Abstracts of Past Talks

• Speaker: Jordan Watts (UIUC)

  Date: May 4, 2015

  Title: The differential structure of an orbifold

  Abstract: There are many ways of viewing an (effective) orbifold besides the classical way: as a Lie groupoid, a stack, a diffeological space, a (Sikorski) differential space, a topological space... Not all of these are equivalent. In fact, in the categorical sense, all of the above categories are generally completely different. However, when restricting our attention to "quotients" of manifolds by proper Lie group actions, there is a chain of functors between the above categories, each of which forgets information along the way. If we ignore "maps between orbifolds" and focus only on a fixed orbifold, one may ask: how far along this chain of functors can one go before losing so much information that our orbifold cannot be recovered? (Going all the way to topological spaces, for example, would be too far.) In this talk I will answer this question with "differential spaces". This category generalises the category of manifolds, and includes as objects arbitrary sets equipped with special rings of functions. I will give a minimal set of invariants required to "remember" the orbifold, and show that these all live in the category of differential spaces. Going back to our chain of categories above, what I will be showing can be restated as follows: there is a functor from orbifolds (as effective proper étale Lie groupoids, say) to differential spaces that is essentially injective.

• Speaker: Derek Krepski (U Manitoba)

  Date: Apr 27, 2015

  Title: Prequantization of the moduli space of flat bundles on 2-manifolds with markings

  Abstract: Let G be a compact Lie group and let S be an oriented 2-manifold. This talk discusses the obstruction to the existence of a prequantization (also called pre-quantum line bundle, a complex line bundle whose Chern class is the symplectic form) of the moduli space M of flat principal G-bundles over S (possibly with marked points and prescribed holonomies around those marked points). The moduli space M is placed in the framework of quasi-Hamiltonian group actions (with Lie group-valued moment maps), where there exists a compatible notion of prequantization expressed in terms of U(1)-gerbes (instead of line bundles). In this context, the obstruction can be described fully if G is simply connected; however if G is not simply connected, less is known.

• Speaker: Stefan Müller (UIUC)

  Date: Apr 20, 2015

  Title: A C^0-characterization of symplectic and contact embeddings

  Abstract: Symplectic (and anti-symplectic) embeddings can be characterized as those embeddings that preserve the symplectic capacity (of ellipsoids). This gives rise to a proof of C^0-rigidity of symplectic embeddings, and in particular, diffeomorphisms. (There are many proofs of rigidity of symplectic diffeomorphisms, but all known proofs of rigidity of symplectic embeddings seem to use capacities.) This talk explains a C^0-characterization of symplectic embeddings via Lagrangian embeddings (of tori); the corresponding formalism is called the shape invariant (discovered by J.-C. Sikorav and Y. Eliashberg). The aforementioned rigidity is again an easy consequence. The shape invariant has two immediate advantages: it avoids the cumbersome distinction between symplectic and anti-symplectic, and the results can be adapted to contact embeddings via coisotropic embeddings (of tori). The adaptation to the contact case is (partly) work in progress. In the talk, I will give proofs of the main results and explain what makes them work (J-holomorphic disc techniques for all deep results).

• Speaker: Reyer Sjamaar (Cornell University)

  Date: Apr 13, 2015

  Title: Induction of representations and Poincare duality

  Abstract: Let G be a group and H a subgroup. Frobenius showed in 1898 how to "enlarge" a representation of H to a representation of G. His method, now called induction, rapidly became a useful tool in algebra and analysis and was adapted by others in various ways. For instance, in 1965 Bott made a systematic study of induction methods based on elliptic differential operators in the context of compact Lie groups, which led to generalizations of the Weyl character formula. I will review and update Bott's work and discuss some applications to K-theory. This is a report on joint work with Greg Landweber.

• Speaker: Ivan Struchiner (U Sao Paulo)

  Date: Apr 6, 2015

  Title: Stability and rigidity of Lie brackets

  Abstract: I will discuss some rigidity and stability results for geometric structures governed by Lie brackets, namely Lie algebroids and their morphisms. Examples include infinitesimal actions of Lie algebras and flat connections. I will focus on providing clear statements of the main results. If time permits, I will discuss some of the ideas in the proofs.

• Speaker: Jo Nelson (IAS/Columbia University)

  Date: Mar 30, 2015

  Title: Cylindrical contact homology: a retrospective

  Abstract: Cylindrical contact homology is arguably one of the more notorious Floer theoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations have tarnished its claim to being a well-defined contact invariant. However, recent work of Hutchings and Nelson has managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. This talk will highlight our implementation of intersection theory, non-equivariant constructions, domain dependent almost complex structures, automatic transversality, and obstruction bundle gluing, yielding a homological contact invariant which is expected to be isomorphic to SH^+ under suitable assumptions, though does not require a filling of the contact manifold. By making use of family Floer theory we obtain a S^1-equivariant theory defined over Z-coefficients, which when tensored with Q yields cylindrical contact homology, now with the guarantee of well-definedness and invariance.

• Speaker: Ioan Marcut (UIUC/Radboud University Nijmegen)

  Date: Mar 16, 2015

  Title: Poisson geometry around transversals

  Abstract: The role of Poisson transversals in Poisson geometry is analogous to the one played by symplectic submanifolds in symplectic geometry, and by transverse submanifolds in foliation theory. They are defined as submanifolds that intersect the symplectic leaves transversally and symplectically. In this talk I will explain how the entire "local Poisson geometry" around a Poisson transversal is encoded by that of the Poisson transversal. This is joint work with Pedro Frejlich.

• Speaker: Ioan Marcut (UIUC/Radboud University Nijmegen)

  Date: Mar 9, 2015

  Title: Lie groupoids and algebroids in geometric mechanics

  Abstract: The role of Poisson transversals in Poisson geometry is analogous to the one played by symplectic submanifolds in symplectic geometry, and by transverse submanifolds in foliation theory. They are defined as submanifolds that intersect the symplectic leaves transversally and symplectically. In this talk I will explain how the entire "local Poisson geometry" around a Poisson transversal is encoded by that of the Poisson transversal. This is joint work with Pedro Frejlich.

• Speaker: Ari Stern (Washington University St. Louis)

  Date: Mar 9, 2015

  Title: Lie groupoids and algebroids in geometric mechanics

  Abstract: Geometric mechanics has deep historical connections with symplectic and Poisson geometry, as well as with the geometry of Lie groups and algebras. More recently, the theory of Lie groupoids and algebroids has led to important generalizations and increased understanding of earlier work, particularly relating to symmetries, reduction, and discretization. In this talk, I will provide an overview of this topic and share some recent results.

• Speaker: Susan Tolman (UIUC)

  Date: Mar 2, 2015

  Title: Non-Hamiltonian actions with isolated fixed points

  Abstract: Let a circle act symplectically on a closed symplectic manifold . If the action is Hamiltonian, we can pass to the reduced space; moreover, the fixed set largely determines the cohomology and Chern classes of . In particular, symplectic circle actions with no fixed points are never Hamiltonian. This leads to the following important question: What conditions force a symplectic action with fixed points to be Hamiltonian? Frankel proved that Kahler circle actions with fixed points on Kahler manifolds are always Hamiltonian. In contrast, McDuff constructed a non-Hamiltonian symplectic circle action with fixed tori. Despite significant additional research, the following question is still open: Does there exists a non-Hamiltonian symplectic circle action with isolated fixed points? The main goal of this talk is to answer this question by constructing a non-Hamiltonian symplectic circle action with exactly 32 fixed points on a closed six-dimensional symplectic manifold. Based in part on joint work with J. Watts.

• Speaker: Donghoon Jang (UIUC)

  Date: Feb 23, 2015

  Title: Multigraphs to symplectic circle actions

  Abstract: Consider a symplectic circle action on a compact symplectic manifold with isolated fixed points. We can associate a directed multigraph to the manifold. Y. Karshon proved that if the dimension of the manifold is four, this multigraph completely determines the manifold up to equivariant symplectomorphism. We prove that we can associate a multigraph that does not have any loops. As an application, we complete the proof of the symplectic Petrie's conjecture for eight dimensional manifolds. This is a joint work with S. Tolman. arXiv:1408.6580.

• Speaker: Alex Tumanov (UIUC)

  Date: Feb 9, 2015

  Title: Symplectic non-squeezing in Hilbert space

  Abstract: The celebrated Gromov's non-squeezing theorem of 1985 says that the unit ball B^{2n} in R^{2n} can be symplectically embedded in the "cylinder" rB^2\times R^{2n-2} of radius r only if r\geq 1. I present a generalization of this theorem for Hilbert space. The result can be applied to symplectic flows of Hamiltonian PDEs. This work is joint with Alexander Sukhov.

• Speaker: Daniele Sepe (Universidade Federal Fluminense)

  Date: Jan 26, 2015

  Title: Singular integral affine structures and focus-focus singularities

  Abstract: The classification of completely integrable Hamiltonian systems is a driving question in Hamiltonian mechanics and symplectic geometry as these can be thought of as (examples of) complexity zero Hamiltonian R^n-actions on symplectic 2n-dimensional manifolds. Away from the locus of singularities (which, mechanically, correspond to equilibria of the system), Duistermaat's approach has revealed the importance of integral affine structures in the study of such systems. A natural question is whether the relation between integral affine geometry and completely integrable Hamiltonian systems can be extended to singularities, which contain important geometric and dynamical information. The aim of this talk is to present a differential-geometric notion of singular integral affine structures by illustrating its role in the classification of neighbourhoods of focus-focus singular fibres, which are analogous to nodal fibres in Lefschetz fibrations. This is ongoing joint work with Rui Loja Fernandes.

• Speaker: Pierre Albin (UIUC)

  Date: Dec 8, 2014

  Title: Linear analysis on manifolds with cylindrical ends

  Abstract: It seems that manifolds with cylindrical ends show up naturally in symplectic geometry. I will recall how analysis on these manifolds compares to analysis on closed manifolds.

• Speaker: Ana Rechtman (Université de Strasbourg)

  Date: Dec 1, 2014

  Title: The dynamics of the minimal set of Kuperberg's plug

  Abstract: In 1993 K. Kuperberg constructed examples of C8 and real analytic flows without periodic orbits on any closed 3-manifold. These examples continue to be the only known examples with such properties. A plug is a manifold with boundary of the type D2 x [0, 1] endowed with a flow that enters through D2 x{0}, exits through D2 x{1} and is parallel to the rest of the boundary. Moreover, it has the particularity that there are orbits that enter the plug and never exit, that is there are trapped orbits. The closure of a trapped orbit limits to a compact invariant set contained entirely within the interior of the plug. This compact invariant set contains a minimal. The first construction of a plug without periodic orbits was done by P. Schweitzer. This plug is constructed from the minimal set that is the Denjoy flow on the torus, implying that the flow of the plug is only C1 . Kuperberg's construction is completely different in nature. I will present a study of the minimal set, its dynamics and topology. (Joint work with Steven Hurder (University of Illinois at Chicago).)

• Speaker: Camilo Arias Abad (University of Toronto)

  Date: Nov 17, 2014

  Title: Flat connections and loop spaces

  Abstract: If $E\rightarrow X$ is a flat vector bundle, and $\pi: LX \rightarrow X$ is the map given by evaluation at $1 \in S^1$, then the pull back bundle $\pi^*E$ is a flat bundle equipped with a canonical automorphism given by the holonomy. I will explain that this construction naturally generalizes to the case of flat $\mathbb{Z}$-graded connections on $X$. Moreover, the restriction of the holonomy automorphism to the based loop space provides a representation of the Pontryagin algebra $C_*(\Omega^{\mathsf{M}} X)$. I will describe how this construction fits into the general story of higher dimensional local systems. The talk is based on joint work in progress with Florian Sch\"atz.

• Speaker: Matias del Hoyo (IMPA)

  Date: Nov 10, 2014

  Title: Metric on Stacks

  Abstract: The Linearization Theorem for Lie groupoids provides an organizing framework for classic results on the geometry of fibrations, actions and foliations. It was conjectured by A. Weinstein, who also gave a Morita invariance argument reducing the problem to the fixed point case, later solved by N. Zung. In a joint work with R. Fernandes we approach the linearization problem from a new perspective, developing a notion of metrics on Lie groupoids, achieving a simpler proof and a stronger theorem. Morita invariance is not needed in our approach, but a version of it still holds, leading to a definition of metrics on stacks. I will recall the interplay between groupoids and stacks, discuss the theory of Riemannian groupoids, and present some of our next results.

• Speaker: Henrique Bursztyn (IMPA)

  Date: Nov 3, 2014

  Title: Poisson geometry and Lie theory of vector bundles

  Abstract: In this talk I will discuss vector bundles in the realm of Lie groupoids (resp. Lie algebroids), known as VB-groupoids (resp. VB-algebroids). Just as Lie groupoids are common generalizations of manifolds and Lie groups (often thought of as models for singular spaces), VB-groupoids encompass vector bundles and linear representations (and can be thought of as "categorified" vector bundles over singular spaces). I will explain the Lie theory relating VB-groupoids and VB-algebroids, including examples and applications. Poisson structures enter the picture as objects dual to Lie algebroids, and I will point out the role of this dual viewpoint in our results. This is joint work with A. Cabrera and M. del Hoyo.

• Speaker: Rajan Mehta (Smith College)

  Date: Oct 27, 2014

  Title: Linear groupoid structures and representations up to homotopy

  Abstract: The notion of representation for a Lie groupoid has the annoying problem that it isn't generally possible to define a good adjoint representation. To fix this problem, Arias Abad and Crainic introduced the notion of "representation up to homotopy". In this talk, I will show how 2-term representations up to homotopy are related to linear groupoid structures, which play the role of semidirect products. There is a one-to-one correspondence at the level of isomorphism classes, but at the level of objects, the correspondence is noncanonical, so it is possible for certain constructions to be "natural" in one perspective but not the other.

  A key example that illustrates the value of linear groupoids is the adjoint representation. To define the adjoint representation up to homotopy of a Lie groupoid G, one needs to choose a distribution transverse to the source fibers. On the other hand, the linear groupoid that corresponds to the adjoint representation is canonical; it is simply the tangent bundle TG.

  This talk is based on joint work with Alfonso Gracia-Saz (arXiv:1007.3658).

• Speaker: Markus Pflaum (University of Colorado - Boulder)

  Date: Oct 20, 2014

  Title: Inertia spaces and the cyclic homology of convolution algebras over proper Lie groupoids

  Abstract: The inertia space of a Lie groupoid encodes interesting topological, geometric, and analytic information about the original Lie groupoid. It is the goal of the talk to explain this point of view using as example the cyclic homology theory of the convolution algebra of a proper Lie groupoid. To this end, the inertia groupoid associated to a proper Lie groupoid is first defined. We show that it is a differentiable stratified groupoid, and non-singular only in exceptional cases. The corresponding quotient space, the inertia space, possesses a Whitney stratification, and is triangulable. Finally, horizontal and basic forms over the inertia space are constructed, and a Hochschild-Kostant-Rosenberg type theorem for the convolution algebra of a proper Lie groupoid is indicated.

• Speaker: Michael Khanevsky (University of Chicago)

  Date: Oct 13, 2014

  Title: The Hofer length spectrum of symplectic surfaces

  Abstract: In Riemannian geometry the length spectrum is a rich source of invariants of the manifold. In the symplectic setting there is no notion of length, hence no possibility to define the length spectrum. Frederic Le Roux proposed the following construction: pick a ball of a fixed radius and translate it by a Hamiltonian isotopy along a given homotopy (or homology) class. The minimal Hofer energy required for such translation behaves in a very similar way to the Riemannian length spectrum. We will discuss some estimates for this energy in the two-dimensional case.

• Speaker: Rui Loja Fernandes (UIUC)

  Date: Oct 6, 2014

  Title: An h-principle for symplectic foliations

  Abstract: I will explain how a classical result of Gromov in symplectic geometry extends to the context of symplectic foliations, which we regard as a h-principle for (regular) Poisson geometry. Joint work with Pedro Frejlich.

• Speaker: Seth Wolbert (UIUC)

  Date: Sept 22, 2014

  Title: Parallel Transport and Principal Bundles for Geometric Stacks

  Abstract: It has been known for about 60 years that parallel transport for principal $G$-bundles with equivariant connections is characterized by holonomy homomorphisms and that these maps can be used to reconstruct not only the associated connection, but also the relevant bundle. This relationship has been most recently restated as an equivalence of categories between bundles with connections over a fixed manifold and a category of objects known as transport functors. I will describe recent work, done with Eugene Lerman and Brian Collier, to prove this equivalence is natural; furthermore, I will describe how this naturality allows us to extend the above characterization of parallel transport to bundles over Lie groupoids and, more generally, geometric stacks.

• Speaker: Ely Kerman (UIUC)

  Date: Sept 15, 2014

  Title: On the peristence of Reeb orbits

  Abstract: Reeb flows on contact manifolds are conjectured to always have closed orbits (the Weinstein Conjecture) and in many settings nontrivial lower bounds for their numbers are also expected. One obstacle to detecting many distinct closed orbits is the difficulty, well known from the study of closed geodesics, of distinguishing between simple and multiply covered orbits. In this talk I will introduce a $C^0$-distance between Reeb flows in the spirit of the Hofer norm and will describe a Floer theoretic proof that certain clusters of simple closed Reeb orbits must persist over surprisingly long distances. Among other things, this allows one to reprove a classic result of Ekeland and Lasry concerning multiple closed characteristics on convex hypersurfaces pinched between spheres of radius 1 and $\sqrt{2}$.

• Speaker: James Pascaleff (UIUC)

  Date: Sept 8, 2014

  Title: Symplectic cohomology and its role in mirror symmetry

  Abstract: Symplectic cohomology is a Floer homology theory for convex symplectic manifolds, a class that includes cotangent bundles and Stein manifolds. Geometrically, it detects the existence of closed characteristics in a hypersurface "at infinity." On the other hand, it plays a crucial role in homological mirror symmetry, where it becomes the "closed-string sector of the A-model." In this talk I will explain these connections, including some of my results and directions for future research.

• Speaker: Milena Pabiniak (IST)

  Date: May 5, 2014

  Title: Symplectic toric manifolds as centered reductions of products of weighted projective spaces

  Abstract: We prove that every symplectic toric orbifold is a "centered" symplectic reduction of a Cartesian product of weighted projective spaces. Reduction is centered if the level set contains central Lagrangian torus fiber of the product of weighted projective spaces. In that case one can deduce certain information about non-displaceable sets or existence of quasimorphisms. For example, a theorem of Abreu and Macarini shows that if the level set of the reduction passes through a non-displaceable set then the image of this set in the reduced space is also non-displaceable. Using this theorem and our result we reprove that every symplectic toric orbifold contains a non-displaceable fiber and identify this fiber. Joint work with Aleksandra Marinkovic.

• Speaker: Sam Evans (Notre Dame)

  Date: April 28, 2014

  Title: Lie algebra cohomology and a degenerate cup product on the flag manifold

  Abstract: Belkale and Kumar introduced a degeneration of the usual cup product on $H^*(G/P)$ which gives an optimal solution to the geometric Horn problem. In this talk, I will explain joint work with Bill Graham where we realize the Belkale-Kumar product using relative Lie algebra cohomology. We do this using a family in the variety of Lagrangian subalgebras.

• Speaker: Ana Cannas da Silva (ETH)

  Date: April 21, 2014

  Title: Lagrangian correspondences - a toric case study

  Abstract: What lagrangians in a symplectic reduced space admit a (one-to-one transverse) lifting to the original symplectic manifold? I will discuss this question (going back to work of Werheim and Woodward) through examples and counterexamples (joint work with Meike Akveld).

• Speaker: David Martinez Torres (PUC)

  Date: April 14, 2014

  Title: Transverse Geometry of Codimension one Foliations Calibrated by Closed 2-Forms

  Abstract: A codimension one foliation is (topologically) taut if it admits a closed 1-cycle everywhere transverse to the foliation. The theory of taut foliations is extremely rich in dimension 3, however, it less satisfactory in higher dimensions. In this talk we will discuss a different generalization of 3-dimensional taut foliations to higher dimensions inspired in symplectic geometry. These are codimension one foliations which admit a closed 2-form which makes every leaf a symplectic manifold. Our main result is that on an ambient closed manifold a foliation (of class at least C^1 in the transverse direction) admitting a 2-calibration has its transverse geometry encoded in a 3-dimensional foliated submanifold. This is joint work with Álvaro del Pino and Francisco Presas (ICMAT, Madrid).

• Speaker: Michael Bailey (CIRGET/UQAM/McGill)

  Date: April 7, 2014

  Title: Integration of generalized complex structures

  Abstract: Generalized complex geometry is a generalization of both symplectic and complex geometry, proposed by Nigel Hitchin in 2002, which is of particular interest in string theory and mirror symmetry. Modulo a parity condition, generalized complex manifolds locally "look like" holomorphic Poisson manifolds, though globally they may not admit a complex structure at all. Therefore, locally they should integrate to holomorphic symplectic groupoids. One can take the global integration if one passes to holomorphic "symplectic" stacks. Earlier work by Crainic defined an integration for generalized complex structures which did not capture the holomorphic nature.

• Speaker: Miguel Abreu (IST)

  Date: March 31, 2014

  Title: Dynamical convexity and elliptic orbits for Reeb flows

  Abstract: A classical conjecture states that any convex hypersurface in even-dimensional euclidean space carries an elliptic closed orbit of its characteristic flow. Dell'Antonio-D'Onofrio-Ekeland proved it in 1995 for antipodal invariant convex hypersurfaces. In this talk I will present a generalization of this result using contact homology and a notion of dynamical convexity first introduced by Hofer-Wysocki-Zehnder for contact forms on the 3-sphere. Applications include certain geodesic flows, magnetic flows and toric contact manifolds. This is joint work with Leonardo Macarini.

• Speaker: Eric Zaslow (Northwestern)

  Date: March 17, 2014

  Title: Legendrian Knots and Constructible Sheaves

  Abstract: We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra of the knot. This sounds harder than it is. Briefly-- INPUT: Knot diagram, OUTPUT: Category. I will illustrate the above with simple examples. This work is joint with David Treumann and Vivek Shende.

• Speaker: Eric Zaslow (Northwestern)

  Date: March 17, 2014

  Title: Legendrian Knots and Constructible Sheaves

  Abstract: We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra of the knot. This sounds harder than it is. Briefly-- INPUT: Knot diagram, OUTPUT: Category. I will illustrate the above with simple examples. This work is joint with David Treumann and Vivek Shende.

• Speaker: Roy Wang (Utrecht University)

  Date: March 10, 2014

  Title: Local Rigidity and Nash-Moser Methods

  Abstract: J. Conn used analytic methods to prove his theorem on the linearization of Poisson structures. For some time that proof was heuristically interpreted as a local rigidity result for linear, compact, semi-simple Poisson structures. In his thesis I. Marcut made this interpretation rigorous, which lead to surprising new results. In collaboration we aim to isolate the method and formulate a local rigidity theorem, which we apply to other geometrical structures. As an example I sketch a proof of the Newlander-Nirenberg theorem.

• Speaker: Daniel Hockensmith (UIUC)

  Date: March 3, 2014

  Title: Folding Symplectic Reduction

  Abstract: The Marsden-Weinstein-Meyer reduction theorem is an indispensable tool for the study of Hamiltonian group actions on symplectic manifolds. It gives an explicit recipe for the construction of a symplectic reduced space using only regular values of the moment map and the group action. I will prove that if one replaces symplectic manifolds with oriented, folded-symplectic manifolds in the statement of the MWM reduction theorem then a reduced space with a natural folded-symplectic form is obtained in the same way. I will then argue that the assumptions of this generalized theorem are too strong, leading us towards a more robust set of assumptions for a folded-symplectic reduction theorem.

• Speaker: Dana Balibanu (Utrecht University)

  Date: February 24, 2014

  Title: Convexity Theorems for Semisimple Symmetric Spaces

  Abstract: N/A

• Speaker: Alexander Caviedes Castro (University of Toronto)

  Date: February 17, 2014

  Title: Upper bounds for the Gromov width of coadjoint orbits of compact Lie groups

  Abstract: I will show how to find an upper bound for the Gromov width of coadjoint orbits with respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten invariants. The approach presented here is closely related to the one used by Gromov in his celebrated Non-squeezing theorem.

• Speaker: Laura Schaposnik Massolo (UIUC)

  Date: February 10, 2014

  Title: Real slices of the moduli space of Higgs bundles

  Abstract: After introducing Higgs bundles and their moduli space, through the natural hyperkaehler structure of the moduli space of Higgs bundles for complex groups we shall construct three anti-holomorphic involutions whose fixed points in the moduli space give branes in the A-model and B-model. After defining what those branes are, we shall attempt to relate them to log-symplectic structures and their invariants.

• Speaker: Jose Mourao (IST)

  Date: February 3, 2014

  Title: Imaginary time flow in geometric quantization and in Kahler geometry, degeneration to real polarizations and tropicalization

  Abstract: We will recall the problem of dependence of quantization of a symplectic manifold on the choice of polarization and study its relation with geodesics in the space Kahler metrics. Complex one parameter subgroups of the "group" of complexified hamiltonian symplectmorphisms appear naturally in this context. For some classes of symplectic manifolds we will describe geodesic rays of Kahler structures degenerating to real polarizations and study the associated metric collapse. Each such ray selects a basis of holomorphic sections which converge to distributional sections supported on Bohr-Sommerfeld fibers as the geodesic time goes to infinity. The same geodesic rays lead to tropicalization of toric varieties and of hypersurfaces on toric varieties.

• Speaker: Susan Tolman (UIUC)

  Date: December 9, 2013

  Title: Morse Theory and the Moduli Space of Curves

  Abstract: Based on joint work with Bott and Weitsman, we will explain how to use Morse theory to calculate the Betti number of reduced spaces for proper Hamiltonian loop-group actions, such as the moduli space of curves.

• Speaker: Eugene Lerman (UIUC)

  Date: December 2, 2013

  Title: Relative equilibria and vector fields on stacks

  Abstract: N/A

• Speaker: Ioan Marcut (UIUC)

  Date: November 18, 2013

  Title: A normal form theorem around symplectic leaves

  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.

• Speaker: Xiang Tang (WUSTL)

  Date: November 11, 2013

  Title: Integration of Exact Courant algebroids

  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.

• Speaker: Daniele Sepe (Utrecht University)

  Date: November 4, 2013

  Title: Semi-toric systems as Hamiltonian S^1-spaces

  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.

• Speaker: Donghoon Jang (UIUC)

  Date: October 30, 2013

  Title: Fixed points of symplectic circle actions

  Abstract: The study of fixed points of maps is a classical and important topic in geometry and topology. During this talk, we focus on the fixed points of maps in the case where manifolds admit symplectic structures and circle actions on the manifolds preserve the symplectic structures. We discuss main theorems on fixed points of symplectic circle actions and discuss techniques to study, ABBV Localization formula and Atiyah-Singer index formula.

• Speaker: Ely Kerman (UIUC)

  Date: October 28, 2013

  Title: All boundaries of contact type can keep secrets

  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.

• Speaker: Peter Spaeth (Penn State University)

  Date: October 21, 2013

  Title: On the Topological Dynamics Arising from a Contact Form

  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.

• Speaker: Augustin Banyaga (Penn State University)

  Date: October 14, 2013

  Title: A generalization of the group of Hamiltonian homeomorphisms

  Abstract: 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).

• Speaker: Songhao Li (WUSTL)

  Date: October 7, 2013

  Title: Category of Lie groupoids

  Abstract: For an integrable Lie algebroid, we define the category of integrating Lie groupoids. If we choose an open cover of the base with respect to the orbits of the Lie algebroid, we may reduce the problem finding the category of integrations to that of finding the category of integrations over each open set of the cover. We illustrate this by classifying the Hausdorff symplectic groupoids of a proper log symplectic manifold. Based on joint work with Marco Gualtieri.

• Speaker: Peter Weigel (Purdue)

  Date: September 30, 2013

  Title: Positive loops and orderability in contact geometry

  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).

• Speaker: Stefan Müller (UIUC)

  Date: September 23, 2013

  Title: Topological Hamiltonian and contact dynamics, part II: applications

  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).

• Speaker: Stefan Müller (UIUC)

  Date: September 16, 2013

  Title: Topological Hamiltonian and contact dynamics, part I: an introduction

  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.

• Speaker: Alvaro Pelayo (WUSTL)

  Date: September 9, 2013

  Title: Some interactions between classical, semiclassical, and random symplectic geometry

  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.

• Speaker: Jordan Watts (UIUC)

  Date: August 26, 2013

  Title: De Rham complexes for orbit spaces and symplectic quotients

  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 defined by Sjamaar.