We discuss the proof of the following Theorem
Assume $E$ is a $C^{p}$ real Banach manifold, and $f:E\circlearrowleft$, $f\circ f=f$ is a $C^{p}$ retraction, where $1\leq p\leq +\infty$. Then the retract $f(E)$ is a $C^{p}$ sub Banach manifold of $E$.
The Jones polynomial was first defined by Vaughan Jones as a "trace function" on an algebra discovered via operator algebras. It was discovered that the polynomial satisfies certain skein relations. The HOMFLY polynomial was discovered through both skein relations and a "lift" of the trace function on the Jones algebra to the "Hecke algebra". Another 2-variable polynomial called the Kauffman polynomial was discovered purely via skein relations. In this talk, we discuss how the process started by Jones was reversed for this polynomial.
I'll talk about some specialized kirby calculus constructions: immersed surface complements and round handles. I'll prove using kirby calculus that S2xS2 minus an appropriate smooth embedded S2vS2 is diffeomorphic to R4. Maybe that is obvious, but the point is we can find nice diagrams where you see everything explicitly.
TBD
Instanton Floer homology, introduced by Floer in the 1980s, has become a power tool in the study of 3-dimensional topology. Its application has led to significant achievements, such as the proof of the Property P conjecture. While instanton Floer homology with complex coefficients is widely studied and conjectured to be isomorphic to the hat version of Heegaard Floer homology, its counterpart with integral coefficients is less understood.
We say that a Riemannian manifold has good higher expansion if every rationally null-homologous i-cycle bounds an i+1 chain of comparatively small volume. The interactions between expansion, spectral geometry, and topology have long been studied in the settings of graphs and surfaces. In this talk, I will explain how to construct rational homology 3-spheres which are good higher expanders.
In this talk, I will present a geometric algorithm for determining whether a given set of elements in SO+(n,1) generates a discrete subgroup, as well as identifying the relators for the corresponding group presentation. The algorithm constructs certain hyperbolic manifolds that are always complete, a key condition for applying Poincaré Fundamental Polyhedron Theorem and ensuring the algorithm is valid.
Given a Legendrian link L in the contact 3-sphere, one can hope to classify the set of exact Lagrangian fillings of L, i.e. exact Lagrangian surfaces in the symplectic 4-ball with boundary equal to L. Much of the recent progress towards this classification relies on establishing a connection between sheaf-theoretic invariants of Legendrians and cluster algebras. In this talk, I will describe this connection and how these invariants behave with respect to certain symmetries of Legendrian links and their fillings. Parts of this are joint work with Agniva Roy.
In a recent note Francesco Lin showed that if a rational homology sphere Y admits a taut foliation then the Heegaard Floer module HF^-(Y) contains a copy of F[U]/U as a summand. This implies that either the L-space conjecture is false or that Heegaard Floer homology satisfies a geography restriction. In a recent paper in collaboration with Fraser Binns we verified that Lin's geography restriction holds for a wide class of rational homology spheres.
In this talk, I will present a complete coarse classification of strongly exceptional Legendrian realizations of the connected sum of two Hopf links in contact 3-spheres. This is joint work with Sinem Onaran.
