masullo (3:00 MR3)
Topological Quantum Computing
Will Berdanier (3:00 MR11)
Coffee, tea and snacks — 4:00-4:15 in the core
Cops and Robber on planar graphs
Yuqing Lin (4:15 MR9)
What makes Quantum Computers tick?
Henrik Dreyer (4:15 MR11)
Smoothed Analysis with applications in Machine Learning
Abigail See (5:15 MR9)

Thursday morning

Number Theory 2 Fluid Dynamics 1 High Energy Physics 1
Modular Polynomials and Complex Multiplication
Nicholas Triantafillou (9:00 MR15)
Downdrafts and Cold Pools in Tropical Thunderstorm Systems
Tom Dobra (9:00 MR14)
The BEC to BCS superconductor crossover
Felix Barber (9:00 MR12)
Orders in Imaginary Quadratic Fields
Jonathan Bootle (10:00 MR3)
Fingering Instabilities in Bi-disperse Granular Flows
Matthew Arran (10:00 MR14)
Real Space Renormalization - XY Model and Polymers
Sami Al-Izzi (10:00 MR12)
Coffee, tea and snacks — 11:00-11:15 in the core
Topology in Physics
Ruben Verresen (11:15 MR12)

Thursday afternoon

Category Theory Analysis General Relativity and Cosmology
Haupsatz, Hercules and Hydras
David Borg (2:00 MR4)
Thoughts about Series that are Divergent
Iddo Rousseau (2:00 MR3)
An Introduction to the Hamiltonian Formulation of General Relativity
Carlos Zapata-Carratala (2:00 MR5)
Enriched Category Theory
Dana Ma (3:00 MR4)
PDEs in General Relativity
Zoe Wyatt (3:00 MR5)
Coffee, tea and snacks — 4:00-4:15 in the core
Homology Stability
Nina Friedrich (4:15 MR4)
The Spectral Theorem - An Intuition
James Kilbane (4:15 MR13)
A Tale of Two Galaxies: New Evidence For Modified Gravity
Indranil Banik (4:15 MR5)
Dependent type theory and univalent foundations
Auke Bart Booij (5:15 MR4)
Topology of the space of test functions and of the space of distributions.
Jo Evans (5:15 MR13)

Friday morning

Algebraic Geometry Fluid Dynamics 2 High Energy Physics 2
Viscous fingering
Panayiota Katsamba (9:00 MR15)
Locomotion of E.coli
Justas Dauparas (10:00 MR15)
Coffee, tea and snacks — 11:00-11:15 in the core
Derived Categories in Mirror Symmetry: Stability Conditions
Kevin Carlson (11:15 MR13)
Elastocapillary Coalescence of Wet Hair
James Munro (11:15 MR11)
Superconformal Field Theory
Ana Raclariu (11:15 MR4)
Two-descent on the Jacobians of hyperelliptic curves
Stiofain Fordham (12:15 MR13)
Stringy Connections: Relating Classical Gauge Theory Solutions to String Theory
Sean Pohorence (12:15 MR4)

Friday afternoon

Algebra Probability and Geometry
Mixing Properties of Markov Processes
James Bell (NB 1:00 MR5)
Brown Representability and the Triangulated Adjoint Functor Theorem
Alexander Betts (2:00 MR5)
Exploring Conformal Loop Ensembles
Noela Müller (2:00 MR4)
Linear Algebraic Groups
Nicolas Dupre (3:00 MR11)
The surface subgroup theorem: ergodic theory in hyperbolic geometry
Salil Bhate (3:00 MR4)
Coffee, tea and snacks — 4:00-4:15 in the core
Algorithms to determine finite dimensional simple Lie algebras over GF(2).
Malalanirainy (4:15 MR14)
Party!!! 5:30 in the Part III room

Combinatorics and Graph Theory

Wednesday afternoon. Group leader: Bhargav Narayanan (bp338).

Random processes on isoradial graphs — Will Rafey (wr248) — 2:00 MR9
An isoradial graph is a graph with a planar embedding such that a circle of constant radius circumscribes each face. I hope to introduce this class of graphs, illustrate some of its geometric properties, and discuss how -- as the most general family on which we can define discrete analytic functions -- it provides a natural setting in which to study conformally invariant scaling limits of random walks and clusters.

Cops and Robbers on diameter two graphs — Zsolt Adam Wagner (zaw22) — 3:00 MR9
The game of Cops and Robbers is a combinatorial game where a set of cops try to catch a robber, while moving along the edges of a fixed graph. The most well-known conjecture in this area states that the number of cops needed is at most the square root of the number of vertices. Even in very special graphs, like graphs of diameter two, the best upper bound is unknown. I will give the best known bound in the diameter two case and describe one possible attempt at obtaining the correct upper bound, show why the proof is incomplete, and where the gaps are that I couldn't fill in.

Cops and Robber on planar graphs — Yuqing Lin (yfl30) — 4:15 MR9
The game of cops and robber takes place on a graph.  The cops and robber take turn moving along the edges of the graph, with the cops winning if they can eventually catch the robber, and the robber winning if he can avoid the cops indefinitely.  Meyniel conjectured that in general, the number of cops needed to guarantee capture of the robber is around the square root of the size of the graph.  Though recent results have advanced progress far beyond Aigner and Fromme's 30-year-old paper, for a gentle introduction to this problem I will focus on their introductory paper, culminating in their result that on any planar graph, only three cops are required to catch the robber. I will also attempt to draw the underlying ideas of how cops win and how the robber wins.

Smoothed Analysis with applications in Machine Learning — Abigail See (aes65) — 5:15 MR9
Sometimes there are gaps between theory and practice - such as when an algorithm works well in practice, but has exponential worst-case complexity. In these cases we would like to know why the algorithm tends to work well, so we can better understand the problem. Smoothed Analysis was developed by Spielman and Teng to attempt to bridge this gap between theory and practice. In this talk I'll give an introduction to Smoothed Analysis and present its application to two problems in Machine Learning (k-means method and the perceptron algorithm).


Number Theory 1

Wednesday afternoon. Group leader: Jack Lamplugh (jl478).

BSD conjecture for CM elliptic curves — Kwok Wing Tsoi (kwt26) — 2:00 MR3
This is going to be an advertising talk on elliptic curves with complex multiplication. The goal of the talk is to formulate the Main Conjecture of Iwasawa Theory and describe its connection with the Birch and Swinnerton-Dyer conjecture (attached with 1 million dollars). If time permits, I will (up to what I know) describe some recent progress towards the B-SD conjecture.

Towards Perfectoid Spaces — Alessandro Maria Masullo (amgeam2) — 3:00 MR3
I will discuss some results in "almost mathematics", in order to talk about a nice (category theoretically) correspondence between certain fields of mixed characteristic and equal characteristic, and spaces over them. In order to do that, I will try to give a sufficiently briad introduction to adic spaces, subordinate to the available time. I plan to end with an explicit example of the aforementioned correspondence, which gives some idea of how the general theory could indeed work. The theory of Perfectoid Spaces was introduced by Peter Scholze about two years ago, following a hint of Rapoport, who believed (apparently correctly) that the weight-monodromy conjecture (by Deligne) in the mixed characteristic case, could have been reduced to the already established equal characteristic case via a highly ramified base change.


Quantum computing

Wednesday afternoon. Group leader: Imdad Sardharwalla (isbs2).

Topological Quantum Computing — Will Berdanier (wvb22) — 3:00 MR11
In the late 1990’s, quantum computers were plagued with the problems of noise and decoherence. It was not possible to implement a perfect unitary gate – all gates were noisy – and interactions of qubits with their environments led to decoherence. Things seemed unstable, potentially fatally so. Quantum error-correcting codes had been shown to exist, but required that the errors be incredibly small, of the order of 10^−4. However, in 1997, Kitaev proposed using anyons, two-dimensional quasiparticles whose world lines can braid around each other in three-dimensional spacetime, to form the logic gates making up a quantum computer that is fundamentally stable. This seminar will explore this idea in some detail.

What makes Quantum Computers tick? — Henrik Dreyer (hd337) — 4:15 MR11
It is generally believed that Quantum Computers will be more powerful than their classical counterparts. Is that true? And if so, what is it that renders them more capable?


Number Theory 2

Thursday morning. Group leader: Jack Lamplugh (jl478).

Modular Polynomials and Complex Multiplication — Nicholas Triantafillou (ngt24) — 9:00 MR15
Modular polynomials parameterize pairs of elliptic curves are related by a cyclic isogeny of given degree, and hence provide a tool for studying when an elliptic curve has a "large" endomorphism ring, i.e. when the elliptic curve has complex multiplication (CM). We will define these polynomials, prove some of their basic properties, and use them to discuss some important connections between CM elliptic curves and class field theory.

Orders in Imaginary Quadratic Fields — Jonathan Bootle (jpb72) — 10:00 MR3
Rings of algebraic integers in number fields are well understood. But what happens if we remove the condition of integral closure? This gives rise to rings called orders. In this talk, I will prove some of the properties of orders in imaginary quadratic fields, and explain why they are useful objects of study.


Fluid Dynamics 1

Thursday morning. Group leader: Tom Eaves (tse23).

Downdrafts and Cold Pools in Tropical Thunderstorm Systems — Tom Dobra (ted29) — 9:00 MR14
In tropical regions, such as West Africa, millions of people are dependent of the rainfall of large convective systems, consisting of thunderstorms. Evidence clearly points towards new storms being influenced by older storms; however, there is no good theory to make them predictable. One key process is through downdrafts and cold pools leading to the triggering of a new convective cell. I will give a brief overview of the system and then explore the dynamics of these two fluid flows, aiming to explain how, where and when these will go on to initiate a new storm.

Fingering Instabilities in Bi-disperse Granular Flows — Matthew Arran (mia31) — 10:00 MR14
Classic models exist for geophysical granular flows, such as avalanches, rockslides, and pyroclastic flows, but currently neglect the effect of the segregation of particles by size. Recent work shows that these segregation effects can be important in determining the large scale structure of the flow, and in particular that front instabilities can arise as a result. I will introduce the mechanism by which this happens, discuss recently developed theoretical models for understanding and quantifying the effect, and present experimental results from this term. Of especial practical interest are effects on the damage caused by such mass flow.


High Energy Physics 1

Thursday morning. Group leader: Adam Solomon (as2089).

The BEC to BCS superconductor crossover — Felix Barber (fpjkb2) — 9:00 MR12
High Temperature Superconductors have potential to fill a range of demands within today's forms of energy use, however, fundamental understanding of these materials remains a challenging area of current research. Here conventional superconductivity is first discussed in the context of BCS theory, coupled with a brief description of the weakly interacting Bose gas, for the benefit of those with little background in the area. We will then demonstrate how to consider these two states of matter as two extreme ends of a spectrum, which can be moved between by varying the interaction strength present within the system. The concept of a Feshbach resonance will be introduced to illustrate how this crossover is experimentally studied, with reference to actual experiments which have been performed on cold fermionic gases. I hope to see you there.

Real Space Renormalization - XY Model and Polymers — Sami Al-Izzi (sca32) — 10:00 MR12
A brief introduction to Real Space Renormalization is presented for those who did not take Stat Field Theory. These methods are then applied to the XY model and used to study the KT transition close to the line of fixed points. Its relation to the Sine-Gordon Model and various condensed matter systems is briefly discussed. The methods are then applied to the case of a self avoiding chain (polymer), first by decimation along a polymer and then by taking the limiting case of a self avoiding walk on a triangular lattice. The critical index of the rms radius of the end-to end distribution is also calculated to a first approximation and compared with vairous other calculations.

Topology in Physics — Ruben Verresen (rv279) — 11:15 MR12
Both accessible and relevant for physicists and mathematicians alike. In recent decades topology (the mathematical study of how things are connected) has slowly but surely started to permeate modern physics. In particular I will focus on (1) how topology characterizes new phases of matter ("topological order"), and (2) how links and knots are related to the theories we use to describe fundamental particles.


Category Theory

Thursday afternoon. Group leader: Filip Bar (fb383).

Haupsatz, Hercules and Hydras — David Borg (dib26) — 2:00 MR4
This talk will be about the seemingly eternal, but actually finite, battle between Hercules and various species of Hydra. I will describe the classical Kirby-Paris Hydra game, and some variations. I will then talk about how we can prove that they terminate and what this result has got to do with proving the consistency of PA.

Enriched Category Theory — Dana Ma (dm537) — 3:00 MR4
In many examples of categories, the hom-sets of pairs of objects often possess additional structure; we studied this explicitly in our investigation of preadditive categories in the Category Theory Part III course. This approach yields some pretty awesome examples of enriched categories, such as metric spaces as categories enriched over the non-negative reals. I'll be introducing a number of enriched categorical notions and exploring them in the context of some specific examples.

Homology Stability — Nina Friedrich (nf302) — 4:15 MR4
We will discuss the general idea of showing homology stability and prove it for the case of symmetric groups (Nakaoka’s Theorem).

Dependent type theory and univalent foundations — Auke Bart Booij (abb40) — 5:15 MR4
In everyday mathematics, we usually don't distinguish between isomorphism (whether of groups, rings, topological spaces, or whatever) and strict equality of mathematical objects because all concepts we're interested in are preserved under isomorphisms. One can look for an alternative foundation of mathematics (rather than ZFC) in which isomorphism and equality are identified (i.e. they are the same thing). I will justify why set theory isn't a very appropriate language for such a foundation, and introduce dependent type theory as an alternative.


Analysis

Thursday afternoon. Group leader: Kostas Papafitsoros (kp366).

Thoughts about Series that are Divergent — Iddo Rousseau (ibr20) — 2:00 MR3
An introduction to the rich branch of divergent series and summability.

The Spectral Theorem - An Intuition — James Kilbane (jk511) — 4:15 MR13
This seminar will aim to give an *intuition* behind the Spectral Theorem. No proofs will be given, but the talk will be of general interest to those who found the last lecture of functional analysis crazy and a little too fast. My goal will be to go through spectral theorems for finite dimensional spaces, compact operators and finally a general normal operator. The purpose is to persuade the audience that the spectral theorems are natural, if unnatural looking.

Topology of the space of test functions and of the space of distributions. — Jo Evans (jahe2) — 5:15 MR13
I will talk about how to define the space of distributions as the analytic dual of an appropriate function space. I will talk about the topology of this function space and the different possible dual topologies, how these interact with an intuitive notion of what distributions are and hopefully how these analytic ideas can help prove some interessting/useful things about distributions.


General Relativity and Cosmology

Thursday afternoon. Group leader: Davide Gerosa (dg438).

An Introduction to the Hamiltonian Formulation of General Relativity — Carlos Zapata-Carratala (cz288) — 2:00 MR5
The precise physical interpretation of GR - and, indeed, of any generally covariant theory - in the Hamiltonian formalism has been historically elusive and not universally agreed upon until rencently. The aim of this talk is to give an account of the main issues that prevented the comunity of physicists and philosophers of physics to find a satisfatory formulation during the 20th century. To this end I will briefly introduce the theory of constrained Hamiltonian dynamics in an intrinsic fashion and show how the usual variational formulation of GR fits in this context. After this purely technical exposition I will introduce the notion of observable in the context of constrained Hamiltonian systems and I will discuss the physical and conceptual background that motivates the definition of it. Finally, I will briefly sketch how this notion of observable may lead to the "frozen time" paradox when we consider the case of Hamiltonian GR.

PDEs in General Relativity — Zoe Wyatt (zw253) — 3:00 MR5
The Einstein equations are a set of 10 coupled elliptic-hyperbolic nonlinear partial differential equations. To understand what this actually means, the first part of this talk will define and discuss some properties of elliptic, hyperbolic and parabolic PDEs. I will also discuss the importance of the Wick rotation with respect to PDEs, and some of its other applications as in the Hawking temperature. The last part of the talk will look at ways to solve the vacuum Einstein equations using harmonic coordinates and the De Turck trick.

A Tale of Two Galaxies: New Evidence For Modified Gravity — Indranil Banik (ib310) — 4:15 MR5
Observations on galactic scales reveal a huge discrepancy with General Relativity, a classical gravity theory. I will begin by describing why the fundamental assumption of classical gravity theories – that quantum mechanics is unimportant – likely fails in galaxies. Thus, I will introduce an empirical modification to gravity. This 30 year old theory predicted many of the observed properties of galaxies very well (see my handout outside the CMS reception). It also predicted an ancient flyby of Andromeda. I will be presenting recently gathered evidence, some of it my own, which suggests this really did happen. Attempts to explain these observations in the standard model - where General Relativity is saved by hypothetical invisible matter - run into severe difficulties. So far, these can be resolved very naturally by abandoning dark matter in all galaxies. Moreover, 30 years of null detections in direct searches for dark matter and recent LHC results argue against its existence and have ruled out dark matter with certain properties. Still, one day, it may be found. But it may not.


Algebraic Geometry

Friday morning. Group leader: Carmelo di Natale (cd475).

Derived Categories in Mirror Symmetry: Stability Conditions — Kevin Carlson (kdc27) — 11:15 MR13
The derived category of an abelian category A is the universal way of turning quasi-isomorphisms into isomorphisms. So, less precisely, it's the coarsest category that captures all homological information A. This is also one of the two first examples of a triangulated category, which is more generally what you get from homotopy theory if you make suspension invertible (there probably won't be homotopy theory in this talk.) 
We'll be interested in the derived category of the category of coherent sheaves on an algebraic variety, which has begun to attract a range of new applications in recent decades. Specifically, we'll investigate the work of Bridgeland on stability conditions. A stability condition is a way of of grading an abelian category by the real numbers compatibly with extensions. This compatibility forces the grading to be sufficiently coarse that it factors through the Grothendieck group, an invariant of the derived category, which is why stability conditions are studied on the derived (and more general triangulated) category. There is a moduli space of all sufficiently reasonable stability conditions on a triangulated category which is an interesting invariant of varieties for which it can be computed, e.g. projective curves, K3 surfaces, and to some extent Calabi-Yau varieties. This hints at a connection to mirror symmetry, where stability conditions on the derived category of coherent sheaves on a variety should give ones on the derived Fukaya category of the mirror variety-the latter thing is much more mysterious, which generates some of the interest in this topic.

Two-descent on the Jacobians of hyperelliptic curves — Stiofain Fordham (sdf27) — 12:15 MR13
The Mordell-Weil theorem states that for an abelian variety over a number field $K$, then the group of $K$-rational points is a finitely-generated abelian group. It would be nice to know the rank of this group; the method of descent allows one to find the rank in some cases (the failure of the Hasse principle complicates things). The first half of the talk will discuss the method of two-descent in the case of elliptic curves, with examples. The rest of the talk will describe the method of two-descent in the higher genus case, possibly with an example.
Note: this talk is in the Algebraic geometry section, but it is more number theory focused.


Fluid Dynamics 2

Friday morning. Group leader: Tom Eaves (tse23).

Viscous fingering — Panayiota Katsamba (pa335) — 9:00 MR15
The classical Saffman-Taylor instability is a fundamental instability of a moving fluid-fluid interface. It occurs when a less viscous fluid displaces a more viscous fluid in a flow where the velocity field is proportional to the pressure gradient, such as a Hele-Shaw cell or a porous medium. As a result of the instability, two main types of growth are reached eventually: fingering patterns or fractal structures depending on whether the system is confined or not. The dynamics depend on the relevant length scales of the problem, the geometry, and the presence of any anisotropies in the system. In the first part of the talk we will go through the linear stability analysis for the classical Saffman-Taylor instability in planar and circular geometries. The Saffman-Taylor instability is an archetype for many pattern-forming systems and instabilities of its type can be crucial for many growth processes such as the formation of lungs. However, in most industrial applications, where the instability can occur, such as coating flows, or oil recovery (where water injected from a well is used to displace the oil out of another well), the fingering resulting from the instability is undesirable, and efforts are made to inhibit it. In the second part of the talk, we will go through two possible ways to do this in a Hele-Shaw cell: by altering the geometry or by replacing the top lid by an elastic membrane.

Locomotion of E.coli — Justas Dauparas (jd540) — 10:00 MR15
E.coli (Escherichia coli) is a rod-shaped bacterium that lives in your gut. Most but not all E.coli are friendly and they can benefit you by producing vitamin K2, preventing invasion of the gut by yeast and fungi. This organism is very small but can swim due to rotating helical filaments. What were the physical constraints that E. coli had to meet to be able to swim? How quickly does it swim? How can flagella be tightly packed in a bundle and not get intertwined? I will answer some of these questions in my talk which is perfectly suitable for people with no fluid mechanics experience.

Elastocapillary Coalescence of Wet Hair — James Munro (jpm82) — 11:15 MR11
Hair isn't sticky. Water isn't sticky. So why does wet hair stick together in clumps? The answer lies in a balance between elastic forces in the hair and capillary forces in the water, with a rich mathematical structure undelying the hydrostatic balances reached, the dynamics as the system approaches them and the resulting distribution of clumps. This talk will describe recent research in elastocapillary coalescence, the collapse of elastic boundaries due to capillary forces. I'll focus on a recent theoretical model that describes the flow between the boundaries with lubrication theory (Singh et al. 2013), and its application in the form of a numerical model.


High Energy Physics 2

Friday morning. Group leader: Andrew Singleton (as921).

Superconformal Field Theory — Ana Raclariu (amr68) — 11:15 MR4
Any realistic quantum field theory is bound to be globally invariant under the Poincare group. It is generally very hard to solve a QFT exactly. However, once other symmetries such as scale invariance and/or supersymmetry are included, the QFT becomes more tractable and often exactly solvable.
I will start off by introducing the conformal group and its representations and explain how it can be extended to include supersymmetry. I will then show how one can use superconformal symmetry to understand the AdS/CFT correspondence relating N=4 Super Yang Mills in 4D to supergravity in Anti de Sitter space.

Stringy Connections: Relating Classical Gauge Theory Solutions to String Theory — Sean Pohorence (swp32) — 12:15 MR4
Yang-Mills theory is a non-abelian gauge theory which has become important in many different areas of both physics and mathematics. Physically, it provides the basis for much of the standard model. Mathematically, it is related to open problems in geometry and topology, such as the classification of smooth 4-manifolds. In this talk, I will focus on classical Yang-Mills solutions, called instantons. After getting more familiar with these objects, I will describe an algebraic method for constructing instantons known as the ADHM construction. Then, I will show how this construction can be realized as the vacuum of a certain system from String Theory. If there is any extra time, I will say a little about how one can view the instanton moduli space as a hyperkähler quotient, and how its singularities are related to the String Theory picture.


Algebra

Friday afternoon. Group leader: Carmelo di Natale (cd475).

Brown Representability and the Triangulated Adjoint Functor Theorem — Alexander Betts (lb497) — 2:00 MR5
Triangulated categories, originally a concept from homotopy theory, arise naturally in both algebraic topology and algebraic geometry. However, it is only with the relatively new approach of Neeman (1996) that mathematicians have realised how the general theory of such categories may be employed to simplify and generalise many existing proofs. We will begin with a lightning introduction to triangulated categories, and develop some useful homotopy-theoretic constructions. From there, we will prove an adjoint functor theorem for triangulated categories, going via the work of Brown. Finally, we will sketch some applications of these ideas, focusing on algebro-geometric uses including constructing Grothendieck's exceptional inverse image functor.

Linear Algebraic Groups — Nicolas Dupre (nd332) — 3:00 MR11
A linear algebraic group is a group that is also an affine variety. More concretely, one may think of algebraic groups as subgroups of GL(V) for some finite dimensional vector space V, which satisfy some polynomial identities on the matrix entries. It turns out that the examples of GL(V) and some of its subgroups allow one to get a good intuition and understanding for the general, more abstract theory. Indeed, any linear algebraic group can be embedded as a closed subgroup of some GL(V). After introducing algebraic groups, the talk will focus on their subgroup structure. In particular, I will discuss Borel subgroups (maximal closed connected subgroups), beginning with the Borel fixed-point theorem, and hopefully proving a few of their properties.

Algorithms to determine finite dimensional simple Lie algebras over GF(2). — Malalanirainy (tam54) — 4:15 MR14
We want to find simple Lie algebras over GF(2) of finite (low) dimension. This problem can easily be implemented in a computer. Indeed, as our Lie algebra is finite dimensional, any Lie product can be expressed as a linear combination of the elements of its basis. Then, because our Lie algebra is simple, it is also semisimple and therefore is isomorphic to its adjoint representation. But an adjoint representation is just an associative algebra of matrices! Therefore, our problem is reduced to a problem on finite dimensional square matrices whose entries are in GF(2). I will try to describe two different solutions to the problem: the first solution is due to Eick and the second one to Vaughan-Lee.


Probability and Geometry

Friday afternoon. Group leader: Tengyao Wang (tw389).

Mixing Properties of Markov Processes — James Bell (jhb43) — NB 1:00 MR5
I will introduce Poincare inequalities for Markov chains then apply them and Diaconis and Saloff-Coste's method of canonical paths to put upper and lower bounds on the mixing time of a basic example. If time allows I shall show how these ideas extend to give bounds on the mixing of a simple diffusion process.

Exploring Conformal Loop Ensembles — Noela Müller (nsm31) — 2:00 MR4
Many two-dimensional statistical physics models can be interpreted as random collections of non-self-intersecting, disjoint loops on a planar lattice. Sometimes, it is natural to expect the collection of all loops to have a scaling limit. A natural candidate for this scaling limit are the Conformal Loop Ensembles (CLEs), which are collections of random loops in the complex plane that combine conformal invariance and a natural restriction property. CLEs can be viewed as "SLE loops" through a a measure on pinned loops, that is constructed via an exploration mechanism for CLEs. In the talk, I will focus on this exploration mechanism and mention some consequences for the classification of CLEs.

The surface subgroup theorem: ergodic theory in hyperbolic geometry — Salil Bhate (ssb37) — 3:00 MR4
The surface subgroup theorem says that every closed irreducible 3-manifold with infinite fundamental group admits a surface which immerses injectively on fundamental groups. The last remaining case (assuming the Geometrisation theorem) was of hyperbolic 3 manifolds, which was proved in 2012. A key aspect of the proof is the interplay between dynamical properties of geometric flows on the manifold and the topological features that are their consequences. Depending on the audience, I will give an account of the basics of ergodic theory in the context of hyperbolic geometry and aim to outline the key ingredients of the proof of the surface subgroup theorem for closed hyperbolic 3 manifolds. Basic ideas from differential geometry and topology (part II would be enough) will be useful.