Department of

Mathematics


Seminar Calendar
for Symplectic & Poisson Geometry Seminar events the year of Thursday, September 12, 2013.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2013           September 2013          October 2013    
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Monday, August 26, 2013

10:00 am in 145 Altgeld Hall,Monday, August 26, 2013

de Rham Complexes on Orbit Spaces and Symplectic Quotients

Jordan Watts   [email] (UIUC Math)

Abstract: Let G be a Lie group acting on a manifold M. If the action is proper and free, then M/G is a manifold which admits a de Rham complex isomorphic to the subcomplex of basic forms on M. We will introduce the notion of a diffeology in order to extend this result to all proper actions. Time permitting, we will then compare this definition to a de Rham complex on a symplectic quotient as defined by Sjamaar.

Monday, September 9, 2013

10:00 am in 145 Altgeld Hall,Monday, September 9, 2013

Some interactions between classical, semiclassical, and random symplectic geometry

Alvaro Pelayo (Washington University Math)

Abstract: I will describe some recent results about classical and quantum integrable systems, emphasizing the interplay between symplectic geometry and semiclassical analysis. I will also briefly describe some random counterparts of classical results in symplectic geometry.

Monday, September 16, 2013

10:00 am in 145 Altgeld Hall,Monday, September 16, 2013

Topological Hamiltonian and contact dynamics, part I: an introduction

Stefan Mueller (UIUC Math)

Abstract: In classical mechanics, the dynamics of a Hamiltonian vector field models the motion of particles in phase space, and the dynamics of a contact vector field play a similar role in geometric optics (in the mathematical model of Huygens' principle). Topological Hamiltonian dynamics and topological contact dynamics are relatively recent theories that explore natural questions regarding the regularity of such dynamical systems (on an arbitrary symplectic or contact manifold). In a nutshell, Hamiltonian and contact dynamics admit genuine generalizations to non-smooth dynamical systems with non-smooth generating (contact) Hamiltonian functions. The talk begins with examples that illustrate the central ideas and lead naturally to the key definitions. The main technical ingredient is the well-known energy-capacity inequality for displaceable subsets of a symplectic manifold. We use it to prove an extension of the classical 1-1 correspondence between isotopies and their generating Hamiltonians. This crucial result turns out to be equivalent to certain rigidity phenomena for smooth Hamiltonian and contact dynamical systems. We then look at some of the foundational results of the new theories. The end of the talk touches upon sample applications to topological dynamics and to Riemannian geometry, which will be explored further in a second talk.

Monday, September 23, 2013

10:00 am in 145 Altgeld Hall,Monday, September 23, 2013

Hamiltonian and contact dynamics, part II: applications

Stefan Mueller (UIUC Math)

Abstract: After recalling the precise definition of a topological Hamiltonian dynamical system, I will sketch the proof of the 1-1 correspondence between topological Hamiltonian isotopies and topological Hamiltonian functions. I also show that this result has non-empty content by constructing a non-smooth topological Hamiltonian dynamical system (with support in a Darboux chart). We then shift gears and focus on two sample applications to 1) hydrodynamics (topological character of the helicity invariant, which measures the average asymptotic linking number of the flow lines of a divergence-free vector field) and to 2) Riemannian geometry (C^0-rigidity of the geodesic flows associated to a sequence of weakly uniformly converging Riemannian metrics).

Monday, September 30, 2013

10:00 am in 145 Altgeld Hall,Monday, September 30, 2013

Positive loops and orderability in contact geometry

Peter Weigel (Purdue University Math)

Abstract: Orderability of contact manifolds is related in some non-obvious ways to the topology of a contact manifold V. We know, for instance, that if V admits a 2-subcritical Stein filling, it must be non-orderable. By way of contrast, in this talk I will discuss ways of modifying Liouville structures for high-dimensional V so that the result is always orderable. The main technical tool is a Morse-Bott Floer theoretic growth rate, which has some parallels with Givental's nonlinear Maslov index. I will also discuss a generalization to the relative case, and applications to bi-invariant metrics on Cont(V).

Monday, October 7, 2013

10:00 am in 145 Altgeld Hall,Monday, October 7, 2013

Lie Algebroid Spray

Songhao Li (Washington University Math)

Abstract: Analogous to the spray in Riemannian geometry, we introduce the Lie algebroid spray, or A-spray. A special case is the Poisson spray as introduced by Crainic and Marcut. As an application, we show that the source-simply-connected symplectic groupoid of a log symplectic surface is diffeomorphic to the cotangent bundle in such a way that the source map coincide with the bundle projection. (Joint work in progress with Marco Gualtieri)

Monday, October 14, 2013

10:00 am in 145 Altgeld Hall,Monday, October 14, 2013

A generalization of the group of Hamiltonian homeomorphisms

Augustin Banyaga (Pennsylvania State University Math)

Abstract: The Eliashberg-Gromov rigidity theorem implies that Symplectic Geometry underlines a topology. This talk is about the automorphism groups of this "continuous" symplectic topology. The group of symplectic homeomorphisms (Sympeo) has a remarkable subgroup: the group of Hamiltonian homeomorphisms (Hameo), defined by Oh and Müller using the $L^{(1q,\infty)}$ Hofer norm. We introduce a generalization of Hameo, called the group of strong symplectic homeomorphisms (SSympeo), using a generalization of the Hofer norm from the group of Hamiltonian diffeomorphisms to the whole group of symplectic diffeomorphisms. Each group Hameo and SSympeo has also a $L^\infty$ version. The two versions coincide (Müller, Banyaga-Tchuiaga).

Monday, October 21, 2013

10:00 am in 145 Altgeld Hall,Monday, October 21, 2013

On the Topological Dynamics Arising from a Contact Form

Peter Spaeth (Pennsylvania State University Math)

Abstract: Stefan Müller and Yong-Geun Oh introduced the Hamiltonian metric on the group of Hamiltonian isotopies of a symplectic manifold, and with it defined the groups of topological Hamiltonian isotopies and homeomorphisms. With Augustin Banyaga we introduced the contact metric on the space of strictly contact isotopies of a contact manifold, and defined the groups of topological strictly contact isotopies and homeomorphisms in a similar manner. In the talk I will explain how the one to one correspondence between smooth strictly contact isotopies and generating contact Hamiltonian functions extends to their topological counterparts when the contact form is regular. I will also prove that the group of diffeomorphisms that preserve a contact form is rigid in the sense of Eliashberg-Gromov. This last result is joint with Müller.

Monday, October 28, 2013

10:00 am in 145 Altgeld Hall,Monday, October 28, 2013

All boundaries of contact type can keep secrets

Ely Kerman (UIUC Math)

Abstract: Let $(M, \omega)$ be a symplectic manifold with nonempty boundary, $W$. The restriction of $\omega$ to $W$, $\omega_W$, has a one dimensional kernel which defines the characteristic foliation of $W$. If $W$ is a boundary of contact type then it admits a tubular neighborhood comprised of hypersurfaces whose characteristic foliations are all conjugate to those of $W$. Since these hypersurfaces lie in the interior one might guess (or hope) that the interior of $(M, \omega)$ determines $omega_W$ or at least some of its symplectic invariants. Several questions in this direction were raised by Eliashberg and Hofer in the early nineties. In this talk I will describe the resolution of some of these questions. I will prove that neither $\omega_W$ or its action spectrum is determined by the interior of $(M, \omega)$. This involves the construction of a new dynamical symplectic plug. The construction uses only soft techniques (Moser's method) and so should hopefully be accessible to all.

Monday, November 4, 2013

10:00 am in 145 Altgeld Hall,Monday, November 4, 2013

Semi-toric systems as Hamiltonian S^1-spaces

Daniele Sepe (Utrecht University Math)

Abstract: The classification of completely integrable Hamiltonian systems on symplectic manifolds is a driving question in the study of Hamiltonian mechanics and symplectic geometry. From a symplectic perspective, such systems correspond to Hamiltonian R^n-actions which are locally toric. The class of integrable Hamiltonian systems on 4-dimensional symplectic manifolds corresponding to Hamiltonian S^1 x R actions (with some extra assumptions on the singularities) is known as semi-toric: it was introduced by Vu Ngoc, and Pelayo and Vu Ngoc obtained a classification for `generic' semi-toric systems. From such a system one obtains a 4-dimensional manifold with a Hamiltonian S^1-action by restricting the action: when the underlying symplectic manifold is closed, Karshon classified these spaces in terms of a labelled graph. This talk aims at explaining how, starting from a semi-toric system on a closed 4-dimensional symplectic manifold, Karshon's invariants of the underlying Hamiltonian S^1-space can be recovered using the notion of `polygons with monodromy' introduced by Vu Ngoc. This should be thought of as analogous to the procedure to obtain Karshon's invariants from Delzant polygons in the case of symplectic toric manifolds. This is joint work with Sonja Hohloch (EPFL) and Silvia Sabatini (IST Lisbon), and part of a longer term project to study Hamiltonian S^1 x R actions on closed 4-dimensional manifolds.

Monday, November 11, 2013

10:00 am in 145 Altgeld Hall,Monday, November 11, 2013

Integration of Exact Courant algebroids

Xiang Tang (Washington University Math)

Abstract: In this talk, we will discuss some recent progress about the problem of integration of exact Courant algebroids. We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a ``Lagrangian" sub-2-groupoid of this symplectic 2-groupoid.

Monday, November 18, 2013

10:00 am in 145 Altgeld Hall,Monday, November 18, 2013

A normal form theorem around symplectic leaves

Ioan Marcut (UIUC Math)

Abstract: In this talk, I will discuss a normal form result in Poisson geometry, which generalizes Conn's theorem from fixed points to arbitrary symplectic leaves. The local model, at least in the integrable case, coincides with the local model of a free and proper Hamiltonian action around the zero set of the moment map. The result is joint work with Marius Crainic.

Monday, December 2, 2013

10:00 am in Altgeld Hall,Monday, December 2, 2013

Relative equilibria and vector fields on stacks

Eugene Lerman (UIUC Math)

Abstract: TBA