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.
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.
Rainbow connectivity is a new concept for measuring the connectivity of a graph which was introduced in 2008 by Chartrand, Johns, McKeon and Zhang. In a graph G with a given edge colouring, a rainbow path is a path all of whose edges have distinct colours. The minimum number of colours required to colour the edges of G so that every pair of vertices is joined by at least one rainbow path is called the rainbow connection number rc(G) of the graph G.
For any graph G, rc(G) >= diam(G). We will discuss rainbow connectivity in the random graph setting and present the result that for random graphs, rainbow connectivity 2 happens essentially at the same time as diameter 2. In fact, in the random graph process, with high probability the hitting times of diameter 2 and of rainbow connection number 2 coincide
Multi-layered cylinders, or 'multitubes', are ubiquitous throughout the biological world, from microscopic axons to plant stems. Whilst these structures share an underlying common geometry, each one fulfils a different key role in its relevant environment. For example plant stems provide a transport network for nutrients within the organism, whilst the tongue of a chameleon is used for prey capture. This talk will be concerned with the mechanical stability of multitubes. How do the material properties, applied tractions and geometry of elastic rods and tubes influence their critical buckling pressure and mode of buckling? We will discuss the phenomenon of differential growth, an important factor in the mechanical behaviour of such systems and introduce a mathematical framework, which can be used to model differential growth in soft tissues and predict the onset of buckling. We will also present a small number of applications for this research.
A nice bed-time story to end the term. It is often said that ideas like the group law or isogenies on elliptic curves were 'known to Fermat' or are 'found
in Diophantus', but this is rarely properly explained. I will discuss the first work on rational points on curves from the point of view of modern number
theory, asking if it really did anticipate the methods we use today.
Consider a diagram of the unknot with c crossings. There is a
sequence of Reidemeister
moves taking this to the trivial diagram. But how many moves are required?
In my talk, I will give
an overview of my recent proof that there is there is an upper bound on the
number of moves, which
is a polynomial function of c.
The lecture will discuss some applications of topology to a number of interesting physical systems:
1. Classifications of Phases, 2. Classifications of one-dimensional textures in Nematics and Superfluid HE-3,
3. Classification of defects, 4. Phase transition in Liquid membranes.
The solution of these problems leads to interesting mathematics but the talk will also include some historical remarks.
We consider the fractional Laplacian perturbed by the gradient operator b(x)\nabla for various classes of vector fields b. We construct end estimate the corresponding semigroup.
We derive an exact implied volatility expansion for any model whose European call price can be expanded analytically around a Black-Scholes call price. Two examples of our framework are provided (i) exponential Levy models and (ii) CEV-like models with local stochastic volatility and local stochastic jump-intensity.
The University of Oxford’s modelling4all software is a wonderful tool to simulate early medieval populations and their cemeteries in order to evaluate the influence of palaeodemographic variables, such as mortality, fertility, catastrophic events and disease on settlement dispersal. In my DPhil project I will study archaeological sites in Anglo-Saxon England and the German south-west in a comparative approach. The two regions have interesting similarities in their early medieval settlement pattern and include some of the first sites where both cemeteries and settlements were completely excavated.
An important discovery in bioarchaeology is that an excavated cemetery is not a straightforward representation of the living population. Preservation issues and the limitations of age and sex estimation methods using skeletal material must be considered. But also the statistical procedures to calculate the palaeodemographic characteristics of archaeological populations are procrustean. Agent-based models can help archaeologists to virtually bridge the chasm between the excavated dead populations and their living counterparts in which we are really interested in.
This approach leads very far away from the archaeologist’s methods and ways of thinking and the major challenge therefore is to balance innovative ideas with practicability and tangibility.
Some of the problems for the workshop are:
1.) Finding the best fitting virtual living populations for the excavated cemeteries
2.) Sensitivity analyses of palaeodemographic variables
3.) General methodologies to evaluate the outcome of agent based models
4.) Present data in a way that is both statistically correct and up to date & clear for archaeologists like me
5.) Explore how to include analytical procedures in the model to present the archaeological community with a user-friendly and not necessarily overwhelming toolkit