## Symplectic Seminar

The symplectic geometry seminars are held Mondays from 2:10--3:00 PM in BA 6183, unless otherwise indicated.

### Lower bounds for Gromov width of coadjoint orbits in U(n).

by Milena Pabiniak | Cornell University
Time: 03:10 — 04:00  (Friday, Jul. 22, 2011)
Location: BA6183, Bahen Center, 40 St. George St.
Abstract:
We use the Gelfand Tsetlin pattern to construct an effective Hamiltonian, completely integrable action of a torus $T=T^D$ on an open dense subset of a coadjoint orbit of $U(n)$ and we obtain a proper Hamiltonian $T$-manifold centered around a point in $\lie{t}^*$. The result of Karshon and Tolman says that such a manifold is equivariantly symplectomorphic to a particular subset of $R^{2D}$. This fact enables us to construct symplectic embeddings of balls into a class of coadjoint orbits of $U(n)$ and therefore obtain a lower bound for their Gromov width. Using the identification of $\lie{u}(n)^*$ with the space of n x n Hermitian matrices, the main theorem states that for a coadjoint orbit through $\lambda=diag(\lambda_1, \ldots , \lambda_n) \in =\lie{u}(n)^*$, with at most one eigenvalue $\lambda_k$ repeated, the lower bound for Gromov width is equal to the minimum of differences $\lambda_i-\lambda_j$, over all $\lambda_i \lambda_j$. For a generic orbit, with additional integrality conditions, this minimum was proved to be exactly the Gromov width of the orbit. For nongeneric orbits this lower bound is new.

At 1:30pm we will meet at Simon Sushi (409 Spadina Ave) for lunch.

#### Dates in this series

· Monday, Jan. 11, 2010: Construction of conformally compact Einstein manifolds (Dezhong Chen)
· Monday, Jan. 18, 2010: Homotopy Gerstenhaber algebras in toric topology (Matthias Franz)
· Monday, Jan. 25, 2010: Geography of symplectic 4-manifolds (Doug Park)
· Monday, Feb. 01, 2010: Lefschetz fibrations on cotagent bundles and Fukaya categories (Joseph Johns)
· Monday, Feb. 08, 2010: Varieties of complex Lie algebras (Tatyana Foth)
· Thursday, Mar. 04, 2010: A C*-algebraist looks at [Q,R]=0 (Nigel Higson)
· Monday, Mar. 15, 2010: Toric degenerations and quantization of toric manifolds (Mark Hamilton)
· Monday, Mar. 22, 2010: Eta-invariants and anomalies in U(1)-Chern-Simons theory (Brendan McLellan)
· Monday, Mar. 29, 2010: Ricci solitons (Andrew Dancer)
· Monday, Apr. 19, 2010: TBA (Octav Cornea)
· Monday, Apr. 26, 2010: B-fields, quantization and Morita equivalence (Henrique Bursztyn)
· Monday, Sep. 13, 2010: Remarks on symplectic realizations (Alejandro Cabrera)
· Monday, Sep. 20, 2010: On symplectic forms in the Soliton theory (Fedor Soloviev)
· Monday, Sep. 27, 2010: On the topology of fillings of contact manifolds (joint work with A. Oancea) (Claude Viterbo)
· Monday, Oct. 04, 2010: Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications (joint work with C. Vizman) (Feride Tiglay)
· Monday, Oct. 18, 2010: Embedded Lagrangian cobordism (Octav Cornea)
· Monday, Oct. 25, 2010: Hamiltonian Spaces with Group Valued Moment Maps (David Li-Bland)
· Monday, Nov. 01, 2010: Non-holonomic geometry and singularity theory (Alex Castro)
· Monday, Nov. 08, 2010: Homotopy Poisson actions (Rajan Mehta)
· Monday, Nov. 15, 2010: Kaehler metrics on spherical varieties (Victor Batyrev)
· Wednesday, Nov. 24, 2010: Geometric representation of closed 2-forms (Patrick Iglesias-Zemmour)
· Monday, Nov. 29, 2010: The graph complex and the Grothendieck-Teichmüller Lie algebra (Thomas Willwacher)
· Wednesday, Dec. 01, 2010: Steps towards the symplectization of Frölicher spaces (Augustin Batubenge)
· Wednesday, Dec. 08, 2010: Examples of co-Higgs bundles (Steven Rayan)
· Monday, Jan. 10, 2011: Braids and the Grothendieck-Teichmuller Group (Dror Bar-Natan)
· Monday, Jan. 17, 2011: Constructions of symplectic groupoids by blow-up's (Travis Li)
· Monday, Jan. 24, 2011: Abelian Chern-Simons theory: topological boundary conditions and surface operators (Natalia Saulina)
· Monday, Jan. 31, 2011: Variational Methods for the N-body problem (Richard Montgomery)
· Wednesday, Feb. 02, 2011: Rational Drinfeld associators and their denominators (Pavol Severa)
· Monday, Feb. 07, 2011: Quantization of manifolds with f-structure (Sean Fitzpatrick)
· Monday, Feb. 14, 2011: Geodesics on the group of symplectomorphisms (David Ebin)
· Monday, Feb. 28, 2011: Hamiltonian Torus Actions on Orbifolds (Tomoo Matsumura)
· Monday, Mar. 07, 2011: Infinitesimal Hilbert 16th problem (Dmitry Novikov)
· Monday, Mar. 14, 2011: The Modular Class of Poisson submersion (John Arlo Caine)
· Monday, Mar. 21, 2011: Poisson and Symplectic Geometry on b-manifolds (Ana Rita Pires)
· Monday, Mar. 28, 2011: Implosion constructions in hyperkahler geometry (Andrew Dancer)
· Monday, Apr. 04, 2011: Delocalized twisted equivariant cohomology (Ping Xu)
· Friday, May. 20, 2011: Toroidal orbifold compactifications of reductive groups and moduli of framed bundles (Johan Martens)
· Friday, Jul. 22, 2011: Lower bounds for Gromov width of coadjoint orbits in U(n). (Milena Pabiniak )
· Monday, Aug. 08, 2011: Real and Kaehler Quantization of Flag Manifolds (Mark Hamilton)
· Monday, Aug. 29, 2011: Local obstructions to approximating tropical curves in surfaces (Kristin Shaw)
· Monday, Sep. 12, 2011: Deforming Poisson varieties (Marco Gualtieri)
· Monday, Sep. 19, 2011: Degeneracy loci of Poisson varieties (Brent Pym)
· Monday, Sep. 26, 2011: Quantization and automorphic forms (Tatyana Barron)
· Monday, Oct. 03, 2011: Equivariant K-theory of the based loop group of SU(2) (Paul Selick)
· Wednesday, Oct. 12, 2011: Todd genera of complex torus manifolds (Hiroaki Ishida)
· Monday, Oct. 17, 2011: Compatible Poisson structures in cluster algebras (Michael Gekhtman)
· Wednesday, Oct. 19, 2011: Quantization and automorphic forms - 2 (Tatyana Barron)
· Wednesday, Oct. 19, 2011: The embedded contact homology of circle bundles over Riemann surfaces (David Farris)
· Monday, Oct. 24, 2011: Action homomorphisms, moment maps and quasimorphisms (Egor Shelukhin)
· Monday, Oct. 31, 2011: Betti numbers of twisted Higgs moduli spaces through quivers (Steven Rayan)
· Wednesday, Nov. 09, 2011: Toric Integrable Geodesic Flows (Susan Tolman)
· Monday, Nov. 14, 2011: $C^0$ limits of Hamiltonian paths and spectral invariants (Sobhan Seyfaddini)
· Monday, Nov. 21, 2011: Kirwan polyhedron of holomorphic coadjoint orbits (Guillaume Deltour)
· Monday, Dec. 05, 2011: Tau function on the spaces of admissible covers (Dmitry Korotkin)
· Wednesday, Jan. 18, 2012: Morse theory and invariants of (almost) symplectic manifolds, (Tara Holm)
· Monday, Jan. 23, 2012: Equivariant cohomology, syzygies, and orbit structure (Matthias Franz)
· Wednesday, Jan. 25, 2012: Kirwan polyhedron of holomorphic coadjoint orbits (Guillaume Deltour)
· Wednesday, Feb. 01, 2012: dg-Categories in Geometry (Jonathan Block)
· Monday, Feb. 06, 2012: A higher Riemann-Hilbert correspondence (Aaron Smith)
· Monday, Feb. 13, 2012: Quantum noise and symplectic rigidity (Leonid Polterovich)
· Wednesday, Feb. 22, 2012: Lower bounds for Gromov width of the unitary and special orthogonal coadjoint orbits (Milena Pabiniak)
· Wednesday, Mar. 07, 2012: The Kirwan method in hyperkahler geometry (Jonathan Fisher)
· Monday, Mar. 12, 2012: Positivity in the symplectic category (Rebecca Goldin)
· Monday, Mar. 19, 2012: Lagrangian Floer homology and toric oribfolds (Cheol-Hyun Cho)
· Monday, Mar. 26, 2012: On translated points of contactomorphisms (Sheila Margherita Sandon)
· Monday, Apr. 02, 2012: Foliated Groupoids and their infinitesimal data (Madeleine Jotz)
· Monday, Apr. 09, 2012: Unifying hypercomplex and holomorphic structures (Mathieu Stienon)
· Thursday, Apr. 12, 2012: Geometry of heterotic string compactifications (Xenia de la Ossa)
· Thursday, Apr. 12, 2012: Periods of Calabi-Yau manifolds in physics and number theory (Philip Candelas)
· Monday, Apr. 30, 2012: Combinatorics of polytopes and a model of nonrational toric varieties (Dan Zaffran)
· Wednesday, Jul. 18, 2012: Bohr-Sommerfeld-Heisenberg quantization (Richard Cushman)
· Friday, Jul. 20, 2012: Geometry of pseudodifferential algebra bundles (Mathai Varghese)
· Monday, Sep. 10, 2012: Generalizations of the Hunter-Saxton equation (Klas Modin)
· Monday, Sep. 24, 2012: Non-squeezing in $R^{2n}\times S^1$ (Maia Fraser)
· Monday, Oct. 01, 2012: Upper bound for the Gromov width of coadjoint orbits (Alexander Caviedes Castro)
· Monday, Oct. 15, 2012: Hydrodynamics and the skew-mean-curvature flow (Boris Khesin)
· Monday, Oct. 22, 2012: Root systems of torus graphs and characterization of extended actions of torus manifolds,  (Shintaro Kuroki)
· Wednesday, Oct. 24, 2012: Doing differential geometry on the irrational torus: a rush introduction to diffeology (Enxin Wu)
· Monday, Oct. 29, 2012: The discriminant metric on the contactomorphism group (Sheila Margherita Sandon)
· Monday, Nov. 12, 2012: Behavior of Welschinger invariants under Morse simplification (Erwan Brugallé)
· Monday, Nov. 19, 2012: On moduli spaces (David Li-Bland)
· Monday, Jan. 14, 2013: Compactness results for the quilted Atiyah-Floer conjecture (David Duncan)
· Monday, Jan. 21, 2013: Schubert calculus of weighted Grassmannians (Tomoo Matsumura)
· Monday, Jan. 28, 2013: A convexity theorem For symplectomorphism groups (Mehdi Mousavi)
· Monday, Feb. 04, 2013: On the Linearization of Poisson Lie group structures (Eckhard Meinrenken)