Date and location
Saturday 24 March 2018
Malott Hall 532 (Mathematics Lounge)
Cornell University
Ithaca, New York
Schedule
| 09:00 am–09:30 am | Coffee and bagels |
| 09:30 am–10:00 am | My Huynh, Cornell University |
| The Gromov Width of Symplectic Cuts of Complex
Grassmannians
There has been extensive research on the Gromov width of many classes of symplectic manifolds, including $4$-dimensional toric manifolds, complex Grassmannians, and some polygon spaces. Eugene Lerman developed the technique of symplectic cuts, which cuts a symplectic manifold into two "smaller ones." It is natural to ask whether or not the Gromov width of a symplectic cut of a symplectic manifold is at most the Gromov with of the original manifold. In the case of complex Grassmannians, I will describe how the theory of J-holomorphic curves lets us find obstructions to the symplectic embedding of balls into a symplectic cut of a complex Grassmannian. I compute the exact Gromov width of these symplectic cuts and answer positively the above question for certain cuts of all Grassmannian manifolds. This work is a part of my dissertation. |
|
| 10:00 am–10:15 am | Coffee |
| 10:15 am–10:45 am | Zelin Yi, Pennsylvania State University |
| Spinors and the tangent groupoid
The $C^*$-algebra of the tangent groupoid of a smooth manifold $M$ decomposes into a copy of the compact operators on $L^2(M)$ for each nonzero $t$, and a copy of the $C^0$ functions on $T^*M$ for $t=0$. But the (densely defined) trace on compact operators has no good continuation at $t=0$. Now suppose that $M$ is a Riemannian spin manifold and let $S$ be its spinor bundle. We shall construct from $S$ a coefficient bundle for the tangent groupoid. The space of smooth compactly supported sections has a convolution algebra structure whose restriction to nonzero t gives the $C^*$-algebra of compact operators on $L^2(M,S)$, and whose restriction to $t=0$ gives the horizontal $C^0$ differential forms for the projection from $T^*M$ to $M$, with an interesting convolution structure. The supertrace on compact operators has a continuation to $t=0$. |
|
| 10:45 am–11:15 am | Coffee |
| 11:15 am–12:15 pm | Richard Hind, University of Notre Dame |
| Embedding and Packing Lagrangian tori
There has been much work recently on embedding and packing problems for symplectic manifolds, especially when the domain is an ellipsoid. We can ask how far an ellipsoid needs to be scaled such that it admits a symplectic embedding, or how many copies of a fixed ellipsoid can be embedded with disjoint images. I will describe joint work with Ely Kerman and Emmanuel Opshtein considering the case when the domain is a Lagrangian torus. The same kinds of questions still make sense provided we put constraints on the area class of the torus. In dimension $4$ we can find optimal embeddings, but natural packings turn out not to be maximal. |
|
| 12:15 pm–02:00 pm | Lunch |
| 02:00 pm–03:00 pm | Lisa Traynor, Bryn Mawr College |
| Lagrangian Fillings of Legendrian Submanifolds
A classic question in knot theory is: Given a smooth knot in the $3$-sphere, what surfaces in the $4$-ball can it bound? In symplectic geometry, a natural question is: Given a Legendrian knot, what Lagrangian surfaces can it bound? Whereas any smooth knot can be filled by an infinite number of topologically distinct surfaces, there are classical and non-classical obstructions to the existence of Lagrangian fillings of Legendrian knots. In particular, a polynomial associated to the Legendrian through the technique of generating functions can show that there is no compatible embedded Lagrangian filling. Legendrians that admit generating functions will always admit compatible immersed Lagrangian fillings, and I will describe how this polynomial also gives information about the minimal number and indices of double points in any compatible immersed Lagrangian filling. In addition, I will describe some constructions to realize minimally immersed fillings. This is joint work with Samantha Pezzimenti. |
|
| 03:00 pm–03:30 pm | Tea |
| 03:30 pm–04:30 pm | Kiumars Kaveh, University of Pittsburgh |
| Convergence of polarizations, toric degenerations,
and Newton-Okounkov bodies
I will talk about a recent joint work with Megumi Harada and Mark Hamilton in geometric quantization in connection with degenerations of varieties to toric varieties. The main result shows that, whenever a smooth complex projective algebraic variety together with a pre-quantum data has a "toric degeneration" satisfying certain natural technical assumptions, then its "Kähler polarization" converges to its "real polarization" (in a sense to be made precise). This provides a large class of examples that one can more or less explicitly relate two main constructions of quantization namely Kähler and real quantizations. Our work builds upon and relies on the previous work of Baier-Florentino-Mourão-Nunes for integrable systems on toric manifolds and the work of Hamilton-Konno for Gelfand-Zetlin integrable system. |
|
| 06:30 pm | Dinner |