The Mathematical Colloquia held in 2012 at Linköping University

Organized by Anders Björn, Milagros Izquierdo Barrios, Vladimir Kozlov, and Hans Lundmark.

• Wednesday 25 January 2012, Stefan Rauch, MAI
  Triangular systems of Newton equations
  Abstract: Triangular form of Newton equations is a strong property. Together with the existence of a single quadratic (with respect to velocities) integral of motion, it usually implies the existence of n − 1 further integrals that are also quadratic. These integrals make the triangular system separable in a new type of coordinates. The separation coordinates are built of quadric surfaces that are nonorthogonal and nonconfocal and can intersect along lower dimensional singular manifolds. We present here the main structural theorems of the theory for n-dimensional triangular systems and discuss the structure of separation coordinates in two and three dimensions.
• Wednesday 1 February 2012, Alexander I. Nazarov, Saint Petersburg State University, Russia
  Qualitative properties for solutions to elliptic and parabolic equations with divergence-free lower-order coefficients
  Abstract: We consider uniformly elliptic and uniformly parabolic equations of divergence type: $Lu \equiv -D_i (a_{ij}(x) D_j u) + b_i(x) D_i u = 0$, $Mu \equiv \partial_t u - D_i (a_{ij}(x;t) D_j u) + b_i(x;t) D_i u = 0$, with additional structure condition $\mathrm{div} (b_i) \le 0$ (*) in the sense of distributions. The equations with the lower-order coefficients satisfying this structure condition arise in some applications, in particular in hydrodynamics. We deal with classical properties of solutions, namely, strong maximum principle, Hölder estimates, the Harnack inequality and the Liouville Theorem. We show that under condition (*) the assumptions on $(b_i)$ which ensure these properties can be considerably weakened in the scale of Morrey spaces. The talk is based on a joint paper with N. N. Ural'tseva.
• Wednesday 8 February 2012, Joakim Arnlind, MAI
  Poisson algebraic and non-commutative geometry
  Abstract: Non-commutative geometry has been a fruitful field, both in pure mathematics and in its applications to physics. Ordinary (commutative) geometry can be studied in terms of the algebra of functions on, for instance, a manifold, and one tries to extend the algebraic formulation to non-commutative algebras.
  In the context of mechanics, the algebra of functions is endowed with another structure – the Poisson bracket. When trying to consider a quantum mechanical analogue of the system, one maps functions to operators such that the Poisson bracket corresponds to the commutator of operators. Thus, it becomes important to understand how geometry can be described in terms of the Poisson algebra of smooth functions (on a manifold).
  In this talk, I will give an overview of an approach to non-commutative geometry in terms of matrix limits, together with some physical motivation and general ideas of non-commutative geometry, and explain how one can formulate Riemannian geometry in a Poisson algebraic way.
• Wednesday 15 February 2012, Irina Asekritova, MAI
  On invertibility of linear operators in interpolation spaces
  Abstract: Let $A$ be a linear bounded operator from a Banach couple $\overrightarrow{X}=(X_{0},X_{1})$ to a Banach couple $\overrightarrow{Y}=(Y_{0},Y_{1})$ such that the restrictions of $A$ to the spaces $X_{0}$ and $X_{1}$ have bounded inverses. This condition does not imply that the restriction of the operator $A$ to the real interpolation space $(X_{0},X_{1})_{\theta ,q}$ has a bounded inverse for all values of the parameters $\theta $ and $q$. I plan to discuss the following problem: how can we describe all spaces $(X_{0},X_{1})_{\theta ,q}$ such that the operator $A:(X_{0},X_{1})_{\theta ,q}\rightarrow (Y_{0},Y_{1})_{\theta ,q}$ is invertible? The talk is based on joint work with N. Kruglyak.
• Wednesday 29 February 2012, Niklas Lundström, Umeå universitet
  Estimates for $p$-harmonic functions vanishing on a flat
  Abstract: We study $p$-harmonic functions in a domain $\Omega \subset \mathbf{R}^n$ near an $m$-dimensional plane (an $m$-flat) $\Lambda_m$, where $0 \leq m \leq n-1$. In particular, let $u$ be a positive $p$-harmonic function, with $n < p \leq \infty$, vanishing on a portion of $\Lambda_m$, and suppose that $\beta = (p-n+m)/(p-1)$, with $\beta = 1$ if $p = \infty$. We prove, using certain barrier functions, that $$ u \approx d(x,\Lambda_m)^{\beta} \quad \textrm{near} \quad \Lambda_m.$$ The lower bound holds also in the range $n - m < p \leq \infty$.
• Monday 5 March 2012, Per Enflo, Kent State University, USA
  Orbits of diagonal operators
  Abstract: We will discuss hyperful orbits of operators, i.e. orbits where every subsequence of the orbit spans the whole space. A cyclic vector is a vector whose orbit spans the whole space. Among other things we will show that for diagonal operators on Hilbert space either every cyclic vector has a hyperful orbit or no cyclic vector has a hyperful orbit.
• Wednesday 7 March 2012, Sergey Vakulenko, Institute of Mechanical Engineering Problems and University of Technology and Design, Saint Petersburg, Russia
  Flexibility and robustness under fluctuations of genetic networks
  Abstract: We consider networks with two types of nodes. The v-nodes, called centers, are hyperconnected and interact one to another via u-nodes, called satellites. This centralized architecture realizes a bow-tie scheme and possesses interesting properties. Namely, this organization creates feedback loops that are capable to generate any prescribed patterning dynamics, chaotic or periodic, and create a number of equilibrium states. We show that activation or silencing of a node can sharply switch the network attractor, even if the activated or silenced node is weakly connected. Centralized networks can keep their flexibility, and still be protected against environmental noise. Finding an optimized network that is both robust and flexible is a computationally hard problem in general, but nonetheless this problem is feasible when the number of satellites is large. This finding reduces to a minimization of some spin glass Hamiltonian. This is a joint work with Ovidiu Radulescu (Montpellier, France) .
• Wednesday 14 March 2012, Andrés Navas, Universidad de Santiago de Chile
  On groups generated by elements near rotations
  Abstract: We will show that for a group of circle diffeomorphisms, having a system of generators close (in the $C^2$ topology) to rotations imposes several dynamical restrictions. The most important one goes back to Dumniny: such a group cannot be of "second kind" (it cannot admit a minimal invariant Cantor set). We will see that among subgroups of the Möbius group, the critical case corresponds to the classical (2,3) group.
• Wednesday 21 March 2012, Rögnvaldur G. Möller, University of Iceland
  Symmetry in the theory of infinite graphs
  Abstract: Symmetry has a strong hold on the human mind and is also a fundamental concept in mathematics. In this talk I will discuss symmetry in the context of infinite graphs. More specifically I want to describe various classes of graphs possessing a very high degree symmetry and constructions and classification results of such graphs. The study of these classes of graphs and their automorphism groups has connections to logic, group theory (relate to many different aspects of group theory), graph theory, probability theory and analysis.
• Wednesday 28 March 2012, Ryszard Rubinsztein, Uppsala universitet
  Knots, quandles and connections
  Abstract: Quandles are spaces acting on themselves according to certain rules. Examples are given by symmetric manifolds and conjugacy classes in groups. I shall explain how one can use quandles to construct invariants of knots. I shall then discuss how, on the other hand, these invariants can, in some cases, be interpreted in terms of moduli spaces of flat connections on a punctured 2-dimensional sphere.
• Wednesday 4 April 2012, Michelle Bucher, University of Geneva, Switzerland
  Title: Volumes in geometry and topology
  Abstract: The simplicial volume of manifolds was introduced by Gromov in the beginning of the 80's to give a topological description of the volume of (families of) Riemannian manifolds. Applied to hyperbolic manifolds, this led Gromov to a new proof of Mostow rigidity. In fact the simplicial volume of any Riemannian manifold is proportional to its Riemannian volume by a constant depending only on the universal cover. This phenomenon is reminiscent of the Hirzebruch proportionality principle between Euler characteristic and Riemannian volume, and in fact Euler characteristic and simplicial volume share important properties such as that their positivity implies the positivity of the minimal volume. In this talk, I will review positivity results for the simplicial volume and its relations to Riemannian volume and Euler characteristic.
• Wednesday 11 April 2012, Sergey Nazarov, Russian Academy of Sciences, Saint Petersburg, Russia
  Spectral gaps for periodically perturbed cylindrical waveguides
  Abstract: The band-gap structure of the spectrum in a periodic waveguide permits for the opening of a spectral gap that is an interval of the real positive semi-axis which is free of the spectrum but has both the endpoints in it. The simplest way to indicate spectral gaps is to consider periodic perturbations of a cylindrical waveguide and to apply asymptotic methods for studying eigenvalues of the model problem in the periodicity cell. In the talk some new approaches will be demonstrated to detect spectral gaps and open questions will be formulated as well.
• Wednesday 18 April 2012, Martin Bridson, University of Oxford, UK
  Rigidity, mapping class groups and automorphism groups of free groups
  Abstract: I shall begin with a discussion about the universe of discrete groups and explain why lattices in semisimple Lie groups, mapping class groups of surfaces, and automorphism groups of free groups have a special place in it. Then, developing the deep analogy between these three types of groups, I shall describe several results that extend ideas of rigidity (à la Mostow and Margulis) from the classical setting to the setting of mapping class groups and automorphism groups of free groups. For example, if n is at least 3, the SL(n,Z) cannot act with infinite image on a compact surface or on a non-abelian free group, nor can it act by homeomorphisms on a sphere of dimension less than n-1.
• Wednesday 25 April 2012, Montserrat Casals-Ruiz, University of Oxford, UK
  First-order theories and Tarski problems
  Abstract: Equations are present, implicitly or explicitly, in most branches of mathematics. The first ones to be formalised were the diophantine equations – equations with integer coefficients and integer solutions. Hilbert's tenth problem asks to construct an algorithm to decide whether or not a diophantine equation is compatible. In 1970, combined work of Davis, Putnam, Robinson and Matiyasevich culminated in a proof of the algorithmic undecidability of this problem.
  Nowadays, Hilbert's tenth problem can be formulated for arbitrary structures and in a more general setting in terms of first-order theories. In the case when the structures under consideration are free groups, this problem is known as Tarski's problem. In contrast to the diophantine case, Makanin devised an algorithm to solve the compatibility problem for systems of equations with coefficients and solutions over a free group. The theory developed to solve Tarski problems has established different connections between model theory, geometry and group theory. In this talk, we will present these connections, introduce some of the key techniques and discuss some new directions in this area.
• Wednesday 2 May 2012, Sebastian Hensel, Universität Bonn, Germany
  Geometry of mapping class groups
  Abstract: The mapping class group of a surface is one of the central objects in low dimensional topology and geometry. As a finitely generated group, it carries a natural metric whose geometry is by now well-understood. The geometry of other mapping class groups, however, is much less studied. In this talk we present joint work with Ursula Hamenstädt on the mapping class groups of handlebodies and doubled handlebodies and their relations.
• Wednesday 9 May 2012, Ruth Kellerhals, University of Fribourg, Switzerland
  Minimal volume tesselations in hyperbolic space
  Abstract: After a short introduction to hyperbolic tesselations, orbifolds, simple constructions and properties, we consider those with many symmetries and try to rank them by means of their volumes. We discuss known results in dimensions below five and present then new developments in hyperbolic 5-space by restricting ourselves to the arithmetic, oriented case. This is joint work with Vincent Emery (MPI Bonn).
• Friday 11 May 2012, Maria del Carmen Reguera Rodriguez, Lunds universitet
  Weights that avoid the cancellative properties of singular integrals
  (Joint with the Analysis seminar series.)
  Abstract:In this talk, we will present a family of weights that avoid the cancellative properties of Singular Integrals. These weights first appeared in the speaker's thesis to provide a counterexample to a dyadic version of the so called Muckenhoupt-Wheeden Conjecture, a weighted weak type estimate for Singular Integrals at the end point $p= 1$. The construction presented in this talk is a simplified version of the original one and it allows to disprove the full Conjecture. This is joint work with C. Thiele. In recent work with J. Scurry, we find applications to another question of Muckenhoupt and Wheeden in the two weight setting.