Past

Abstract: This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. 
The classical McKay correspondence relates the geometry of so-called 
Kleinian surface singularities with the representation theory of finite 
subgroups of SL(2,C). M. Auslander observed an algebraic version of this 
correspondence: let G be a finite subgroup of SL(2,K) for a field K whose
characteristic does not divide the order of G. The group acts linearly on 
the polynomial ring S=K[x,y] and then the so-called skew group algebra
A=G*S can be seen as an incarnation of the correspondence. In particular
A is isomorphic to the endomorphism ring of S over the corresponding 
Kleinian surface singularity.
Our goal is to establish an analogous result when G in GL(n,K) is a finite 
subgroup generated by reflections, assuming that the characteristic
of K does not divide the order of the group. Therefore we will consider a 
quotient of the skew group ring A=S*G, where S is the polynomial ring in n 
variables. We show that our construction yelds a generalization of 
Auslander's result, and moreover, a noncommutative resolution of the 
discriminant of the reflection group G.

Abstract: Joint work with Carlson, Grodal, Nakano. In this talk we will
present some recent results on an 'important' class of modular 
representations for an 'important' class of finite groups. For the 
convenience of the audience, we'll briefly review the notion of an 
endotrivial module and present the main results pertaining endotrivial 
modules and finite reductive groups which we use in our ongoing work.

I shall describe recent work with Srikanth Iyengar, Henning 
Krause and Julia Pevtsova on the representation theory and cohomology
of finite group schemes and finite supergroup schemes. Particular emphasis 
will be placed on the role of generic points, detection of projectivity
for modules, and detection modulo nilpotents for cohomology.

 

Topological field theories (TFT's) are physical theories depending only on the topological properties of spacetime as opposed to also depending on the metric of spacetime.  This talk will introduce topological field theories, and the work of Freed and Hopkins on how a class of TFT's called "invertible" TFT's describe certain states of matter, and are classified by maps of spectra.  Constructions of field theories corresponding to specific maps of spectra will be described.
 

This half-day research workshop will address issues at the intersection between network science, matrix theory and mathematical physics.

Network science is producing a wide range of challenging research problems that have diverse applications across science and engineering. It is natural to cast these research challenges in terms of matrix function theory. However, in many cases, closely related problems have been tackled by researchers working in statistical physics, notably quantum mechanics on graphs and quantum chaos. This workshop will discuss recent progress that has been made in both fields and highlight opportunities for cross-fertilization. While focusing on mathematical, physical and computational issues, some results will also be presented for real data sets of relevance to practitioners in network science.

Fibrillation is a chaotic, turbulent state for the electrical signal fronts in the heart. In the ventricle it is fatal if not treated promptly. The standard treatment is by an electrical shock to reset the cardiac state to a normal one and allow resumption of a normal heart beat.

The fibrillation wave fronts are organized into scroll waves, more or less analogous to a vortex tube in fluid turbulence. The centerline of this 3D rotating object is called a filament, and it is the organizing center of the scroll wave.

The electrical shock, when turned on or off, creates charges at the conductivity discontinuities of the cardiac tissue. These charges are called virtual electrodes. They charge the region near the discontinuity, and give rise to wave fronts that grow through the heart, to effect the defibrillation. There are many theories, or proposed mechanisms, to specify the details of this process. The main experimental data is through signals on the outer surface of the heart, so that simulations are important to attempt to reconstruct the electrical dynamics within the interior of the heart tissue. The primary electrical conduction discontinuities are at the cardiac surface. Secondary discontinuities, and the source of some differences of opinion, are conduction discontinuities at blood vessel walls.

In this lecture, we will present causal mechanisms for the success of the virtual electrodes, partially overlapping, together with simulation and biological evidence for or against some of these.

The role of small blood vessels has been one area of disagreement. To assess the role of small blood vessels accurately, many details of the modeling have been emphasized, including the thickness and electrical properties of the blood vessel walls, the accuracy of the biological data on the vessels, and their distribution though the heart. While all of these factors do contribute to the answer, our main conclusion is that the concentration of the blood vessels on the exterior surface of the heart and their relative wide separation within the interior of the heart is the factor most strongly limiting the significant participation of small blood vessels in the defibrillation process.

Since the pioneering work of Hodgkin and Huxley , we know that electrical signals propagate along a nerve fiber via ions that flow in and out of the fiber, generating a current. The voltages these currents generate are subject to a diffusion equation, which is a reduced form of the Maxwell equation. The result is a reaction (electrical currents specified by an ODE) coupled to a diffusion equation, hence the term reaction diffusion equation.

The heart is composed of nerve fibers, wound in an ascending spiral fashion along the heart chamber. Modeling not individual nerve fibers, but many within a single mesh block, leads to partial differential equation coupled to the reaction ODE.

As with the nerve fiber equation, these cardiac electrical equations allow a propagating wave front, which normally moves from the bottom to the top of the heart, giving rise to contractions and a normal heart beat, to accomplish the pumping of blood.

The equations are only borderline stable and also allow a chaotic, turbulent type wave front motion called fibrillation.

In this lecture, we will explain the 1D traveling wave solution, the 3D normal wave front motion and the chaotic state.

The chaotic state is easiest to understand in 2D, where it consists of spiral waves rotating about a center. The 3D version of this wave motion is called a scroll wave, resembling a fluid vortex tube.

In simplified models of reaction diffusion equations, we can explain much of this phenomena in an analytically understandable fashion, as a sequence of period doubling transitions along the path to chaos, reminiscent of the laminar to turbulent transition.

  Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with even first Betti number. However, not all solutions of the Einstein-Maxwell equations on such manifolds arise in this way. In this lecture, I will describe a construction of new compact examples that are Hermitian, but not Kaehler.
 

A main part of the proof uses forcing to establish a Ramsey theorem on a new type of tree, though the result holds in ZFC.  The space of such trees almost forms a topological Ramsey space.

The talk will focus on five items:

Theorem 1. It is ZFC-independent whether every locally compact, $\omega_1$-compact space of cardinality $\aleph_1$  is the union of countably many countably compact spaces.

Problem 1. Is it consistent that every locally compact, $\omega_1$-compact space of cardinality $\aleph_2$  is the union of countably many countably compact spaces?

[`$\omega_1$-compact' means that every closed discrete subspace is countable. This is obviously implied by being the union of countably many countably compact spaces, but the converse is not true.]

Problem 2. Is ZFC enough to imply that there is  a normal, locally countable, countably compact space of cardinality greater than $\aleph_1$?

Problem 3. Is it consistent that there exists a normal, locally countable, countably compact space of cardinality greater than $\aleph_2$?

The spaces involved in Problem 2 and Problem 3 are automatically locally compact, because by "space" I mean "Hausdorff space" and so regularity is already enough to give every point a countable countably compact (hence compact) neighborhood.

Theorem 2. The axiom $\square_{\aleph_1}$ implies that there is a normal, locally countable, countably compact space of cardinality $\aleph_2$.

This may be the first application of $\square_{\aleph_1}$ to construct a topological space whose existence in ZFC is unknown.

Oxford Mathematics Public Lectures

The Law of the Few - Sanjeev Goyal

The study of networks offers a fruitful approach to understanding human behaviour. Sanjeev Goyal is one of its pioneers. In this lecture Sanjeev presents a puzzle:

In social communities, the vast majority of individuals get their information from a very small subset of the group – the influencers, connectors, and opinion leaders. But empirical research suggests that there are only minor differences between the influencers and the others. Using mathematical modelling of individual activity and networking and experiments with human subjects, Sanjeev helps explain the puzzle and the economic trade-offs it contains.

Professor Sanjeev Goyal FBA is the Chair of the Economics Faculty at the University of Cambridge and was the founding Director of the Cambridge-INET Institute.

28 June 2017, 5.00-6.00pm, Lecture Theatre 1, Mathematical Institute Oxford.

Please email @email to register

In this lecture, we present  practical and provably
secure (authenticated) key exchange protocol and password
authenticated key exchange protocol, which are based on the
learning with errors problems. These protocols are conceptually
simple and have strong provable security properties.
This type of new constructions were started in 2011-2012.
These protocols are shown indeed practical.  We will explain
that all the existing LWE based key exchanges are variants
of this fundamental design.  In addition, we will explain
some issues with key reuse and how to use the signal function
invented for KE for authentication schemes.

Starting from an example in quantum chemistry, we explain the techniques of Numerical Tensor Calculus with particular emphasis on the convolution operation. The tensorisation technique also applies to one-dimensional grid functions and allows to perform the convolution with a cost which may be much cheaper than the fast Fourier transform.

Lisa Lamberti

Geometric models in algebra and beyond

Many phenomena in mathematics and related sciences are described by geometrical models.

In this talk, we will see how triangulations in polytopes can be used to uncover combinatorial structures in algebras. We will also glimpse at possible generalizations and open questions, as well as some applications of geometric models in other disciplines.

Jaroslav Fowkes

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Optimization Challenges in the Commercial Aviation Sector

The commercial aviation sector is a low-margin business with high fixed costs, namely operating the aircraft themselves. It is therefore of great importance for an airline to maximize passenger capacity on its route network. The majority of existing full-service airlines use largely outdated capacity allocation models based on customer segmentation and fixed, pre-determined price levels. Low-cost airlines, on the other hand, mostly fly single-leg routes and have been using dynamic pricing models to control demand by setting prices in real-time. In this talk, I will review our recent research on dynamic pricing models for the Emirates route network which, unlike that of most low-cost airlines, has multiple routes traversing (and therefore competing for) the same leg.

In 1933, lattice theory was a new subject, put forth by Garrett Birkhoff. In contrast, in 1940, it was already a mature subject, worth publishing a book on. Indeed, the first monograph, written by the same G. Birkhoff, was the result of these 7 years of working on a lattice theory. In my talk, I would like to focus on this fast development. I will present the notion of a theory not only as an actors' category but as an historical category. Relying on that definition, I would like to focus on some collaborations around the notion of lattices. In particular, we will study lattice theory as a meeting point between the works of G. Birkhoff and two other mathematicians: John von Neumann and Marshall Stone.

Let X be a smooth, complete geometrically connected curve defined over a one variable function field K over a finite field. Let G be a subgroup of the points of the Jacobian variety J of X defined over a separable closure of K with the property that G/p is finite, where p is the characteristic of K. Buium and Voloch, under the hypothesis that X is not defined over K^p, give an explicit bound for the number of points of X which lie in G (related to a conjecture of Lang, in the case of curves). In this joint work with Pazuki, we extend their result by requiring just that X is non isotrivial.

Groups which are "attached" to theories of fields, appearing in models of the theory  
as the automorphism groups of intermediate fields fixing an elementary submodel are called geometrically represented. 
We will discuss the concept ``geometric representation" in the case of pseudo finite fields.  Then will show that any group which is geometrically represented in a complete theory of a pseudo-finite field must be abelian. 
This result also generalizes to bounded PAC fields. This is joint work with Zoe Chatzidakis.
 

Kim's iterative non-abelian reciprocity laws carve out a sequence of subsets of the adelic points of a suitable algebraic variety, containing the global points. Like Ellenberg's obstructions to the existence of global points, they are based on nilpotent approximations to the variety. Systematically exploiting this idea gives a sequence starting with the Brauer-Manin obstruction, based on the theory of obstruction towers in algebraic topology. For Shimura varieties, nilpotent approximations are inadequate as the fundamental group is nearly perfect, but relative completions produce an interesting obstruction tower. For modular curves, these maps take values in Galois cohomology of modular forms, and give obstructions to an adelic elliptic curve with global Tate module underlying a global elliptic curve.

Despite its fame there appears to be little literature outlining Lurie's proof sketched in his expository article "On the classification of topological field theories." I shall embark on the quixotic quest to explain how the cobordism hypothesis is formalised and give an overview of Lurie's proof in one hour. I will not be able to go into any of the motivation, but I promise to try to make the talk as accessible as possible. 

Understanding the evolution of a solidification front is important in the study of solidification processes. Mathematically, self-similar solutions exist to the Stefan problem when the liquid domain is assumed semi-infinite, and such solutions have been extensively studied in the literature. However, in the case where the liquid region is finite and sufficiently small, such of solutions no longer hold, as in this case two solidification fronts will move toward each other and interact. We present an asymptotic analysis for the two-front Stefan problem with a small amount of constitutional supercooling and compare the asymptotic results with numerical simulations. We finally discuss ongoing work on the same problem near the time when the two fronts are close to colliding.
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Silicon is produced from quartz rock in electrode-heated furnaces by using carbon as a reduction agent. We present a model of the heat and mass transfer in an experimental pilot furnace and perform an asymptotic analysis of this model. First, by prescribing a steady state temperature profile in the furnace we explore the leading order reactions in different spatial regions. We next utilise the dominant behaviour when temperature is prescribed to reduce the full model to two coupled partial differential equations for the time-variable temperature profile within the furnace and the concentration of solid quartz. These equations account for diffusion, an endothermic reaction, and the external heating input to the system. A moving boundary is found and the behaviour on either side of this boundary explored in the asymptotic limit of small diffusion. We note how the simplifications derived may be useful for industrial furnace operation.

We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath, Jarrow and Morton (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability.  In this framework, we  derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes. This is based on joint work with Thorsten Schmidt.