Past

No Seminar this week

We first provide a brief overview of some of the key properties of the space $\textrm{BV}(\Omega;\mathbb{R}^{N})$ of functions of Bounded Variation, and the motivation for its use in the Calculus of Variations. Now consider the variational integral

\[

F(u;\Omega):=\int_{\Omega}f(Du(x))\,\textrm{d} x\,\textrm{,}

\]

where $\Omega\subset\mathbb{R}^{n}$ is open and bounded, and $f\colon\mathbb{R}^{N\times n}\rightarrow\mathbb{R}$ is a continuous function satisfying the growth condition $0\leq f(\xi)\leq L(1+|\xi|^{r})$ for some exponent $r$. When $u\in\textrm{BV}(\Omega;\mathbb{R}^{N})$, we extend the definition of $F(u;\Omega)$ by introducing the functional

\[

\mathscr{F}(u,\Omega):= \inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(Du_{j})\,\textrm{d} x\, \left|

\!\!\begin{array}{r}

(u_{j})\subset W_{\textrm{loc}}^{1,r}(\Omega, \mathbb{R}^{N}) \\

u_{j} \stackrel{\ast}{\rightharpoonup} u\,\,\textrm{in }\textrm{BV}(\Omega, \mathbb{R}^{N})

\end{array} \right. \bigg\} \,\textrm{.}

\]

\noindent For $r\in [1,\frac{n}{n-1})$, we prove that $\mathscr{F}$ satisfies the lower bound

\[

\mathscr{F}(u,\Omega) \geq \int_{\Omega} f(\nabla u (x))\,\textrm{d} x + \int_{\Omega}f_{\infty} \bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|\,\textrm{,}

\]

provided $f$ is quasiconvex, and the recession function $f_{\infty}$ ($:= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$) is assumed to be finite in certain rank-one directions. This result is a natural extension of work by Ambrosio and Dal Maso, which deals with the case $r=1$; it involves combining work of Kristensen, Braides and Coscia with some new techniques, including a polyhedral approximation result and a blow-up argument that exploits fine properties of BV functions.

I will be looking at some conjectures and theorems closely related to the h-cobordism theorem and will try to show some connections between them and some group theoretic conjectures.

The homological dimension of a group can be computed over any coefficient ring $K$.
It has long been known that if a soluble group has finite homological dimension over $K$
then it has finite Hirsch length and the Hirsch length is an upper bound for the homological
dimension. We conjecture that equality holds: i.e. the homological dimension over $K$ is
equal to the Hirsch length whenever the former is finite. At first glance this conjecture looks
innocent enough. The conjecture is known when $K$ is taken to be the integers or the field
of rational numbers. But there is a gap in the literature regarding finite field coefficients.
We'll take a look at some of the history of this problem and then show how some new near complement
and near supplement theorems for minimax groups can be used to establish the conjecture
in special cases. I will conclude by speculating what may be required to solve the conjecture outright.

We will first review the construction of N =1

supersymmetric Yang-Mills theory in three dimensions. Then we will

construct a superloop space formulation for this super-Yang-Mills

theory in three dimensions.Thus, we will obtain expressions for loop

connection and loop curvature in this superloop space. We will also

show that curvature will vanish, unless there is a monopole in the

spacetime. We will also construct a quantity which will give the

monopole charge in this formalism. Finally, we will show how these

results hold even in case of deformed superspace.

In this talk we show how degree N maps of the form $u_{N}(z) = \frac{z^{N}}{|z|^{N-1}}$ arise naturally as stationary points of functionals like the Dirichlet energy. We go on to show that the $u_{N}$ are minimizers of related variational problems, including one whose associated Euler-Lagrange equation bears a striking resemblance to a system studied by N. Meyers in the 60s, and another where the constraint $\det \nabla u = 1$ a.e. plays a prominent role.

We define a McCool group of G as the group of outer automorphisms of G acting as a conjugation on a given family of subgroups. We will explain that these groups appear naturally in the description of many natural groups of automorphisms. On the other hand, McCool groups of a toral relatively hyperbolic group have strong finiteness properties: they have a finite index subgroup with a finite classifying space. Moreover, they satisfy a chain condition that has several other applications.
This is a joint work with Gilbert Levitt.

• Kiran Singh - Multi-body dynamics in elastocapillary systems
• Graham Morris - Investigating a catalytic mechanism using voltammetry
• Thomas Woolley - Cellular blebs: pressure-driven axisymmetric, membrane protrusions

Problems of packing shapes with maximal density, sometimes into a

container of restricted size, are classical in discrete

mathematics. We describe here the problem of packing a given set of

ellipsoids of different sizes into a finite container, in a way that

allows overlap but that minimizes the maximum overlap between adjacent

ellipsoids. We describe a bilevel optimization algorithm for finding

local solutions of this problem, both the general case and the simpler

special case in which the ellipsoids are spheres. Tools from conic

optimization, especially semidefinite programming, are key to the

algorithm. Finally, we describe the motivating application -

chromosome arrangement in cell nuclei - and compare the computational

results obtained with this approach to experimental observations.

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This talk represents joint work with Caroline Uhler (IST Austria).

A refinement of the Pandharipande-Thomas stable pair invariants for local toric Calabi-Yau threefolds is defined by what we call the virtual Bialynicki-Birula decomposition. We propose a product formula for the generating function for the refined stable pair invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local ${\bf P}^1$. I will also describe how the proposed product formula is related to the wall crossing in my first talk. This is joint work with Sheldon Katz and Albrecht Klemm.

We study the birational relationship between the moduli spaces of $\alpha$-stable pairs and the moduli space $M(d,1)$ of stable sheaves on ${\bf P}^2$ with Hilbert polynomial $dm+1$. We explicitly relate them by birational morphisms when $d=4$ and $5$, and we describe the blow-up centers geometrically. As a byproduct, we obtain the Poincare polynomials of the moduli space of stable sheaves, or equivalently the refined BPS index. This is joint work with Kiryong Chung.

I will describe the history of the homotopy groups of spheres, and some of the many different roles they have come to play in mathematics.

I'll discuss some results about lattices in totally
disconnected locally compact groups, elaborating on the question:
which classical results for lattices in Lie groups can be extended to
general locally compact groups. For example, in contrast to Borel's
theorem that every simple Lie group admits (many) uniform and
non-uniform lattices, there are totally disconnected simple groups
with no lattices. Another example concerns with the theorem of Mostow
that lattices in connected solvable Lie groups are always uniform.
This theorem cannot be extended for general locally compact groups,
but variants of it hold if one implants sufficient assumptions. At
least 90% of what I intend to say is taken from a paper and an
unpublished preprint written jointly with P.E. Caprace, U. Bader and
S. Mozes. If time allows, I will also discuss some basic properties
and questions regarding Invariant Random Subgroups.

Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of Aut(G) on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev and Saitô for lattices in nilpotent Lie groups and in solvable Lie groups of real type. We prove that the deformation space of every Zariski-dense lattice Γ in G is finite and Hausdorff, provided that the maximal nilpotent normal subgroup of G is connected.  I will introduce all necessary notions and try to motivate and explain this result.

Most results about the Cayley graph of a hyperbolic surface group can be replicated in the context of more general hyperbolic groups. In this talk I will discuss two results about such Cayley graphs which I do not know how to replicate in the more general context.

• Victor Burlakov - Understanding the growth of alumina nanofibre arrays
• Brian Duffy - Measuring visual complexity of cluster-based visualisations
• Chris Bell - Autologous chemotaxis due to interstitial flow

In this talk, we will look at two non-linear wave equations in 2+1 dimensions, whose elliptic parts exhibit conformal invariance.

These equations have their origins in prescribing the Gaussian and mean curvatures respectively, and the goal is to understand well-posedness, blow-up and bubbling for these equations.

This is a joint work with Sagun Chanillo.

(Joint work with Nikolaos Galatos.) Proof-theoretic methods provide useful tools for tackling problems for many classes of algebras. In particular, Gentzen systems admitting cut-elimination may be used to establish decidability, complexity, amalgamation, admissibility, and generation results for classes of residuated lattices corresponding to substructural logics. However, for classes of algebras bearing a family resemblance to groups, such methods have so far met only with limited success. The main aim of this talk will be to explain how proof-theoretic methods can be used to obtain new syntactic proofs of two core theorems for the class of lattice-ordered groups: namely, Holland's result that this class is generated as a variety by the lattice-ordered group of order-preserving automorphisms of the real numbers, and the decidability of the word problem for free lattice-ordered groups.

We will start with the square torus, move on to all regular polygons, and then look at a large family of flat surfaces called Bouw-Möller surfaces, made by gluing together many polygons. On each surface, we will consider the action of a certain shearing action on geodesic paths on the surface, and a certain corresponding sequence.

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of L\'evy processes that is based on the Wiener-Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. We also provide here a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant. We find that the rate of convergence is uniformly with respect to the ``jump activity'' (e.g. characterised by the Blumenthal-Getoor index).

Mesocosm experiments provide a major test bed for models of plankton, greenhouse gas export to the atmosphere, and changes to ocean acidity, nitrogen and oxygen levels. A simple model of a mesocosm plankton ecology is given in terms of a set of explicit natural population dynamics rules that exactly conserve a key nutrient. These rules include many traditional population dynamics models ranging from Lotka-Volterra systems to those with more competitors and more trophic levels coupled by nonlinear processes. The rules allow a definition of an ecospace and an analysis of its behaviour in terms of equilibrium points on the ecospace boundary.

Ecological issues such as extinctions, plankton bloom succession, and system resilience can then be analytically studied. These issues are understood from an alternative view point to the usual search for interior equilibrium points and their classification, coupled with intensive computer simulations. Our approach explains why quadratic mortality usually stabilises large scale simulation, but needs to be considered carefully when developing the next generation of Earth System computer models. The ‘Paradox of the Plankton’ and ‘Invasion Theory’ both have alternative, yet straightforward explanations within these rules.

(Joint work with Artem Chernikov.) In the talk, we will first recall some basic results on valued difference fields, both from an algebraic and from a model-theoretic point of view. In particular, we will give a description, due to Hrushovski, of the theory VFA of the non-standard Frobenius acting on an algebraically closed valued field of residue characteristic 0, as well as an Ax-Kochen-Ershov type result for certain valued difference fields which was proved by Durhan. We will then present a recent work where it is shown that VFA does not have the tree property of the second kind (i.e., is NTP2); more generally, in the context of the Ax-Kochen-Ershov principle mentioned above, the valued difference field is NTP2 iff both the residue difference field and the value difference group are NTP2. The property NTP2 had already been introduced by Shelah in 1980, but only recently it has been shown to provide a fruitful ‘tameness’ assumption, e.g. when dealing with independence notions in unstable NIP theories (work of Chernikov-Kaplan).

The periodic orbits of a discrete dynamical system can be described as

permutations. We derive the composition law for such permutations. When

the composition law is given in matrix form the composition of

different periodic orbits becomes remarkably simple. Composition of

orbits in bifurcation diagrams and decomposition law of composed orbits

follow directly from that matrix representation.

This talk surveys the well known relationship between half-flat SU(3) structures on 6-manifolds M and metrics with holonomy in G_2 on Mx(a,b), focusing on the case in which M=S3xS3 with solutions invariant by SO(4).

We introduce a deformation process of universal enveloping algebras of Borcherds-Kac-Moody algebras, which generalises quantum groups' one and yields a large class of new algebras called coloured Borcherds-Kac-Moody algebras. The direction of deformation is specified by the choice of a collection of numbers. For example, the natural numbers lead to classical enveloping algebras, while the quantum numbers lead to quantum groups. We prove, in the finite type case, that every coloured BKM algebra have representations which deform representations of semisimple Lie algebras and whose characters are given by the Weyl formula. We prove, in the finite type case, that representations of two isogenic coloured BKM algebras can be interpolated by representations of a third coloured BKM algebra. In particular, we solve conjectures of Frenkel-Hernandez about the Langland duality between representations of quantum groups. We also establish a Langlands duality between representations of classical BKM algebras, extending results of Littelmann and McGerty, and we interpret this duality in terms of quantum interpolation.

We describe our current efforts to develop finite volume

schemes for solving PDEs on logically Cartesian locally adapted

surfaces meshes. Our methods require an underlying smooth or

piecewise smooth grid transformation from a Cartesian computational

space to 3d surface meshes, but does not rely on analytic metric terms

to obtain second order accuracy. Our hyperbolic solvers are based on

Clawpack (R. J. LeVeque) and the parabolic solvers are based on a

diamond-cell approach (Y. Coudi\`ere, T. Gallou\"et, R. Herbin et

al). If time permits, I will also discuss Discrete Duality Finite

Volume methods for solving elliptic PDEs on surfaces.

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To do local adaption and time subcycling in regions requiring high

spatial resolution, we are developing ForestClaw, a hybrid adaptive

mesh refinement (AMR) code in which non-overlapping fixed-size

Cartesian grids are stored as leaves in a forest of quad- or

oct-trees. The tree-based code p4est (C. Burstedde) manages the

multi-block connectivity and is highly scalable in realistic

applications.

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I will present results from reaction-diffusion systems on surface

meshes, and test problems from the atmospheric sciences community.

Our starting point is a recent characterisation of one-dimensional, time-homogeneous diffusion in terms of its distribution at an exponential time. The structure of this characterisation leads naturally to the idea of measuring `how far' a diffusion is away from being a martingale diffusion in terms of expected local time at the starting point. This work in progress has a connection to finance and to

a Skorokhod embedding.

A subgroup $H$ of a group $G$ is said to be engulfed if there is a
finite-index subgroup $K$ other than $G$ itself such that $H<K$, or
equivalently if $H$ is not dense in the profinite topology on $G$.  In
this talk I will present a variety of methods for showing that a
subgroup of a discrete group is engulfed, and demonstrate how these
methods can be used to study finite-sheeted covering spaces of
topological spaces.

Several shape optimization problems, e.g. in image processing, biology, or discrete geometry, involve the Willmore functional, which is for a surface the integrated squared mean curvature. Due to its singularity, minimizing this functional under constraints is a delicate issue. More precisely, it is difficult to characterize precisely the structure of the minimizers and to provide an explicit

formulation of their energy. In a joint work with Giacomo Nardi (Paris-Dauphine), we have studied an "integrated" version of the Willmore functional, i.e. a version defined for functions and not only for sets. In this talk, I will describe the tools, based on Young measures and varifolds, that we have introduced to address the relaxation issue. I will also discuss some connections with the phase-field numerical approximation of the Willmore flow, that we have investigated with Elie Bretin (Lyon) and Edouard Oudet (Grenoble).

Let G be a permutation group acting on a set Omega. For g in G, let pi(g) denote the partition of Omega given by the orbits of g. The set of all partitions of Omega is naturally ordered by refinement and admits lattice operations of meet and join. My talk concerns the groups G such that the partitions pi(g) for g in G form a sublattice. This condition is highly restrictive, but there are still many interesting examples. These include centralisers in the symmetric group Sym(Omega) and a class of profinite abelian groups which act on each of their orbits as a subgroup of the Prüfer group. I will also describe a classification of the primitive permutation groups of finite degree whose set of orbit partitions is closed under taking joins, but not necessarily meets.

This talk is on joint work with John R. Britnell (Imperial College).

Beautiful results of Deligne-Illusie, Sabbah, and Ogus-Vologodsky show that certain modifications of the de Rham complex (either the usual one, or twisted versions of it that appear in the study of the cyclic homology of categories of matrix factorizations) are formal in positive characteristic. These are the crucial steps in proving algebraic analogues of the Hodge theorem (again, either in the ordinary setting or in the presence of a twisting). I will present these results along with a new approach to understanding them using derived intersection theory. This is joint work with Dima Arinkin and Marton Hablicsek.