Correction: SGGTC seminars this week
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Robert Lipshitz
Apr 26, 2012, 11:37:55 AM4/26/12
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Hi folks,
Sorry -- I missed a speaker from the schedule. We also have Jo Nelson discussing "Cylindrical contact homology as a well-defined homology theory?" That should be great, too; contact homology is one of the great tools in contact topology, and not seen here as often as it deserves.
So, again:
Friday, April 27, 9:30 a.m. in Math 520
Jo Nelson, "Cylindrical contact homology as a well-defined homology theory?"
Abstract: Symplectic field theory has been around for more than a decade but significant analytic obstacles remain and very few rigorous proofs have appeared in the literature. Cylindrical contact homology is one of the "simplest" invariants coming from this framework, but even the full details of its construction have not been lovingly worked out. In fact it has recently come to light that the usual assumptions on the Conley-Zehnder indices of contractible closed Reeb orbits do not ensure d^2=0 due to the existence of multiply covered cylinders and their branched covers. In this talk I will explain these issues with concrete examples and explore what stronger conditions are necessary on the growth rates of the indices of simple contractible orbits to obtain a homological invariant. I will also sketch a method in progress that seems to avoid these issues for prequantization spaces and certain S^1 bundles over nicely behaved symplectic orbifolds.
Friday, April 27, 10:45 a.m., in Math 520
Cotton Seed, "Twisting Szabo's geometric spectral sequence"
Abstract: Recently, by studying suitably twisted complexes, a number of knot homology theories have been formulated in terms of complexes generated by the spanning trees of a knot. In this talk, I will describe a twisted version of Szabo's geometric spectral sequence in Khovanov homology. To begin, I will review related constructions: Roberts' totally twisted Khovanov homology, a twisted variant of the spectral sequence from Khovanov homology to the double-branched cover, and Szabo's geometric spectral sequence. I will present my construction and give some computational results. Finally, I will describe some natural directions for future work.
Friday, April 27, 1:15 p.m. in Math 520
Haydee Aguilar Cabrera “New open book decompositions in Singularity Theory”
Abstract: Given a real analytic $d$-regular function from $\mathbb{R}^n$ to $\mathbb{R}^p$, ($n \geq p$), with an isolated singularity at the origin, a refinement of the Milnor fibration Theorem associates an open book decomposition of $\mathbb{S}^{n-1}$ to the singularity.
We present a family of real analytic $d$-regular functions $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated singularity at the origin. Let $L_f$ be the link of the singularity of $f$. From $f$ we define a function $F$ from $\mathbb{R}^6$ to $\mathbb{R}^2$, with link $L_F$, such that $F$ is $d$-regular, $F$ has an isolated singularity at the origin and the knot $(\mathbb{S}^{5},L_F)$ is a cyclic suspension of $(\mathbb{S}^{3},L_f)$.
Best,
Robert
Sorry -- I missed a speaker from the schedule. We also have Jo Nelson discussing "Cylindrical contact homology as a well-defined homology theory?" That should be great, too; contact homology is one of the great tools in contact topology, and not seen here as often as it deserves.
So, again:
Friday, April 27, 9:30 a.m. in Math 520
Jo Nelson, "Cylindrical contact homology as a well-defined homology theory?"
Abstract: Symplectic field theory has been around for more than a decade but significant analytic obstacles remain and very few rigorous proofs have appeared in the literature. Cylindrical contact homology is one of the "simplest" invariants coming from this framework, but even the full details of its construction have not been lovingly worked out. In fact it has recently come to light that the usual assumptions on the Conley-Zehnder indices of contractible closed Reeb orbits do not ensure d^2=0 due to the existence of multiply covered cylinders and their branched covers. In this talk I will explain these issues with concrete examples and explore what stronger conditions are necessary on the growth rates of the indices of simple contractible orbits to obtain a homological invariant. I will also sketch a method in progress that seems to avoid these issues for prequantization spaces and certain S^1 bundles over nicely behaved symplectic orbifolds.
Friday, April 27, 10:45 a.m., in Math 520
Cotton Seed, "Twisting Szabo's geometric spectral sequence"
Abstract: Recently, by studying suitably twisted complexes, a number of knot homology theories have been formulated in terms of complexes generated by the spanning trees of a knot. In this talk, I will describe a twisted version of Szabo's geometric spectral sequence in Khovanov homology. To begin, I will review related constructions: Roberts' totally twisted Khovanov homology, a twisted variant of the spectral sequence from Khovanov homology to the double-branched cover, and Szabo's geometric spectral sequence. I will present my construction and give some computational results. Finally, I will describe some natural directions for future work.
Friday, April 27, 1:15 p.m. in Math 520
Haydee Aguilar Cabrera “New open book decompositions in Singularity Theory”
Abstract: Given a real analytic $d$-regular function from $\mathbb{R}^n$ to $\mathbb{R}^p$, ($n \geq p$), with an isolated singularity at the origin, a refinement of the Milnor fibration Theorem associates an open book decomposition of $\mathbb{S}^{n-1}$ to the singularity.
We present a family of real analytic $d$-regular functions $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated singularity at the origin. Let $L_f$ be the link of the singularity of $f$. From $f$ we define a function $F$ from $\mathbb{R}^6$ to $\mathbb{R}^2$, with link $L_F$, such that $F$ is $d$-regular, $F$ has an isolated singularity at the origin and the knot $(\mathbb{S}^{5},L_F)$ is a cyclic suspension of $(\mathbb{S}^{3},L_f)$.
Best,
Robert
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