Dublin Area Mathematics Colloquium
Andras Szenes (Geneve)
Y-systems and dilogarithm identities
Thursday 27 October 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
In 1991, Zamolodchikov discovered an intriguing rational recursion called the Y-system. The recursion has some remarkable periodicity properties, and relations to number theory, representation theory, and several other fields of mathematics. The Y-systems gave rise to identities among the values of the dilogarithm function, and also served as one of the first examples of an important new notion of modern algebra: the cluster algebras. In this talk, we will review recent results and developments in the field.
Stephanie Nivoche (Nice)
Proof of a conjecture of Zahariuta concerning
a problem of Kolmogorov on the epsilon-entropy
Thursday 13 October 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
The problem of Kolmogorov on the epsilon-entropy of some classes of analytic functions, is in connection
with the 13th problem of Hilbert about complexity of functions' spaces.
The idea is to find a determination of the precise asymptotic behaviour of the epsilon-entropy of functions
of n variables defined on a bounded domain in R^n which extend analytically to some domain in C^n.
This is possible, with technics of pluripotential theory, by solving a conjecture of Zahariuta about
approximation of plurisubharmonic functions.
Eric Loubeau (Brest)
Biharmonic maps and tress-energy tensors
Friday 10 June 2011, 4.00 pm, Mathematics Seminar Room (2nd floor), Hamilton Building, TCD.
Abstract.
I will try and explain how the idea of a stress-energy tensor introduced in General relativity can be applied to variational problems in Differential Geometry, first for the classical case of harmonic maps and then its more recent generalization, called biharmonic maps.
I will derive the basic properties of such stress-energy tensors and show how they can be useful to study critical points of either functional.
Chris Wood (York)
Harmonic Vector Fields
Thursday 9 June 2011, 5.00 pm, Mathematics Seminar Room (2nd floor), Hamilton Building, TCD.
Abstract.
We address the question of what are the "best" vector fields (equivalently, flows) on a Riemannian manifold. The idea is to look at the energy of the vector field, viewed as a mapping into the tangent bundle, and apply the theory of harmonic maps. This was first mooted back in the 1980s, but unfortunately suffered an early blow when it was observed that on a compact manifold any such "harmonic vector field" is necessarily parallel (Ishihara, Nouhaud, W). This was partially overcome by restricting the energy functional to unit vector fields (Wiegmink, Vanhecke, Gil-Medrano, W, et al), which has led to quite a rich theory. However, it is not applicable manifolds of non-zero Euler characteristic. In this talk I will show how a fairly tightly prescribed perturbation of the background geometry of the tangent bundle, away from the Sasaki metric, allows us to formulate a general theory of harmonicity for vector fields, which to a large extent preserves the existing theory of harmonic unit fields. As a criterion for optimality, it produces some interesting results when applied to vector fields on space forms, and raises some interesting questions. This is joint work with Eric Loubeau and Michelle Benyounes.
Roman Fedorov (MPI Bonn)
Categorical geometric Langlands duality
Thursday 7 April 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
The categorical Langlands duality is a certain equivalence of categories (still very conjectural). It is a far-reaching generalization of Fourier-Mukai transform for abelian varieties. This equivalence is the strongest form of the geometric Langlands duality, and also it is the easiest to formulate. I shall explain the statement in detail and shall explain why it is useful. Then I shall discuss some established cases.
Martin Kolar (Brno)
Chern-Moser operators and symmetries of CR manifolds
Friday 1 April 2011, 4.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
The talk will consider an analytic approach to the local equivalence problem in CR geometry,
originating in the work of Poincare. We
will explain the main ideas of the Chern-Moser normal form
construction and its generalization to degenerate manifolds. Then we
use this approach to study symmetry groups, and classify manifolds
admitting nonlinear symmetries.
Loïc Foissy (Reims)
Combinatorial Dyson-Schwinger equations
Thursday 31 March 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
In quantum field theory, the propagators satisfy a system of Dyson-Schwinger equations in a Hopf algebra of Feynman graphs. Using a universal property, these systems can be lifted to the Hopf algebra of decorated rooted trees with the help of grafting operators. It turns out that in the known cases, the subalgebra generated by the solution of this system is Hopf, a result that is false for an arbitrary system. By the Milnor-Moore theorem, this Hopf algebra is dual to the enveloping algebra of a Lie algebra. Using combinatorial and graph-theoretical methods, we shall give a complete description of Dyson-Schwinger systems with this properties, as well as the associated Lie algebras.
Nordine Mir (Rouen)
Algebraic approximation in CR geometry
Thursday 24 March 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
In this talk, I will provide a CR version of Artin's
approximation theorem for holomorphic mappings between real-algebraic
sets in complex space. The result can be seen as a PDE version of AAT
involving systems of complex vector fields.
Jasmin Raissy (Milan)
Normal forms in complex dynamics
Thursday 10 March 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
Normal forms are a very important tool in several branches of mathematics. I shall discuss the normalization and linearization problems for germs of biholomorphisms in several complex variables with an isolated fixed point, starting from the classical Poincaré-Dulac procedure, going through small divisors problem, and ending with more geometric approaches I recently introduced. I shall also present some new applications to the study of complex local dynamics.
Paolo Piccione (São Paulo)
Equivariant stability and bifurcation in geometric variational problems
Friday 18 February 2011, 3.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
I will discuss an abstract formulation of an equivariant
implicit function theorem, and present some applications to geometric variational problem, such as
closed geodesics in (pseudo)-Riemannian manifolds, harmonic maps, CMC and
minimal hypersurfaces. These variational problems are invariant by a Lie group
of symmetries, whose action is not necessarily differentiable.
David Quinn (Queen's University Belfast)
Incidence algebras of posets and acyclic categories
Thursday 17 February 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
There exists a close relationship between the topology of a simplicial complex and the
algebraic structure of the incidence algebra of its face poset. In particular the incidence
algebra is Koszul if and only if the complex is Cohen-Macaulay. Acyclic categories are
small categories which can be considered as a generalization of posets, and one can
also define their incidence algebras and semi-simplicial complexes. In this talk I will
present two new results. First, the Koszul equivalence can be generalised to acyclic
categories, and second, there exists a similar equivalence if one replaces the condition
of Cohen-Macaulay with the stronger (but purely combinatorial) condition of
lexicographic shellability.
David Jordan (Sheffield)
Poisson algebras and noncommutative algebras
Thursday 10 February 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
For the purposes of this talk, a Poisson algebra is a commutative associative algebra over the complex numbers with a so-called Poisson bracket, under which it is a Lie algebra in a manner compatible with the associative structure. The examples that I mention will be mostly polynomial algebras. For many Poisson algebras there is a corresponding noncommutative (or quantised) algebra and, although there is no general proof, the algebraic properties of either the Poisson algebra or the noncommutative algebra reflect those of the other. In some sense, the Poisson algebra is a limit, known as the semiclassical limit, of a family of noncommutative algebras in which, in the limit, the noncommutativity disappears but is captured in the Poisson bracket. I will illustrate these ideas with a case study involving a family of Poisson algebras which have recently led to interesting new noncommutative algebras. These examples arose in the context of quiver mutation and cluster algebras but they can also be presented in an elementary way in terms of recurrence sequences.
Sung-Yeon Kim (Kangwon National University, Korea)
Domains with Noncompact Automorphism Groups
Thursday 3 February 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
Let D be a smoothly bounded domain in Cn, n\ge 2; and let Aut(D)
be the group of all biholomorphic self maps of D. Then Aut(D)
together with the map composition is a real Lie group with respect to the compact-open topology.
It is a Greene-Krantz conjecture that if D admits a noncompact automorphism group,
then there is an orbit accumulation boundary point of finite type.
In this talk, we show that every hyperbolic orbit accumulation boundary point of D
is of finite type provided that the Bergman kernel of D
extends smoothly up to the boundary minus the diagonal.
As an application, we show that such a domain admits
a hyperbolic orbit accumulation boundary point if and only
if it is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial.
Alessandra Frabetti (University of Lyon)
Renormalization in quantum field theory and combinatorial groups
Friday 28 January 2011, 4.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
Ten years ago, D. Kreimer and A. Connes efficiently described the renormalization of Feynman graphs using Hopf algebraic tools. In this talk I give an overview of the steps which lead to some groups of series expanded over combinatorial objects and their physical interpretation.
Juan Luis Vázquez (Univ. Autónoma de Madrid)
The Theories of Nonlinear Diffusion
Thursday 27 January 2011, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
We will present the mathematical theory of nonlinear diffusion processes, starting from the physical and mathematical motivations. We will then focus on the models called the porous medium equation and the fast diffusion equation. For these models we will discuss the basic problems, existence, uniqueness and regularity, as well as the main specific features, like the asymptotic behaviour of the solutions for large time. For fast diffusion there are curious phenomena like extinction in finite time tied to Functional Analysis via suitable weighted Sobolev inequalities. If time allows we will present some of the current lines.
Nadia Sidorova (University College London)
A two cities theorem for the parabolic Anderson model
Thursday 2 December 2010, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
The parabolic Anderson problem is the Cauchy problem for the
heat equation on the d-dimensional integer lattice with random potential.
We consider independent and identically distributed potentials, such
that the corresponding distribution function converges polynomially at
infinity. If the solution is initially localised in the origin we show
that, as time goes to infinity, it will be completely localised in two
points almost surely and in one point with high probability. We also
identify the asymptotic behaviour of the concentration sites in terms
of a weak limit theorem.
Esther Vergara (TCD)
Some of The Best Geometric Structures
Thursday 18 November 2010, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
From a very well-known functional of energy - the one for harmonic maps- we introduce the vertical functional of energy and its correspondent associated Euler-Lagrange equation. The critical points of this vertical functional of energy are `harmonic sections'. In particular, we study the harmonic sections of the twistor bundle since they are associated to geometric structures. We rewrite the Euler-Lagrange equation for the vertical functional of energy in terms of the geometric structure only. Equations for harmonicity of almost complex manifolds, almost contact manifolds and f-structures are provided, as well as representative examples in each case.
Leon A. Takhtajan (Stony Brook)
On Bott-Chern forms for holomorphic Hermitian vector bundles and
differential K-theory
Friday 8 October 2010, 4.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
We will discuss a geometric model for differential K-theory for
holomorphic vector bundles which generalizes
the theory defined by J. Simons and D. Sullivan for the smooth complex
vector bundles.
Alexey Bondal (Aberdeen)
Minuscule Varieties
Thursday 30 September 2010, 5.00 pm, Schrödinger Lecture Theatre, Physics Building, TCD.
Abstract.
The introduction to the talk will contain a short description, targeted on
pure mathematicians, of the Standard Model of Particle Physics and its
Grand Unifications. This gives a motivation for a closed study of
minuscule varieties, particularly nice homogeneous spaces. We shall
describe minuscule descent, a transit from one minuscule variety to
another one of lower dimension, which is parallel to symmetry breaking in
Quantum Field Theory. We shall show how mirror symmetry works for
minuscule varieties.
Bruno Vallette (Nice)
Operads in algebra, topology, geometry and mathematical physics
Thursday 22 April 2010, 5.00 pm, TBA, Hamilton Building, TCD.
Abstract.
The purpose of this talk will be to make accessible the
notion of operad, which a mathematical device used to encode
operations with several inputs and one output. This notion is used
nowadays in many fields of mathematics. Time permitting, I will give
examples in algebra, topology, geometry and mathematical physics.
Karin Baur (ETH Zurich)
Geometric constructions of cluster categories
Thursday 25 March 2010, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
Cluster categories arise in the representation theory of
algebras. They are categorical
models of the cluster algebras as defined by Fomin and Zelevinsky
around 2000. Cluster
algebras have beein introduced in connection with the studies of the
dual canonical basis
and with the phenomena of total positivity.
We explain how to geometrically construct cluster categories of type A
(and D) and present
geometrical models for the m-cluster categories. This generalizes work of
Caldero-Chapoton-Schiffler.
Andras Szenes (Geneva)
Residues and Thom polynomials
Thursday 18 March 2010, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
Starting with the work of Whitney, the study of
singularities of smooth maps between manifolds has been an important
question of modern topology. There is a polynomial invariant,
introduced by Thom in the 1950s, which links the enumerative
characteristics of manifolds with the type of singularities which
cannot be avoided in maps between them. In this talk, I will report on
recent progress in calculating these polynomials.
Brendan Owens (Glasgow)
Knots in 4-dimensional topology
Friday 12 March 2010, 4.30 pm, Salmon Lecture Theatre, Hamilton Building, TCD.
Abstract.
Classical knot theory is the study of embedded circles in 3-
dimensional space. The purpose of this talk is to illustrate the rich
give-and-take between knot theory and 4-dimensional topology. I will
discuss the use of knots in descriptions of 3- and 4-dimensional
manifolds. I will also describe how a 4-dimensional point of view of
knots gives rise to a group called the knot concordance group, and
discuss some recent advances in the study of this group.
Vincent Minerbe (Paris)
On ALF gravitational instantons
Thursday 25 February 2010, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
Basically, ALF gravitational instantons are complete
non-compact hyperkähler four-manifolds whose geometry at infinity
is asymptotic to a circle fibration over the Euclidean three-space, with fibres
of asymptotically constant length. This kind of geometry appears naturally
in gauge theory and is also relevant in string theory. In this talk, I will explain
the definition above, describe examples and prove a classification result,
which is part of a broader conjecture.
Tom Lenagan (Edinburgh)
Totally nonnegative matrices
Thursday 11 February 2010, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
A real matrix is totally positive if all of its minors are
positive. More generally, a matrix is totally non-negative if all of
its minors are non-negative. Totally positive/non-negative matrices
arise in many areas; for example, oscillations in mechanical systems,
stochastic processes and approximation theory, planar resistor
networks, ....
This talk will be an elementary introduction to the theory of totally
non-negative matrices and the associated study of the non-negative
real grassmannian.
Wilhelm Kaup (Tuebingen)
Tube realizations for CR-manifolds and maximal abelian subalgebras
Thursday 4 February 2010, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
For every real-analytic CR-manifold M we discuss necessary and
sufficient conditions that M can be locally realized as a tube
submanifold of some C^n. It turns out that the affine equivalence
classes of such tube realizations can be described purely
algebraically by certain maximal abelian subalgebras of a suitable
real Lie algebra. For the special case where M is a non-degenerate
hyperquadric the classification of all affine equivalence classes of
local tube realizations is the same as the classification of all
commutative (associative) nilpotent algebras of finite dimension with
1-dimensional annihilator over the real and the complex field up to
isomorphy.
Filippo Bracci (Rome)
Loewner's theory on complex manifolds
Friday 4 December 2009, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.
Abstract.
Loewner's theory has been introduced in the unit disc by Loewner in the 1930's and it has been mainly developped by Pommerenke. De Branges used such a theory to give a solution to the Bieberbach conjecture about the growth of coefficients of univalent functions. In the late 1990's Schramm introduced such a theory in the realm of stochastic equations and, with Werner and Lawler, solved the Mandelbrot conjecture. In the meantime, Loewner's theory has been generalized and applied to several complex variables by Pfalzgraff, Graham, Kohr, Hamada and others.
In this talk I present a general and unified treatment of Loewner's theory which works on complete hyperbolic manifolds, and give a precise correspondence among Loewner chains, Herglotz vector fields and evolution family.
Iain Gordon (Edinburgh)
Generalisation of Catalan numbers
Thursday 29 October 2009, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
I will discuss a couple of classical definitions for the
Catalan numbers, and then more recent work that relates these numbers
to certain finite reflection groups. Using some results of Opdam and
of Rouquier from the representation theory of Hecke algebras, I will
then show how to construct bigraded versions of these Catalan numbers
for _any_ complex reflection group, answering a couple of
combinatorial questions along the way.
Jean-Louis Loday (Strasbourg)
Parenthesizing and polytopes
Friday 23 October 2009, 4.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
Computing a word in a group or an algebra consists in choosing some parenthesizing and, then, apply the group law or the product several times. The combinatorics underlying these parenthesings is a very intricate and rich matter which leads to the notion of algebra up to homotopy and to a family of polytopes called associahedron or Stasheff polytopes. New results on these objects have recently been discovered, like the strong relationship with the notion of dendriform algebra, that is an associative algebra whose product splits into two other binary operations.
Frederic Chapoton (Lyon)
Exceptional sequences and the dendriform operad
Thursday 8 October 2009, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
Exceptional sequences are used a lot in the study of derived categories.
There is an action of the braid group on exceptional sequences, which is
transitive in some cases. In the case of quivers of type A, exceptional
sequences can be described in a simple way using noncrossing trees.
On the other hand, Loday has introduced a new algebraic structure,
called a dendriform algebra, which is closely related to another kind of
trees, called planar binary trees.
We will explain how these two subjects are related, first at a basic
combinatorial level, then in a more algebraic way by using the setting
of operad theory.
Richard Cleyton (Esbjerg, Denmark)
Weak mirror symmetry on Lie algebras
Thursday 1 October 2009, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.
Abstract.
I shall explain what weak mirror symmetry is, and attempt a brief explanation of how this is related to the 'real' mirror symmetries of physics and homological algebra. I will give an outline of how one can go about getting solutions to weak mirror symmetry, that is, weak mirror pairs consisting of a compact complex manifold on one side and a compact symplectic manifold on the other. If time allows, I'll list some classifications of solutions under certain constraints.