Heidelberg geometry seminar main page

Thu May 12th ∙ 12:15 in room 0.200 ∙

Symplectic toric manifolds are in correspondence with unimodular polytopes. Before stating the Theorem due to Delzant, we will review the essential concepts of both worlds and see examples. In the second part of the talk, we then look at constructions on either side - faces of polytopes, products, symplectic reduction and symplectic cutting - and see to what it translates on the other side of the correspondence.

Wed November 10th ∙ 15:15 in Zoom ∙

In this talk, I, first, will provide a short introduction on Lotka-Volterra and replicator equations followed by some preliminary definitions and observations on Dirac/big-isotropic structures and Hamiltonian inverse problem. Then I will present an algorithm to solve the Hamiltonian inverse problem for a given Lotka-Volterra equation. This algorithm generalizes the well-known use of gauge transformations to skew-symmetrize the interaction matrix of a LV system which leads to a Hamiltonian description of LV systems. In the case of the predator-prey model, our method does allow interaction between different predators and between different preys.

Wed December 1st ∙ 11:15 ∙

I will present a construction of symplectic cohomology due to Zhengyi Zhou and following Zhou I will use this construction to derive some obstructions to Weinstein fillability of ADC contact manifolds. Then, I will also present a construction due to Zhou of ADC contact manifolds of dimensions 4k+1 (for K = 3) for which the above mentioned obstructions to Weinstein fillability does not vanish but the bordism theoretic obstruction to Weinstein fillability found by Bowden-Crowley-Stipsicz does vanish.

Wed December 8st ∙ 11:15 in Zoom ∙

A famous Birkhoff conjecture states that the only integrable convex planar billiards are billiards in an ellipse. We examined two closely related rigidity questions. A rational caustic is a caustic associated to a family of periodic orbits of the same period and the same rotation number. For example, a convex domain with a rational caustic of period two is a domain of a constant width. We investigated a question proposed by Tabachnikov: are there nearly circular domains other than discs with two rational caustics of a prime period p and q? In this talk, I will discuss our following two new results:

- (rigidity) There are no nearly circular domains with two coexisting rational caustics of period two and three.
- (no super-rigidity) There may be infinitely many deformations of the circular domains with two coexisting rational caustics of period three and five with error given by the 15th power of the perturbation parameter.

Wed January 12th ∙ 11:15 in Zoom ∙

In this talk, I will introduce the notion of n-morphisms between two A-infinity algebras. These higher morphisms are such that 0-morphisms correspond to standard A-infinity morphisms and 1-morphisms correspond to A-infinity homotopies. The set of higher morphisms between two A-infinity-algebras then defines a simplicial set which has the property of being a Kan complex. The combinatorics of n-morphisms are moreover encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra. Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A-infinity algebras naturally arises in the context of Morse theory.

Wed January 19th ∙ 11:15 in Zoom ∙

In order to generalize Thurston's laminations to a split real G-higher Teichmüller space, one is naturally led to make use of the (spherical) affine Hecke algebra of G. If as a toy model one instead considers the Hecke algebra of a finite Coxeter system or more generally a symmetric algebra of finite rank, the same construction yields an open-closed 2d topological quantum field theory (TQFT). It associates an element of the base ring to any punctured surface - for example, a Laurent polynomial with integer coefficients when the symmetric algebra is a Hecke algebra. These Laurent polynomials have only positive coefficients when the Coxeter system is of classical type or exceptional type H

Wed February 16th ∙ 11:15 ∙

TBA

Wed July 21 ∙ 11:15 ∙

Let n>1, G=SU(n) and S be a closed surface. The mapping class group of S acts on the G-character variety of S and this action is known to be ergodic. This action induces an action of the Torelli subgroup of S. In this talk, we will introduce these objects and the tools we need to understand why the Torelli group acts ergodically on the G-character varieties. If time allows us, we will explain how this proof can be extended to to G-character varieties for every semi-simple and compact Lie group G.

Wed July 14 ∙ 11:15 ∙

Symplectic capacities are symplectic invariants that can be very useful in the problem of symplectic embeddings. Viterbo's conjecture is the inequality relating volumes of the convex bodies with their symplectic capacities. We consider Viterbo's conjecture for the Ekeland-Hofer-Zehnder capacity which is equal to minimal actions of closed characteristics on the boundaries of convex bodies and some special cases of the bodies for which the conjecture holds.

Wed Nov 11 ∙ 11:15 ∙

Wed Nov 18 ∙ 11:15 ∙

We present some obstructions to the existence of Lagrangian surfaces in \R^4 which can be viewed as cobordisms between links. The obstructions arise from considering moduli spaces of holomorphic disks with boundary on the Lagrangian. We present examples of pairs of knots that cannot be Lagrangian cobordant and knots which cannot bound Lagrangian disks.

Wed Nov 25 ∙ 11:15 ∙

tba

Wed Dec 02 ∙ 11:15 ∙

tba

Wed Dec 09 ∙ 11:15 ∙

tba

Wed Dec 16 ∙ 11:15 ∙

tba

Wed Jan 20 ∙ 11:15 ∙

We show that in the definition of IHSM being simply connected can be replaced by vanishing irregularity. This fits also well with the theory of singular symplectic varieties. The proof uses the decomposition theorem for compact Kähler manifolds with trivial canonical bundle as well as representation theory of finite groups to analyze quotients of complex tori.

Wed Feb 10 ∙ 11:15 ∙

The classical Brunn-Minkovski inequality in the Euclidean space generalizes to Riemannian manifolds with Ricci curvature bounded from below. Indeed this inequality can be used to define the notion of "Ricci curvature bounded from below" for more general metric spaces. A class of spaces which do not satisfy this more general definition is the one of sub-Riemannian manifolds: these can be seen as a limit of Riemannian manifolds having Ricci curvature that is unbounded, whose prototype is the Heisenberg group. In the first part of the talk I will discuss about the validity of a Brunn-Minkovski type inequality in the SR setting. The second part concerns a notion of sub-Riemannian Bakry-Émery curvature and the corresponding comparison theorems for distortion coefficients. The model spaces for comparison are variational problems coming from optimal control theory.

Wed May 27 ∙ 11:15 ∙

The control of quantum systems has several applications in physics, chemistry and engineering, such as spectroscopy and, more recently, quantum sensing and quantum information theory. In this talk we will start by introducing the controllability problem on Lie groups, and analysing in detail the case of SU(n). We will then move to the control of infinite-dimensional systems, and describe a sufficient condition to obtain approximate controllability. Finally, we will apply these techniques to the control of a rotating symmetric molecule.

Wed Feb 5 ∙ 11:15 ∙

Toric manifolds have gained prominence as testing grounds for new theories. Moreover, they have a strong mathematical appeal, inasmuch as they enjoy a slew of properties and they follow a rigid yet rich classification. From the symplectic viewpoint, they are classified by some polytopes (a theorem of Delzant's) and can be handled with action-angle coordinates (an idea rooted in work of Archimedes). After an overview of symplectic toric manifolds, this talk will explain how the language of polytopes and action-angle coordinates allows us to understand lagrangian submanifolds well adjusted to the toric structure.

Mon Feb 2 ∙ 09:15 ∙ SR1 ∙

I will introduce the concept of k-dilation on symplectic cohomology, which generalizes the vanishing of symplectic cohomology and symplectic dilation. The existence of k-dilation is independent of certain fillings for a class of contact manifolds admitting only the trivial augmentation, called asymptotically dynamically convex manifolds. Then I will derive some consequences on uniqueness and existence of fillings, embeddings, and cobordisms.

Wed Jan 29 ∙ 11:15 ∙

In the last few years, the use of symplectic normal forms has lead to very precise spectral asymptotics of the magnetic Laplacian, in the semiclassical limit. I will explain the general idea, which is based on quantizing these normal forms and on performing a quantum Birkhoff procedure, and report on recent advances in the subject.

Wed Jan 15 ∙ 11:15 ∙

Wed Jan 8 ∙ 11:15 ∙

The quantum product on the cohomology of a symplectic manifold X is a deformation of the cup product, or wedge product in the de Rham model. The deformation is given by adding contributions from pseudoholomorphic spheres. Adding a Lagrangian submanifold L, one might consider the relative cohomology H^*(X,L). In a joint work with Jake Solomon, we define a quantum product on H^*(X,L) that combines deformations of the wedge products of differential forms on X and L, with corrections coming from pseudoholomorphic spheres as well as disks with boundary conditions in L. The associativity of this product is equivalent to the open WDVV equations, a PDE in the generating functions of the closed and open Gromov-Witten invariants.

Wed Dec 17 ∙ 11:15 ∙

In the context of star-shaped domains in R

Mon Dec 15 ∙ 11:15 ∙ SR9 ∙

In this talk I will present the computation of the Hofer-Zehnder capacity for magnetic systems on the two-sphere with constant magnetic field. While finding a lower bound for the Hofer-Zehnder capacity is relatively easy, as any admissible Hamiltonian function provides one, finding an upper bound is much harder. By a theorem of G. Lu for closed symplectic manifolds (M,?) an upper bound is given by the symplectic area ?(A) of a homology class A ? H_2(M) that has a non-vanishing Gromov-Witten invariant. Our strategy is therefore, to find an embedding of the magnetic system into a closed symplectic manifold. We will then use the theorem to find an upper bound and explicitly construct an admissible Hamiltonian to find a lower bound of the Hofer-Zehnder capacity.

Wed Dec 4 ∙ 11:15 ∙

After a brief introduction to Engel structures, we will construct infinitely many non-isotopic transverse tori in Engel manifolds that are all smoothly isotopic.

Mon Dec 2, SR1 ∙ 09:15 ∙ SR1 ∙

We explore the topology of real Lagrangian submanifolds in S

Wed Nov 27 ∙ 11:15 ∙

Wed Nov 6 ∙ 11:15 ∙

I will show Barraud's contruction of the fundamental group of a monotone Lagrangian with big minimal Maslov number. This construction uses holomorphic curves. From the construction we will derive restrictions on the fundamental group of a Lagrangian cobordism. This talk is based on joint work with Jean-Francois Barraud.

Wed Oct 30 ∙ 11:15 ∙

Null-geodesics are used in general relativity to describe the motion of light in a Lorentzian spacetime. Inspired by Penrose's work on twistor theory, Low showed that the space of all null-geodesics, provided it is a smooth manifold, carries a natural contact structure. Moreover he observed that in the case of a globally hyperbolic spacetime the space of null-geodesics is always contactomorphic to a standard co-sphere bundle. After a short introduction to Lorentzian geometry I will show how this result can be generalised to certain non-globally hyperbolic spacetimes using Giroux's theory of convex surfaces and results from Chekanov, van Koert and Schlenk.

Wed Oct 16 ∙ 11:15 ∙

Wed July 24 ∙ 11:15 ∙

Define a relation between labeled ideal polygons in the hyperbolic 3-space by requiring that the complex distances (a combination of the distance and the angle) between their respective sides equal a constant c; the complex number c is a parameter of the relation. This defines a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss complete integrability of this family of maps and related topics, including a continuous version of this relation.

Wed July 17 ∙ 16:00 In SRA ∙

Wrinkling (and, relatedly, surgery of singularities) is one of the main tools in the theory of h-principles. In Contact Topology, wrinkling is best known for its role in the study of legendrians: E. Murphy's h-principle for loose legendrians and D. Álvarez-Gavela's simplification of front singularities both rely on it. Just like the space of 1-jets of functions is endowed with a canonical contact structure, higher jet spaces are endowed with other canonical distributions. I will report on joint work with L. Toussaint in which we use wrinkling to construct submanifolds of jet spaces tangent to these canonical distributions (generalising thus the study of legendrians). I will pay particular attention to explaining how the contact case differs from the general setup.

Wed July 17 ∙ 11:15 ∙

Frieze patterns are interesting combinatorial objects introduced by Coxeter. Recently they have attracted much attention due to their relation with the theory of cluster algebras. I shall introduce frieze patterns and prove the theorem of Conway and Coxeter that relates arithmetical frieze patterns with triangulations of polygons. There is an intimate, and somewhat unexpected, relation between three object: frieze patterns, 2nd order linear difference equations, and polygons in the projective line. I shall describe some recent work on frieze patterns, including an interpretation of frieze patterns as a discretization of a coadjoint orbit of the Virasoro algebra.

Fri July 12 ∙ 15:30 ∙ Room INF205 ∙

Let M=M

Wed July 10 ∙ 11:15 ∙

Cf. Part I

Wed July 3 ∙ 11:15 ∙

By “elementary”, I do not mean Euclidean axiomatic high school geometry, nor do I mean that the results that I will discuss are expected or easy to obtain. I use this term to distinguish my topic from differential geometry. I shall present a sampler of recent results that, in most cases, were discovered as a result in computer experiments and were motivated by the theory of completely integrable systems. The topics include the circumcenter of mass of polygons, the locus of centroids of Poncelet polygons, billiards in ellipses and Ivory’s lemma, Poncelet grid, a theorem of Kasner and its generalizations, projective configuration theorems, and a variation on Steiner’s porism. I shall also describe four equivalent properties of completely integrable billiards.

Wed May 22 ∙ 11:00 ∙

TBA.

Wed May 29 ∙ 11:00 ∙

We introduce a simplification to the problem of finding a closed characteristic with minimal action on the boundary of a convex polytope in R

Wed Feb 6 ∙

Consider a compact symplectic manifold of dimension 2n which is acted on by a circle in a Hamiltonian way with isolated fixed points; we refer to it as a Hamiltonian S

[C] Charton, "Hamiltonian manifolds with high index". Master thesis. University of Cologne, 2017.

[GHS] Godinho, von Heymann, Sabatini "12, 24 and beyond ", Advances in Mathematics, 319 (2017), 472 - 521.

[S] Sabatini "On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action", Communications in Contemporary Mathematics, 19, No. 04 (2017).

Wed Dec 12 ∙

Adapting a construction of Viterbo, Abouzaid and Seidel defined a map V between the wrapped Floer homologies of two pairs (M,L)⊂(M',L') where M,M' are Liouville domains and L,L' suitable exact Lagrangians in M and M'. V is an isomorphims if M' is obtained by attaching a subcritical handle on M or a handle on a Legendrian that is loose in the complement of the boundary of L in M. I will talk about the construction of V and some consequences.

Wed Nov 21 ∙

As discovered by Hofer the moduli space of holomorphic discs is related to the question of periodic Reeb orbits. A surgery construction leads to a cobordism theory with amazing properties. Holomorphic discs in the upper boundary yield to statements about non-dense characteristics in the lower boundary.

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