Past

Let \Gamma_{g,1} denote the mapping class group of a genus g surface with one parametrized boundary component.  The group homology H_i(\Gamma_{g,1}) is independent of g, as long as g is large compared to i, by a famous theorem of Harer known as homological stability, now known to hold when 2g > 3i.  Outside that range, the relative homology groups H_i(\Gamma_{g,1},\Gamma_{g-1,1}) contain interesting information about the failure of homological stability.  In this talk, I will discuss a metastability result; the relative groups depend only on the number k = 2g-3i, as long as g is large compared to k.  This is joint work with Alexander Kupers and Oscar Randal-Williams.

The smooth homotopy category is a simultaneous enlargement of the usual homotopy category and of the category of smooth manifolds. Its structure can be described very simply and explicitly by a version of van Est's theorem.  It provides us with an  interpolation between topology and geometry (and with a toy model of derived algebraic geometry and motivic homotopy theory, though I shall not pursue those directions).  My talk will list some situations which the category seems to illuminate: one will be Kapranov's beautiful description of the Lie algebra of the 'group' of based loops in a manifold.
 

After a brief review on the classical Prandtl system, we introduce our recent work on the well-posedness and high Reynolds numbers limit for the MHD boundary layer that shows the tangential magnetic field stabilizes the boundary layer. And then we will discuss some instability phenomena of the shear flow for both the classical Prandtl and MHD boundary layer systems. The talk includes some recent joint works with Chengjie Liu, Yaguang Wang on the classical Prandtl equation, and with Chengjie Liu and Feng Xie on the magnetohydrodynamic boundary layer.

We will discuss models for vehicular traffic flow on networks. The models include both the Lighthill-Whitham-Richards (LWR) model and Follow-the-Leader (FtL) models.
The emphasis will be on the Braess paradox in which adding a road to a traffic network can make travel times worse for all drivers. 
In addition we will present a novel proof how FtL models approximate the LWR model in case of heavy traffic.

Abstract: If A is a finite dimensional algebra, and D(A) the unbounded
derived category of the full module category Mod-A, then it is
straightforward to see that D(A) is generated (as a "localizing
subcategory") by the indecomposable projectives, and by the simple 
modules. It is not so obvious whether it is generated by the 
indecomposable injectives. In 2001, Keller gave a talk in which he 
remarked that"injectives generate" would imply several of the well-known
homological conjectures, such as the Nunke condition and hence the 
generalized Nakayama
conjecture, and asked if there was any relation to the finitistic 
dimension conjecture. I'll show that an algebra that satisfies "injectives 
generate" also satisfies the finitistic dimension conjecture and discuss 
some examples. I'll present things in a fairly concrete way, so most of 
the talk won't assume much knowledge of derived categories.

 

Abstract: In this talk I will discuss the interplay between the local and
the global invariants in modular representation theory with a focus on the
first Hochschild cohomology $\mathrm{HH}^1(B)$ of a block algebra $B$. In
particular, I will show the compatibility between $r$-integrable 
derivations
and stable equivalences of Morita type. I will also show that if
$\mathrm{HH}^1(B)$ is a simple Lie algebra such that $B$ has a unique
isomorphism class of simple modules, then $B$ is nilpotent with an
elementary abelian defect group $P$ of order at least 3. The second part 
is joint work with M. Linckelmann.

Abstract: I will describe how the ADE preprojective algebras appear in 
certain Conformal Field Theories, namely SU(2) WZW models, and explain
the generalisation to the SU(3) case, where 'almost CY3' algebras appear.

Abstract: In this talk, we will introduce new affine algebraic varieties 
for algebras given by quiver and relations. Each variety contains a 
distinguished element in the form of a monomial algebra. The properties 
and characteristics of this monomial algebra govern those of all other 
algebras in the variety. We will show how amongst other things this gives 
rise to a new way to determine whether an algebra is quasi-hereditary. 
This is a report on joint work both with Ed Green and with Ed Green and 
Lutz Hille.

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.

Adjoint derivatives reveal the sensitivity of a computer program's output to changes in its inputs. These derivatives are useful as a building block for optimisation, uncertainty quantification, noise estimation, inverse design, etc., in many industrial and scientific applications that use PDE solvers or other codes.
Algorithmic differentiation (AD) is an established method to transform a given computation into its corresponding adjoint computation. One of the key challenges in this process is the efficiency of the resulting adjoint computation. This becomes especially pressing with the increasing use of shared-memory parallelism on multi- and many-core architectures, for which AD support is currently insufficient.
In this talk, I will present an overview of challenges and solutions for the differentiation of shared-memory-parallel code, using two examples: an unstructured-mesh CFD solver, and a structured-mesh stencil kernel, both parallelised with OpenMP. I will show how AD can be used to generate adjoint solvers that scale as well as their underlying original solvers on CPUs and a KNC XeonPhi. The talk will conclude with some recent efforts in using AD and formal verification tools to check the correctness of manually optimised adjoint solvers.
 

I will talk about the fluid equations used to model pedestrian motion and traffic. I will present the compressible-incompressible Navier-Stokes two phase system describing the flow in the free and in the congested regimes, respectively. I will also show how to approximate such system by the compressible Navier-Stokes equations with singular pressure for the fixed barrier densities and also some recent developments for the barrier densities varying in the space and time.
This is a talk based on several papers in collaboration with: D. Bresch, C. Perrin, P. Degond, P. Minakowski, and L. Navoret.
 

We will follow a suggestion by Udi to construct a decidable field which has an undecidable finite extension.

In my talk I will give a basic introduction to the finiteness properties of groups and their relation to subgroups of direct products of groups. I will explain the relation between such subgroups and fibre products of groups, and then proceed with a discussion of the n-(n+1)-(n+2)-Conjecture and the Virtual Surjections Conjecture. While both conjectures are still open in general, they are known to hold in special cases. I will explain how these results can be applied to prove that there are groups with arbitrary (non-)finiteness properties.

Komlós conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large.  In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify. This is joint work with Hong Liu, Maryam Sharifzadeh and Katherine Staden.

One of the greatest challenges in developing renewable energy sources is finding an efficient energy storage solution to smooth out the inherently fluctuating supply. One cheap solution is lead-acid batteries, which are used to provide off-grid solar energy in developing countries. However, modelling of this technology has fallen behind other types of battery; the state-of-the-art models are either overly simplistic, fitting black-box functions to current and voltage data, or overly complicated, requiring complex and time-consuming numerical simulations. Neither of these methods offers great insight into the chemical behaviour at the micro-scale.

In our research, we use asymptotic methods to explore the Newman porous-electrode model for a constant-current discharge at low current densities, a good estimate for real-life applications. In this limit, we obtain a simple yet accurate formula for the cell voltage as a function of current density and time. We also gain quantitative insight into the effect of various parameters on this voltage. Further, our model allows us to quantitatively investigate the effect of ohmic resistance and mass transport limitations, as a correction to the leading order cell voltage. Finally, we explore the effect on cell voltage of other secondary phenomena, such as growth of a discharge-product layer in the pores and reaction-induced volume changes in the electrolyte.

In the cosmological scheme of conformal cyclic cosmology (CCC), the equations governing the crossover form each aeon to the next demand the creation of a dominant new scalar material that is postulated to be dark matter. In order that this material does not build up from aeon to aeon, it is taken to decay away completely over the history of the aeon. The dark matter particles (erebons) would be expected to behave as essentially classical particles of around a Planck mass, interacting only gravitationally, and their decay would be mainly responsible for the (~scale invariant)

temperature fluctuations in the CMB of the succeeding aeon. In our own aeon, erebon decay ought to be detectable as impulsive events observable by gravitational wave detectors.

The contact line problem in interfacial fluid mechanics concerns the triple-junction between a fluid, a solid, and a vapor phase. Although the equilibrium configurations of contact lines have been well-understood since the work of Young, Laplace, and Gauss, the understanding of contact line dynamics remains incomplete and is a source of work in experimentation, modeling, and mathematical analysis. In this talk we consider a 2D model of contact point (the 2D analog of a contact line) dynamics for an incompressible, viscous, Stokes fluid evolving in an open-top vessel in a gravitational field. The model allows for fully dynamic contact angles and points. We show that small perturbations of the equilibrium configuration give rise to global-in-time solutions that decay to equilibrium exponentially fast.  This is joint with with Yan Guo.

Boundaries of hyperbolic spaces have played a key role in low dimensional topology and geometric group theory.  In 1993, Paulin showed that the topology of the boundary of a (Gromov) hyperbolic space, together with its quasi-mobius structure, determines the space up to quasi-isometry.  One can define an analogous boundary, called the Morse boundary, for any proper geodesic metric space.  I will discuss an analogue of Paulin’s theorem for Morse boundaries of CAT(0) spaces. (Joint work with Devin Murray.)

We consider a class of nonlinear population models on a two-dimensional lattice which are influenced by a small random potential, and we show that on large temporal and spatial scales the population density is well described by the continuous parabolic Anderson model, a linear but singular stochastic PDE. The proof is based on a discrete formulation of paracontrolled distributions on unbounded lattices which is of independent interest because it can be applied to prove the convergence of a wide range of lattice models. This is joint work with Jörg Martin.

Reinhard Farwig and Chenyin Qian

Consider the autonomous quasi-geostrophic equation with fractional dissipation in $\mathbb{R}^2$
  \begin{equation} \label{a}
 \theta_t+u\cdot\nabla\theta+(-\Delta)^{\alpha}\theta=f(x,\theta)
 \end{equation}
in the subcritical case $1/2<\alpha\leq1$, with initial condition $\theta(x, 0)= \theta^{0}$ and given external force $f(x,\theta)$. Here the real scalar function $\theta$ is the so-called potential temperature, and the incompressible velocity field $u=(u_1,u_2)=(-\mathcal {R}_2\theta,\mathcal {R}_1\theta)$ is determined from $\theta$ via Riesz operators.  Our aim is to prove the existence of the compact global attractor $\mathcal{A}$ in the Bessel potential space $H^s(\mathbb{R}^2)$ when $s>2(1-\alpha)$.

The  construction of the attractor is based on the existence of an absorbing set in $L^2(\mathbb{R}^2)$ and $H^s(\mathbb{R}^2)$ where $s>2(1-\alpha)$. A second major step is usually based on compact Sobolev embeddings which unfortunately do not hold for unbounded domains. To circumvent this problem we exploit compact Sobolev embeddings on  balls $B_R \subset \mathbb{R}^2$ and uniform smallness estimates of solutions on $\mathbb{R}^2 \setminus B_R$. In the literature the latter estimates are obtained by a damping term $\lambda\theta$, $\lambda<0$, as part of the right hand side $f$ to guarantee exponential decay estimates. In our approach we exploit a much weaker nonlocal damping term of convolution type $\rho*\theta$ where $\widehat \rho<0$. 

TBC

We describe a general program for the classification of flat connections on topological manifolds. This is motivated by the classification of locally homogeneous geometric structures on manifolds, in the spirit of Ehresmann and Thurston.  This leads to interesting dynamical systems arising from mapping class group actions on character varieties. The mapping class group action is a discrete version of a continuous object, namely the extension of the Teichmueller flow to a  unversal character variety over over the tangent bundle of Teichmuller space. We give several examples of this construction
and discuss joint work with Giovanni Forni on a mixing property of this suspended flow.