Matthew Strom Borman

Persistent Homology for Random Fields and Complexes

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices and, in most detail, the algebraic topology of the excursion sets of random elds.

Quasi-States, Quasi-Morphisms, and the Moment Map

We prove that symplectic quasi-states and quasi-morphisms on a symplectic manifold descend under symplectic reduction on a superheavy level set of a Hamiltonian torus action. Using a construction due to Abreu and Macarini, in each dimension at least four we produce a closed symplectic toric manifold with infinite dimensional spaces of symplectic quasi-states and quasi-morphisms, and a one-parameter family of non-displaceable Lagrangian tori. By using McDuffs method of probes, we also show how Ostrover and Tyomkins method for finding distinct spectral quasi-states in symplectic toric Fano manifolds can also be used to find different superheavy toric fibers.

Displacing Lagrangian toric fibers by extended probes

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff’s original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space P.1;3;5/ after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown. 53D12; 14M25, 53D40

Quasimorphisms on contactomorphism groups and contact rigidity

We build homogeneous quasimorphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental’s nonlinear Maslov index and a contact reduction technique for quasimorphisms. We show how these quasimorphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups. 53D35; 53D12, 53D20 Dedicated with gratitude to our teacher Leonid Polterovich on his 50 th birthday.

Spherical Lagrangians via ball packings and symplectic cutting

In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian,

Bounding Lagrangian widths via geodesic paths

S^{2}

Reduction for quasi-morphisms on contactomorphism groups and contact rigidity

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