# Dublin Area Mathematics Colloquium

## School of Mathematics, Trinity College Dublin

New Colloquium page with more recent talks: http://www.maths.tcd.ie/~mozgovoy/geometry/

Vincent Pilaud (École Polytechnique, CNRS)
The Cambrian algebra.
Thursday March 24, 2.00 pm, East End Lecture Theatre 3, basement of Panoz institute, Hamilton Building, TCD.

Abstract. Cambrian trees provide a natural generalization of binary search trees, where each node can have one parent and two children, or two parents and one child. The talk will present applications of Cambrian trees to combinatorics, algebra, and geometry. In particular, I will present the the Cambrian algebra, generalizing Loday-Ronco's algebra on binary trees, and revisit Hohlweg-Lange's construction of the associahedron. The talk is based on joint works with Grégory Chatel (Univ. Marne la Vallée) and Carsten Lange (Univ. Munich).

Mark Spivakovsky (Toulouse, CNRS)
On the contriubtions of John Nash to algebraic geometry.
Thursday March 10, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. This lecture is a homage to John Forbes Nash, who died tragically on May 23rd, 2015. We will attempt to describe two of John Nash's main contributions to algebraic geometry, dating from the 1960s:
1) Nash blowing-up as a conjectural method for constructing a canonical resolution of singularities of varieties in characteristic zero.
2) The Nash problem on the spaces of arcs on singular algebraic varieties.
All the required notions such as algebraic variety, singularity, blowing up, resolution, arc, etc. will be introduced from scratch. No previous knowledge of algebraic geometry will be assumed.

Linear Algebra over Polynomial Rings
Thursday 29 October, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. The main question I will address in this talk is how the rank of a matrix with entries in a polynomial ring depends on the parameters. In the simplest case of matrices over a field (no parameters), the question is answered by Gaussian elimination, which allows us to compute the row canonical form of the matrix. In the case of one parameter, the polynomial ring is a PID, and Gaussian elimination combined with the Euclidean algorithm for GCDs allows us to compute the Hermite normal form of the matrix. As soon as there are two or more parameters, we need different techniques: one very useful fact is the characterization of the rank in terms of (non-)vanishing of certain minors of the matrix. This leads to the notion of determinantal ideals of a matrix and their Gröbner bases and radicals. Computing determinantal ideals can be very difficult, which leads us to look for canonical or at least reduced forms of matrices. This connects with the notion of Gröbner bases for submodules of free modules over polynomial rings.

Markus Reineke (University of Wuppertal)
Geometry of matrix invariants
Thursday 22 October, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. It is a classical unsolved problem of linear algebra to classify tuples of matrices up to simultaneous conjugacy. Via Geometric Invariant Theory, one constructs moduli spaces of such conjugacy classes, allowing to approach a qualitative understanding of the problem with geometric means. Using techniques from the geometric representation theory of quivers, global invariants of the moduli spaces of matrix tuples can be computed. These constructions and results will be reviewed, and links to combinatorics, probability, BPS state counts and lattice models will be discussed.

Peter Zograf (Steklov Math. Inst. and Chebyshev Lab, SPbU)
Mathematical physics of map enumeration
Friday 9 October, 4.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.

Abstract. Counting problems for maps (ribbon graphs) and hypermaps (Grothendieck's "dessins") naturally appear in various fields of mathematics and mathematical physics. It will be shown that certain generating functions associated with map and hypermap count satisfy a number of remarkable integrability properties, like Virasoro constraints, evolution equation, KP hierarchy and topological recursion in the sense of Eynard-Orantin.

Twisting Elements in Homotopy Gerstenhaber Algebras Controlling Deformations
Thursday 8 October, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. The structure of homotopy G-algebra in the Hochschild complex will be described. This structure will be used to define twisting elements and their transformations, which control Grestenhaber's deformations of associative algebras and Stasheff's A_\infty-deformations of graded algebras.

Ilya Kossovsky (Vienna)
On the classification of second order differential equations
Thursday 1 October, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. We consider the problem of linearization of a second order ordinary differential equation y''=F(x,y,y') under local diffeomorphisms of the plane (x,y). It has been known since the work of Cartan and Tresse that only a very rare such equation can be actually linearized. Cartan and Tresse in also presented certain obstruction for the linearization of an ODE. However, it is of a separate interest to see the possibility to linearize a second order ODE by bringing it to certain normal form, where the ODE becomes "maximally close" to linear, and the non-linear terms in the normal form represent the obstruction for the linearization. Arnold in the 80's made an attempt to construct such a normal form, however, his normal form is known to be incomplete. In our joint work with Zaitsev, we use some recently discovered connection between Several Complex Variables and Dynamical Systems to overcome the difficulties in completing the above Arnold's problem. Namely, we construct a complete convergent normal form for a smooth second order ODE. We also show applications of the normal form for studying symmetries of second order ODEs.

Norbert Poncin (Luxembourg)
Geometry of generalized supermanifolds
Thursday 2 April, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. The aim of the talk is to present a generalization of superalgebra and supergeometry to Z_2^n-gradings, n>1. The corresponding sign rule is not given by the product of the parities, but by the scalar product of the involved Z_2^n - degrees. This Z_2^n - supergeometry exhibits interesting differences with classical supergeometry, provides a sharpened viewpoint, and has better categorical properties. Further, it is closely related to Clifford calculus: Clifford algebras have numerous applications in physics, but the use of Z_2^n - gradings has never been investigated. If time permits, the Z_2^n-Berezinian determinant and the corresponding integration theory will be discussed.

The exterior algebra and central notions in mathematics
Thursday 12 March, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. We show how the exterior algebra and natural structures on it give rise to central notions in combinatorics, topology, Lie theory, mathematical physics, and algebraic geometry.

Francine Meylan (Fribourg)
Holomorphic Approximation in Banach Spaces.
Friday 6 March, 4.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.

Abstract. In this talk, I will give a survey for the Runge approximation problem for a complex Banach space X of infinite dimension and state a conjecture that has been around since the work of L. Lempert in the late nineties.

Victor Ufnarovski (Lund)
Multigraded sequences, systems of Diophantine equations and magic cubes
Thursday 5 March, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. In this talk we would like to connect quite different topics. We start from an algebraic problem of constructing an infinite cube-free word, translate the solution to the system of Diophantine equations and extend the idea to a general method – construction of multigraded sequences. As example of application we develop a method for creating magic squares and cubes.

Aeryeong Seo (KIAS, Seoul)
Proper holomorphic maps between bounded symmetric domains
Thursday 29 January, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. I will explain one way of finding proper holomorphic maps between bounded symmetric domains of type I and present some infinite number of inequivalent proper holomorphic maps between bounded symmetric domains.

Igor Zelenko (Texas A&M)
Canonical frames for filtered structures on manifolds: old and new
Thursday 27 November, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. Filtered structures on manifolds are nonholonomic vector distributions (subbundles of tangent bundles) possibly with additional structures on them like sub-Riemannian, CR, pseudo-product structures etc. Many of such structures appear naturally in Control Theory as sets of admissible velocities for control systems linear with respect to control parameters and Geometric Theory of Differential Equations as natural distributions on submanifolds of jet spaces. We are interested in determining whether one such structure can be transformed to another one by the natural action of the group of diffeomorphisms of the ambient manifold. The goal of this colloquium talk is to give a survey of the classical and recent methods for solving this problem via construction of the canonical frames (the structure of absolute parallelism). A special emphasis will be given to the algebraic version of Cartan's method of equivalence developed by N. Tanaka in 1970s and my recent joint works with Boris Doubrov which combine the Tanaka approach with the ideas from Geometric Control Theory.

Lagrangian inclusions and holomorphic discs
Thursday 20 November, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. Gromov's theorem on the existence of a holomorphic disc attached to a compact Lagrangian submanifold of C^n has had a deep impact on symplectic topology and complex analysis. I will discuss generalizations of this result to singular submanifolds.

Yuri Bilu (Bordeaux)
Runge's method and modular curves
Thursday 10rd April, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. Runge's method is a very old method in Diophantine analysis, going back 19th century. For quite long it was overshadowed by more powerful tools, but it experienced a remarkable comeback in the first decade of this century. I will give a very elementary introduction into Runge's method, and then I will show how it applied to the study of rational points on modular curves.

Masha Vlasenko (UCD)
Counting points over finite fields on Calabi-Yau hypersurfaces
Thursday 3rd April, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. For a family of algebraic varieties defined over integers, we discuss the relation between periods of the family, that is solutions to a Picard-Fuchs differential equation, and zeta functions of fibers reduced modulo a chosen prime p. We explain a basic example with elliptic curves due to Dwork and Tate and state a generalization to families of Calabi-Yau hypersurfaces which we discovered recently. This is a joint result with Anton Mellit.

Leon Takhtajan (Stony Brook)
On Bott-Chern characteristic forms
Friday 7 February, 3.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.

Abstract. Refinement of the Chern-Weil theory for Hermitian vector bundles naturally leads to secondary characteristic forms, introduced by R. Bott and S.S. Chern in 1964. They play an important role in arithmetic intersection theory and geometric stability. In this lecture I will discuss the double descent' construction associated with the Chern-Weil theory for Hermitian vector bundles, and will explain how to compute Bott-Chern forms explicitly.

Dmitry Kaledin (Steklov Mathematical Institute)
Hochschild-Witt complex
Friday 7 February, 4.15 pm, Synge Lecture Theatre, Hamilton Building, TCD.

Abstract. The "de Rham-Witt complex" of Deligne and Illusie is a functorial complex of sheaves $W\Omega^*(X)$ on a smooth algebraic variety X over a finite field computing the cristalline cohomology of X. I am going to present a non-commutative generalization of this: even for a non-commutative algebra $A$, one can define a functorial "Hochschild-Witt complex" with homology $WHH_*(A)$; if $A$ is commutative and smooth, then $WHH_i(A)=W\Omega^i(X)$, $X = Spec A$ (this is analogous to the isomorphism $HH_i(A)=\Omega^i(X)$ discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex.

Ezra Getzler (Northwestern University)
Quasi-invertible elements in a differential graded algebra
Thursday 6 February, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. In high school, we learn about the invertible matrices and the determinant. Later in our mathematical education, we might learn about the de Rham cohomology of the group GL(n) (which is closely related to Bott periodicity and K-theory). In this talk, I would like to introduce the quasi-invertible elements of a differential graded algebra A (elements f in A of degree 0 which have a quasi-inverse g, that is, fg-1 and gf-1 are coboundaries of elements of A of degree -1). I will explain what they form (not quite a group, but a higher generalization called a k-group), and what the correct generalization of the determinant and of the de Rham cohomology is in this setting. The main applications are presently in algebraic geometry, but in fact, some of the idea in this talk first came to light in the theory of supergravity.

Pierre Cartier (CNRS-IHES)
Random walks in Young diagrams
Thursday 16 January, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. The russian school around Vershik studied the limiting form of random Young diagrammes for large size. This can be reformulated using a suitable Markov process. The limit laws so discovered are not of the standard Gaussian type, but rather are connected with the limit laws in the so-called "free probability" of Voiculescu. We present here the combinatorial mathods developed by Ph. Biane, P. Sniatki and V. Feray to calculate the values of the irreducible characters of the symmetric groups in an new exact form leading easily to asymptotic estimates.

Ilya Kossovsky (Vienna)
Divergent equivalences of CR-manifolds and the geometry of singular differential equations
Thursday 14 November, 5.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. One of the remarkable differences between complex analysis in one and several complex variables is contained in the rigidity phenomenon for mappings between real submanifolds in multi-dimensional complex space. A good illustration of the rigidity is the fact that any formal invertible power series mapping between nondegenerate real hypersurfaces in complex space is always a holomorphic mapping. In this talk we establish a connection between real hypersurfaces with degeneracies and singular second order complex differential equations. We then demonstrate how the local holomorphic dynamics of the singular ODEs enables one to construct counterexamples to the above convergence phenomenon.

Chong-Kyu Han (Seoul)
Conservation laws for boundaries
Thursday 21 March 2013, 5.00 pm, Maxwell Lecture Theatre, Hamilton Building, TCD.

Abstract. Given a system of PDEs on a smoothly bounded domain we discuss the conditions that the system imposes on the boundary values of the solutions. In particular, we consider conservation laws for the system in the interior and on the boundary. We review the well known theory due to E. Noether on conservation laws for the variational problems and another approach due to Bryant-Griffiths from the viewpoint of the exterior differential systems. Then we discuss the conditions on the boundary values, extrinsic and intrinsic, and present some examples.

Bernd Kreussler (Limerik)
Means, Braids and Triangle Equations
Thursday 14 March 2013, 5.00 pm, Maxwell Lecture Theatre, Hamilton Building, TCD.

Abstract. The Triangle Equation, also known as the Yang-Baxter equation, was discovered around 1970 by Yang (in quantum field theory) and Baxter (in statistical mechanics). It now plays an important role for quantum groups and was used in the construction of invariants of knots. In this talk I shall start with two elementary topics: the arithmetic-geometric mean and braid groups. The arithmetic-geometric mean is closely related to elliptic integrals (Lagrange and Gauss). From there it is only one step to elliptic functions. Representations of braid groups will lead us to the Triangle Equation. Its classical limit is known as the classical Yang-Baxter equation. I shall explain how vector bundles on elliptic curves give rise to elliptic solutions of this equation. The degeneration of such solutions into trigonometric or rational solutions can be explained geometrically and is linked to the degeneration of an elliptic curve into a nodal or cuspidal rational curve.

Sergey Mozgovoi (TCD)
Topological invariants of the moduli spaces of stable bundles on ruled surfaces
Thursday 7 March 2013, 5.00 pm, Maxwell Lecture Theatre, Hamilton Building, TCD.

Abstract. Moduli spaces of stable vector bundles on complex surfaces is an old and intensively studied subject. By the Kobayashi-Hitchin correspondence these moduli spaces can be identified with moduli spaces of instantons on 4-manifolds. Their invariants play an important role in the S-duality conjecture by Vafa and Witten. However, even for P^2 and the ruled surfaces topological invariants of these moduli spaces are known only for rank one (Goettsche), rank two (Yoshioka), and in some cases for rank three (Manschot). In this talk I will present a recently proved formula for arbitrary rank bundles on a ruled surface. This result confirms the conjecture of Manschot for the Hirzebruch surfaces. If time permits, I will explain how this result together with the wall-crossing formula and the blow-up formula can be used to compute invariants on the projective plane.

Xiaojun Huang (Rutgers)
Local extension of holomorphic isometries of Kahler manifolds
Friday 22 February 2013, 4.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.

Abstract. An isometry in a euclidean plane is a composition of translation, rotation and reflection. Hence, it is always globally defined. What is the nature of this phenomenon for locally defined isometries in the holomorphic (complex variables) category? In this talk, I will discuss theorems, open problems along these lines.

Benjamin McKay (Cork)
Geometry on complex surfaces
Thursday 31 January 2013, 5.00 pm, Maxwell Lecture Theatre, Hamilton Building, TCD.

Abstract. You can picture Riemann surfaces by thinking about the uniformization theorem: they are all quotients of the Riemann sphere, the complex plane or the disk. You can picture many complex surfaces by similar ideas, as quotients of simple geometric objects. This leads to some new theorems and examples of complex analytic geometric structures on complex surfaces.

Vern Paulsen (Houston)
Friday 19 October 2012, 4.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.

Abstract. The Kadison-Singer problem was originally motivated by quantum mechanics, but it is now known to be equivalent to a problem about partitioning certain types of linearly dependent sets into "very" linearly independent sets. In this talk I will make these ideas precise, survey what is known about the Kadison-Singer problem, and present some of the interesting questions that lie at the boundaries of our knowledge.

Francine Meylan (Fribourg)
Around an old result on holomorphic extension of meromorphic maps
Friday 12 October 2012, 4.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.

Abstract. In this talk, we will discuss an old result due to S. Chiappari in 1991, that states the following. Any germ of a meromorphic map from C^n to C^{n'}, sending a real analytic real hypersurface M_{2n+1} into the sphere S_{2n'+1} in C^{n'} is holomorphic.

Teresa Monteiro Fernandes (Lisbon)
Grauert's theorem for real subanalytic sets
Thursday 4 October 2012, 5.00 pm, Synge Lecture Theatre, Hamilton Building, TCD.

Abstract. By open neighborhood of an open subset O of R^n we mean an open subset O' of C^n such that R^n \cap O'=O. A well known result of H. Grauert implies that any open subset of R^n admits a fundamental system of Stein open neighborhoods in C^n. Another way to state this property is to say that each open subset of R^n is Stein. In this talk, besides recalling Grauert's theorem, we shall make an overview of the notion and main results on subanalytic sets with the purpose of showing that a similar result in the subanalytic category is naturally true; so, under the assumption that O is a subanalytic open subset in a paracompact real analytic manifold, we show that O admits a fundamental system of subanalytic Stein open neighborhoods in any of its complexifications. We will refer some very useful applications of this new result. This is a joint work with Daniel Barlet.

Sung Yeon Kim (Kangwon, South Korea)
Compact CR manifolds with noncompact automorphism groups
Friday 1 June 2012, 4.00 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. CR manifolds with noncompact automorphism groups are first studied by Webster. He showed that a strictly pseudoconvex compact CR manifold with a one parameter family of CR automorphisms acting nonproperly is locally spherical. Later Burns, Shnider, Lee, Schoen generalized Webster's result. It is showed that a strictly pseudoconvex CR manifold with noncompact automorphism group is a sphere if compact and is a Heisenberg group if noncompact. In this talk, we study compact pseudoconvex CR manifolds of finite type with noncompact automorphism groups. We give a partial result for their infinitesimal CR automorphism groups.

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.

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.