Nov 14, 2022
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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09:30 AM - 10:30 AM
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Floer Homology and Non-Fibered Knot Detection
Steven Sivek (Max-Planck-Institut für Mathematik; Imperial College, London)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Floer homology can tell whether a knot is fibered, and this has led to proofs that both knot Floer homology and Khovanov homology can positively identify a small handful of knots: the unknot, the trefoils, the figure eight, and the cinquefoil. In this talk, I'll discuss recent work with John Baldwin in which we show for the first time that both invariants can also detect non-fibered knots, including 5_2, and that HOMFLY homology detects infinitely many knots. The key input is a classification of genus-1 knots which are "nearly fibered" from the perspective of knot Floer homology.
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Front Courtyard
- Video
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- Supplements
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11:00 AM - 12:00 PM
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Normally Complex Polynomial Perturbations and Arnold Conjecture
Shaoyun Bai (MSRI / Simons Laufer Mathematical Sciences Institute (SLMath))
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Moduli spaces of J-holomorphic curves usually carry a "normal" complex structure coming from the nature of Cauchy-Riemann-type equations. In recent work joint with Guangbo Xu, we take advantage of such a structure to construct abstract perturbations of J-holomorphic curve equations to define integral Gromov-Witten type invariants and integral Hamiltonian Floer homology for general symplectic manifolds. The latter leads to a solution of the Arnold conjecture over the integers, realizing a conjectural picture of Fukaya-Ono proposed in the 90s. I will survey this circle of ideas and perhaps tell you some possibilities going beyond integer-valued invariants.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: 2nd Floor Deck
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Virtual Equivariant Morse-Floer Theory
Semon Rezchikov (Princeton University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
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- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
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03:30 PM - 04:30 PM
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Localization and Morava K-theory Gromov-Witten Invariants
Tim Large (Columbia University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
The traditional rational-valued Gromov-Witten invariants of a symplectic manifold with a circle or torus action can be often be computed using equivariant localization. In this talk, I will discuss a homotopy-theoretic extension of this, for Abouzaid-McLean-Smith’s Morava K-theory Gromov-Witten invariants.
- Supplements
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Nov 15, 2022
Tuesday
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09:30 AM - 10:30 AM
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Microlocal Sheaves of Spectra Supported on Lagrangians
Xin Jin (Boston College)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
I will present recent results on microlocal sheaves of spectra supported on a smooth Lagrangian in a cotangent bundle, and some of the applications. I will discuss the ideas and techniques of the proofs, including the use of an $(\infty,2)$-category of correspondences, and results from sheaf quantizations.
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Front Courtyard
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Normal Invariant of Nearby Lagrangians via Twisted Generating Functions
Daniel Alvarez-Gavela (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
I will present some new constraints on the smooth structure of nearby Lagrangians. The method of proof uses homotopical methods and Waldhausen's algebraic K-theory of spaces. Joint work in progress with M. Abouzaid, S. Courte and T. Kragh.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: 2nd Floor Deck
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Stable Spaces of Equivariant H-Cobordisms
Mona Merling (University of Pennsylvania)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Waldhausen's algebraic K-theory of manifolds satisfies a homotopical lift of the classical h-cobordism theorem and provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will give an overview of joint work with Goodwillie, Igusa and Malkiewich about a homotopical lift of an equivariant h-cobordism theorem. In this talk I will zoom in on the functoriality of the stable equivariant h-cobordism space and I will give a flavor of some of the geometric ideas that go into these constructions.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
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- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Naturality Issues in Involutive Heegaard Floer Homology
Kristen Hendricks (Rutgers University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Heegaard Floer homology is an invariant of 3-manifolds, and knots and links within them, introduced by P. Oszváth and Z. Szabó in the early 2000s. Because of its relative computability by the standards of gauge and Floer theoretic invariants, it has enjoyed considerably popularity. However, it is not immediately obvious from the construction that Heegaard Floer homology is natural, that is, that it assigns to (eg) a basepointed 3-manifold a well-defined module over an appropriate base ring rather than an isomorphism class of modules, and well-defined cobordism maps to 4-manifolds with boundary. This situation was improved in the 2010s when A. Juhász, D. Thurston, and I. Zemke later showed naturality of the various versions of Heegaard Floer homology. In this talk we consider involutive Heegaard Floer homology, a refinement of the theory introduced by C. Manolescu and I in 2015, whose definition relies on Juhász-Thurston-Zemke naturality but which is itself not obviously natural even given their results. We prove that involutive Heegaard Floer homology is natural and has well-defined maps associated to cobordisms. Despite the apparent technicality of this abstract, this talk is intended to be understandable to an audience not necessarily familiar with Heegaard Floer homology. This is joint work with J. Hom, M. Stoffregen, and I. Zemke.
- Supplements
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Front Courtyard
- Video
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Nov 16, 2022
Wednesday
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09:30 AM - 10:30 AM
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Generalized Holomorphic Curve Counting
Mark McLean (State University of New York, Stony Brook)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
We explain how one can define Gromov Witten invariants of any genus with respect to any generalized cohomology theory in which Chern classes as well as fundamental classes of complex orbifolds make sense.
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Front Courtyard
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Homology Concordance and Knot Floer Homology
Linh Truong (University of Michigan)
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
This talk will focus on the homology concordance group of knots in integer homology 3-spheres bounding integer homology 4-spheres. Using knot Floer homology, we will construct integer-valued homomorphisms from the homology concordance group. This leads to the existence of an infinite rank summand of the quotient group of the homology concordance group modded out by knots in the 3-sphere. This is joint work with Irving Dai, Jennifer Hom, and Matthew Stoffregen.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: 2nd Floor Deck
- Video
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02:00 PM - 03:00 PM
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Stabilizations, Satellites, and Exotic Surfaces
Gary Guth (University of Oregon)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
A long standing question in the study of exotic behavior in dimension four is whether exotic behavior is “stable". For example, in thinking about the four-dimensional h-cobordism theorem, Wall proved that simply connected, exotic four-manifolds always become smoothly equivalent after applying a suitable stabilization operation enough times. Similarly, Hosokawa-Kawauchi and Baykur-Sunukjian showed that exotic surfaces become smoothly equivalent after stabilizing the surfaces some number of times. The question remains, how many stabilizations are necessary, and is one always enough? By considering certain satellite operations, we provide a negative answer to this question in the case of exotic surfaces with boundary. (This draws on joint work with Hayden, Kang, and Park).
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
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- Abstract
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- Supplements
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Nov 17, 2022
Thursday
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09:30 AM - 10:30 AM
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A Knot Floer Stable Homotopy Type
Ciprian Manolescu (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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Given a grid diagram for a knot or link K in the three-sphere, we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Front Courtyard
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Computations of Ribbon Concordances
Sherry Gong (Texas A & M University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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We discuss computations and data regarding ribbon concordances of knots of up to 19 crossings.
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: 2nd Floor Deck
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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One Stabilization is Not Enough for Contractible 4-Manifolds
Sungkyung Kang (Institute for basic science)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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We construct an example of a cork that remains exotic after taking a connected sum with $S^2 \times S^2$. Combined with a work of Akbulut-Ruberman, this implies the existence of an exotic pair of contractible 4-manifolds which remains absolutely exotic after taking a connected sum with $S^2 \times S^2$.
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
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03:30 PM - 04:30 PM
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Annular Links, Double Branched Covers, and Annular Khovanov Homology
Gage Martin (Harvard University; Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.
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Nov 18, 2022
Friday
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09:30 AM - 10:30 AM
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Homogenization of Knot Invariants and Slice-Bennequin Inequalities
Peter Feller (ETH Zurich)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
We will discuss how certain knot invariants can be homogenized to yield invariants for braids. The idea of this type of homogenization has been around for decades, for example in pioneering work by Gambaudo-Ghys on Levine-Tristram signatures.
By focusing on knot invariants stemming from Heegaard Floer theory, we find a new inequality relating the fractional Dehn twist coefficient with the 4-genus of knots, answering a question of Hubbard-Kawamuro-Ceren Kose-Martin-Plamenevskaya-Raoux-Truong-Turner.
In fact, with the tool of homogenization, this new inequality is readily seen to be one of a family of inequalities, the most famous of which is the slice Bennequin inequality established by Kronheimer-Mrowka and Rudolph.
The meta goal of the talk is to invite the audience to consider what homogenization yields for their favorite (Floer homotopy) invariants.
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Front Courtyard
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Equivariant Khovanov Stable Homotopy Types and their Applications
Melissa Zhang (University of California, Davis)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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A few years ago, Matt Stoffregen and I constructed equivariant Khovanov spectra for p-periodic links using Lawson, Lipshitz, and Sarkar's Khovanov stable homotopy type. In this talk, I will sketch our construction and then discuss applications of these spectra in low-dimensional topology. If time permits, I will also suggest further research problems that may require the use of Khovanov spectra or Manolescu-Sarkar's knot Floer spectra.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: 2nd Floor Deck
- Video
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- Abstract
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- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
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- Video
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- Abstract
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- Supplements
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