Summer semester 2023

Wednesday, April 19th ∙ 11:15 hrs ∙ Coffee meeting See you at Görtz

Wednesday, April 26th ∙ 11:15 hrs
Kevin Wiegand (Heidelberg)
On the Origin of Minimal Sets by Means of Feral Curves

Wednesday, May 3rd ∙ 11:15 hrs
Lukas Hahn (Heidelberg)
Aspects of Extended Mirror Symmetry

Wednesday, May 10th ∙ 11:15 hrs
Alec Metsch (Universität Gießen)
Kiruma's Conjecture

Tuesday, May 23rd ∙ 11:15 hrs ∙ Room 00.200
note the unusual time and place!
Lina Deschamps (Nantes)
Existence of Birkhoff sections for geodesic flows on closed orientable Riemannian surfaces

Wednesday, May 24th ∙ 11:15 hrs
Connor Jackman (Heidelberg)
Bicycling curves and the filament hierarchy

A (planar) bicycling curve is the pair of curves traced by the wheels of a bicycle as it rides over the plane. In this talk, we will mention a few relations with bicycling curves to some other --seemingly unrelated-- types of curves: Kirchoff rods, rolling extremals, and orbits of the vortex filament flow. Then we will state/explain some conjectures about these relations, for example: a relation between the hierarchy of integrals associated to the vortex filament flow and the bicycling monodromy.

Wednesday, May 31st ∙ 11:15 hrs
Arthur Limoge (Heidelberg)
Local Floer homology(ies?)

Wednesday, June 14th ∙ 11:15 hrs
Marco Mazzucchelli (École normale supérieure de Lyon)
On the local maximizers of higher capacity ratios

A conjecture due to Viterbo asserts that, on the space of convex bodies of volume one in a symplectic vector space, any symplectic capacity achieves its maximum precisely on the symplectomorphic images of the round ball. A theorem of Abbondandolo and Benedetti, extending earlier work of several authors in the special case of dimension 3, establishes the local version of the Viterbo conjecture: the round balls are local maximizers of any symplectic capacity over the space of smooth convex bodies of volume 1 endowed with the C3 topology. In this talk, I will present a partial extension of this result for the higher Ekeland-Hofer capacities: on the space of 4-dimensional smooth star-shaped domains of volume one, endowed with the C3 topology, the local maximizers of the k-th Ekeland-Hofer capacities are those domains symplectomorphic to suitable rational ellipsoids. The talk is based on joint works with Abbondandolo-Lange, and with Baracco-Bernardi-Lange.

Wednesday, June 21st ∙ 11:15 hrs
Bernhard Albach (RWTH Aachen University)
Polynomial growth of Reeb orbits

Wednesday, June 28th ∙ 11:15 hrs
Michael Rothgang (Humboldt-Universität zu Berlin)
Equivariant transversality for simple holomorphic curves

We study closed holomorphic curves in symplectic G-manifolds, with respect to a G-equivariant almost complex structure. We should not expect the moduli space of such curves to be a manifold (after all, transversality and symmetry are famously incompatible). However, we can hope for a clean intersection condition: the moduli space decomposes into disjoint strata which are smooth manifolds; the dimensions of the strata are explicitly computable. I'll explain this decomposition for simple curves and outline the proof that each stratum is smooth. This is the first step towards a well-behaved theory of equivariant holomorphic curves.

Wednesday, July 5th ∙ 11:15 hrs
Lael Costa (Penn State)
Billiard maps and visualizations

Wednesday, July 12th ∙ 11:15 hrs
Yannis Bähni (Augsburg University)
First Steps in Twisted Rabinowitz-Floer Homology

Rabinowitz-Floer homology is the Morse-Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In our work, we consider a generalisation of this theory to a Rabinowitz-Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces. In particular, our theory applies to lens spaces.

Wednesday, July 19th ∙ 11:15 hrs
Anastasiia Sharipova (Penn State)
Convex Bodies with all Planar Characteristics

I will show that in symplectic space smooth strongly convex bodies with all characteristics planar or all outer billiard trajectories planar are affine symplectic images of a ball.

Wednesday, July 26th ∙ 11:15 hrs
Levin Maier (Heidelberg)
On Manes critical value for the Hunter Saxton system

Wednesday, August 2nd ∙ 11:15 hrs
Otto van Koert (Seoul National University)
Computational symplectic topology and RTBP

Winter semester 2022/23

Wednesday, November 16th ∙ 11:15 hrs ∙ Jakub Tomaszewski (Poznań University of Technology) Dendrite maps are not that phenomenal

It has been a general belief that entropy relates to the complexity of the behavior of a dynamical system. However, it has been shown in 2009 by Harańczyk and Kwietniak that exact maps on an interval can attain lower entropy than pure mixing maps. This phenomenon has been demonstrated to hold also for graph maps by Harańczyk, Kwietniak and Oprocha in 2014. We studied an analogous question whether the same assertion holds for dendrites, and specifically, for the Gehman dendrite. It is known by works of Špitalský that for the Gehman dendrite the infimum of entropies of exact maps is zero and is not attainable. We will present the construction of a family of pure mixing maps on the Gehman dendrite containing maps of arbitrarily small but positive entropy. This shows that the phenomenon "pure mixing forces higher entropy than exactness" no longer holds for dendrites.

Thursday, November 24th ∙ 9:15 hrs ∙ SR2 ∙ note the unusual time and place! ∙ Urs Fuchs (Monash University) TBA

Wednesday, November 30th ∙ 11:15 hrs ∙ Tilman Becker (Cologne) Geodesic and conformally Reeb vector fields on flat 3-manifolds

A vector field of unit length on a Riemannian manifold is called geodesic if all of its integral curves are geodesics. In this talk, I will start by giving an overview of some relevant recent results on geodesic vector fields on space forms and related contact structures. Then, I will show that geodesic vector fields on flat 3-manifolds not equal to E3 are of a simple "1-parametric" type. Using this result, I will derive a (necessary and sufficient) criterion for such a vector field to be the Reeb vector field of a contact form (up to rescaling).

Wednesday, December 7th ∙ 11:15 hrs ∙ Alessio Pellegrini (ETH Zürich) A Bangert-Hingston Theorem for Starshaped Hypersurfaces

In the first part of the talk we will discuss some aspects of a celebrated theorem due to Bangert and Hingston which says the following: on any closed manifold Q, which is not a circle and has fundamental group Z, there exist prime-many geometrically distinct closed geodesics. In the second part we will explain how Bangert and Hingston's theorem can be restated in terms of Hamiltonian dynamics on S*Q and discuss the natural generalization from geodesics to Reeb orbits. Under an additional circle action assumption and the use of Floer theory, we proceed to give a proof of a Bangert and Hingston type result for closed Reeb orbits on non-degenerate starshaped hypersurfaces inside T*Q.

Summer semester 2022

Wednesday, June 22th ∙ 11:15 hrs ∙ Francois Gay-Balmaz (Paris) Coadjoint orbits relevant to ideal fluid dynamics

We explore coadjoint orbits of the group of volume preserving diffeomorphisms that are modeled on nonlinear Grassmannians of submanifolds of codimension one. These coadjoint orbits are associated to classes of singular solutions of the Euler equation of an ideal fluid. We focus on vortex loops in 2D (collections of point vortices) and on vortex sheets in 3D (collections of vortex filaments). We also determine a Feynman-Onsager condition for their prequantization. This is a joint work with Cornelia Vizman.

Thursday, May 12th ∙ 12:15 hrs in room 0.200 ∙ Reto Kaufmann (Zürich) From Symplectic Toric Manifolds to Polytopes and back again

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.

Winter semester 2021/22

Wednesday, November 10th ∙ 15:15 hrs in Zoom ∙ Hassan Najafi Alishah (Minas Gerais) Dirac geometry and conservative Lotka-Volterra equations

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.

Wednesday, December 1st ∙ 11:15 hrs ∙ Samanyu Sanjay (Aachen) Some symplectic obstructions to Weinstein fillability in high dimensions

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.

Wednesday, December 8th ∙ 11:15 hrs in Zoom ∙ Comlan E. Koudjinan (IST Austria) On non coexistence of 2 & 3-rational caustics in nearly circular billiard tables

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.

This is based on a joint work with V. Kaloshin.

Wednesday, January 12th ∙ 11:15 hrs in Zoom ∙ Thibaut Mazuir (Sorbonne) Higher algebra of A-infinity algebras in Morse theory

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.

Wednesday, January 19th ∙ 11:15 hrs in Zoom ∙ Valdo Tatitscheff (Strasbourg) Hecke algebras and two-dimensional topological quantum field theories

In order to generalize Thurston's laminations to a split real G-higher Teichmuller 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 H₃, E₆ and E₇, a fact that we now understand from the representation theory of the corresponding Hecke algebra.

Wednesday, February 16th ∙ 11:15 hrs ∙ Matthias Meiwes TBA

Summer semester 2021

Wednesday, July 21st ∙ 11:15 hrs ∙ Johann Bouilly (Strassbourg) Torelli group action on SU(n)-character varieties

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.

Wednesday, July 14th ∙ 11:15 hrs ∙ Anastasiia Sharipova (Penn State) Viterbo's conjecture for the Ekeland-Hofer-Zehnder capacity

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.

Winter semester 2020/21

Wednesday, November 11th ∙ 11:15 hrs ∙ Valerio Assenza (Heidelberg) Green Bundle for Tonelli Systems, part 1

Wednesday, November 18th ∙ 11:15 hrs ∙ Ipsita Datta (Stanford) Obstructions to the existence of Lagrangians in \R^4

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.

Wednesday, November 25th ∙ 11:15 hrs ∙ Valerio Assenza (Heidelberg) Green Bundle for Tonelli Systems, part 2

Wednesday, December 2nd ∙ 11:15 hrs ∙ Gabriele Benedetti, Davide Legacci (Heidelberg) Zero-sum Evolutionary Games and Convex Hamiltonian systems, part 1

Wednesday, December 09th ∙ 11:15 hrs ∙ Davide Legacci (Heidelberg) Zero-sum Evolutionary Games and Convex Hamiltonian systems, part 2

tba

Wednesday December 16th ∙ 11:15 hrs ∙ Lucas Dahinden (Heidelberg) Unexpected but robust topological entropy

tba

Wednesday, January 20th ∙ 11:15 hrs ∙ Martin Schwald (Essen) On the definition of irreducible holomorphic symplectic manifolds and their singular analogs

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 Kahler manifolds with trivial canonical bundle as well as representation theory of finite groups to analyze quotients of complex tori.

Wednesday, February 10th ∙ 11:15 hrs ∙ Davide Barilari (Padova) The Brunn Minkovski inequality in sub-Riemannian geometry

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.

Summer semester 2020

Wednesday, May 27th ∙ 11:15 hrs ∙ Eugenio Pozzoli (Sorbonne) Controllability of molecular dynamics

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.

Winter semester 2019/20

Wednesday, February 5th ∙ 11:15 hrs ∙ Ana Cannas da Silva (Zürich) Toric lagrangians

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.

Monday, February 2nd ∙ 09:15 hrs ∙ SR1 ∙ Zhengyi Zhou (Princeton) Symplectic fillings of asymptotically dynamically convex manifolds

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.

Wednesday, January 29th ∙ 11:15 hrs ∙ San Vũ Ngọc (Rennes) Symplectic geometry for semiclassical magnetic laplacians

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.

Wednesday, January 15th ∙ 11:15 hrs ∙ Chris Wendl (Berlin) TBA

Wednesday, January 8th ∙ 11:15 hrs ∙ Sara Tukachinsky (Princeton) Quantum product on relative cohomology

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.

Wednesday, December 17th ∙ 11:15 hrs ∙ Jean Gutt (Toulouse) Symplectic homology is a Morse theory

In the context of star-shaped domains in R²ⁿ, Abbondandolo & Majer have defined a Morse theory for the action functional. I shall build a chain complex isomorphism from this Morse complex to the Floer complex which commutes with continuations. This is joint work with Vinicius Ramos.

Monday, December 15th ∙ 11:15 hrs ∙ SR9 ∙ Johanna Bimmermann (Heidelberg) Hofer-Zehnder Capacity for Magnetic Systems on the Two-Sphere

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.

Wednesday, December 4th ∙ 11:15 hrs ∙ Marc Kegel (Berlin) Non-isotopic transverse tori in Engel manifold

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

Monday, December 2nd ∙ 09:15 hrs ∙ SR1 ∙ Joontae Kim (KIAS) Symplectic topology of real Lagrangian tori in S²xS²

We explore the topology of real Lagrangian submanifolds in S²xS² towards their classification. We explain why it is interesting to study real Lagrangian tori in S²xS², and show that the Chekanov torus is not real. This exhibits a genuine real symplectic phenomenon in terms of involutions.

Wednesday, November 27th ∙ 11:15 hrs ∙ Anna-Maria Vocke Electro-magnetic periodic bounce orbits

Wednesday, November 6th ∙ 11:15 ∙ Lara Suarez (Bochum) The fundamental group of monotone Lagrangian cobordisms of big Maslov class

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.

Wednesday, October 30th ∙ 11:15 hrs ∙ Jakob Hedicke (Bochum) Causally simple spacetimes and the contact structure on the space of null-geodesics

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.

Wednesday, October 16th ∙ 11:15 hrs ∙ Kevin Wiegand (Heidelberg) Magnetic cotangent bundles

Summer semester 2019

Wednesday, July 24th ∙ 11:15 hrs ∙ Sergei Tabachnikov (PennState) Cross-ratio dynamics on ideal polygons

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.

Wednesday, July 17th ∙ 16:00 hrs in SRA ∙ Alvaro del Pino (Utrecht) Submanifolds of jet spaces and wrinkling

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. Ãlvarez-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.

Wednesday, July 17th ∙ 11:15 hrs ∙ Sergei Tabachnikov (PennState) Frieze Patterns

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.

Friday, July 12th ∙ 15:30 hrs ∙ Room INF205 ∙ Christian Seidel (Bochum) Closed geodesics in compact Lorentzian manifolds of splitting type

Let M=M₀Ã—S¹ be a compact Lorentzian manifold of splitting type, equipped with a splitting metric. Provided that the fundamental group of M₀ is finite, a result by Candela, Giannoni and Masiello ensures the existence of infinitely many closed timelike geodesics. I will present a quick view on their proof and discuss the case of an infinite fundamental group using some results by Galloway.

Wednesday, July 10th ∙ 11:15 hrs ∙ Sergei Tabachnikov (PennState) Elementary geometry is dead. Long live elementary geometry! (Part II)

Cf. Part I

Wednesday, July 3rd ∙ 11:15 hrs ∙ Sergei Tabachnikov (PennState) Elementary geometry is dead. Long live elementary geometry! (Part I)

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 Ivoryâ€™s lemma, Poncelet grid, a theorem of Kasner and its generalizations, projective configuration theorems, and a variation on Steinerâ€™s porism. I shall also describe four equivalent properties of completely integrable billiards.

Wednesday, May 22nd ∙ 11:00 hrs ∙ Markus Reineke (Bochum) Cohomological Hall Algebras

TBA.

Wednesday, May 29th ∙ 11:00 hrs ∙ Pazit Haim-Kislev (Tel Aviv) Closed characteristics on the boundary of convex polytopes

We introduce a simplification to the problem of finding a closed characteristic with minimal action on the boundary of a convex polytope in R²ⁿ, which yields a combinatorial formula for the EHZ capacity. As an application, we show that the EHZ capacity of a general convex body is sub-additive with respect to hyperplane cuts, and we bound the systolic ratio for simplices in R⁴.

Winter semester 2018/19

Wednesday, February 6th ∙ Silvia Sabatini (Köln) Hamiltonian S¹-spaces with large minimal Chern number

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 S¹-space. In [S] it is proved that the minimal Chern number N is bounded above by n+1, bound which is expected for all positive monotone compact symplectic manifolds. Assuming that the Hamiltonian S¹-space is monotone (i.e. the first Chern class is a multiple of the class of the symplectic form) in [GHS] several bounds on the Betti numbers are proved, these bounds depending on N. I will first discuss the ideas behind the proofs of the aforementioned facts, and then concentrate on N=n+1. In this case my student Isabelle Charton [C] proved that the manifold must be homotopically equivalent to a complex projective space of dimension 2n.

[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).

Wednesday, December 12th ∙ Matthias Meiwes (Tel Aviv) Wrapped Floer homology and surgery

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.

Wednesday, November 21st ∙ Kevin Wiegand (Gießen) Odd-symplectic surgery

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.