| Speaker: Ryan Budney (U.Vic) |
| Title: The type-2 invariant |
| Abstract: I will give a brief discussion of Vassiliev's work on finite-type invariants, Birman's reformulation and a few geometric descriptions of the type-2 invariant of knots. |
| Location: COR B129, 4pm--5pm |
| Speaker: Ryan Budney (U.Vic) |
| Title: A variation on the type-2 invariant |
| Abstract: The idea behind the type-2 invariant, when extrapolated to higher-dimensional manifolds gives rise to an interesting invariant of mapping-class groups of manifolds. This presentation will survey such results. |
| Location: COR B129, 4pm--5pm |
| Speaker: Andrea Marino |
| Title: Embedding Calculus - Taylor Approximations for 'Geometric' Functors |
| Abstract: The talk aims to give an overview of the Embedded Calculus introduced by Goodwillie and Weiss. It is a powerful technique that can be used, among others, to approximate spaces of knots with configuration spaces. The framework is the following. Suppose X is a presheaf of spaces over a manifold M, and assume the restrictions X(U) → X(V) are homotopy equivalences whenever the inclusion V→U is a (sort of) homotopy equivalence. Can we determine the homotopy type of X(M) from the homotopy type of X on small disks? The answer is yes, whenever its “Taylor Tower” converges! We will see how cubical formalism helps us define the n-th derivatives, polynomial degree, and the Taylor Tower of presheaves. If time permits, we will apply the machinery to the space of knots in dimension 4 and higher. With some (deep) formality results, such a technique yields the rational cohomology of knot spaces. |
| Location: COR B129, 4pm--5pm |
| Speaker: Andrea Marino |
| Title: A combinatorial description of the Goodwillie-Sinha Spectral Sequence in terms of Fox-Neuwirth Trees |
| Abstract: Building on the previous seminar, we will present the Sinha’s description of the Goodwillie-Weiss Tower for Knot Spaces in terms of (Compactified) Configuration Spaces. The (co)homology of the latter admit a combinatorial presentation in terms of the so-called Fox-Neuwirth Trees. It turns out that the Sinha Spectral Sequence in (co)homology can be defined itself in terms of a degree-wise finite-dimensional bicomplex built from Fox-Neuwirth Trees. This equivalence stems from a geometric connection between Configuration Spaces and their Compactifications, through a series of Spaces based on "Weighted Fox-Neuwirth Trees". If time permits, we will show the connections of Sinha Spectral Sequence with Vassiliev Invariants and Formality of Little Disk Operads. This is joint work with Paolo Salvatore. |
| Location: COR B129, 4pm--5pm |
| Speaker: Andrea Marino |
| Title: Non-collapse of Sinha Spectral Sequence mod 2 |
| Abstract: The Sinha Spectral Sequence is a central object in Knot Theory, approximating the cohomology of the space of knots modulo immersions and providing insights into finite-type invariants. Exploiting the Fox-Neuwirth stratification of configuration spaces, we construct a combinatorial basis for this spectral sequence. However, the resulting "Barycentric Fox-Neuwirth" structure is prohibitively large for explicit computations. Using homotopy transfer ideas, we deform the bicomplex into a smaller multicomplex over characteristic 2 and ambient dimensions 2, 3. While the resulting differentials become more intricate, they remain tractable enough to demonstrate that the spectral sequence does not collapse at page 2. This result shows that formality techniques used in the rational case cannot be extended to finite-type invariants mod 2. This is joint work with Paolo Salvatore. |
| Location: COR B129, 4pm--5pm |