topology@ub

Monday 19 March 2018, 12:00, IMUB

Abstract: We will recall the definition of dendroidal sets as a generalization of simplicial sets, and present the connection (Quillen equivalence) to connective spectra, which gives a factorization of the so-called K-theory spectrum functor from symmetric monoidal categories to spectra. We will present a common generalization of two results of Thomason: 1) posets model all homotopy types; 2) symmetric monoidal categories model all connective spectra. We will introduce a notion of multiposets (special type of coloured operads) and of the subdivision of dendroidal sets which can be used to show that multiposets model all connective spectra. If time permits we will mention homology of dendroidal sets as it provides a means to define equivalences of multiposets in an internal combinatorial way.

Friday 25 May 2018, 10:00, Room T2

Abstract: Classical obstruction theory studies the extensions of a continuous map along a relative CW-complex in terms of cohomology with local coefficients. In this talk, I will describe a similar obstruction theory for (∞,1)- and (∞, 2)-categories, using cohomology with coefficients in local systems over the twisted arrow category and the `twisted 2-cell category'. As an application, I will give an obstruction-theoretic argument that shows that adjunctions can be made homotopy coherent (as proven by Riehl–Verity). This is joint work with Yonatan Harpaz and Matan Prasma.

Monday 28 May 2018, 12:00, IMUB

Abstract: In 2000 Khovanov associated to each diagram of a knot a familily of chain complexes whose quasi-isomorphism type is a knot invariant. Five years ago Lipschitz and Sarkar found that this knot invariant can be refined to a family of cellular spectra whose cellular cochain complexes are the Khovanov complexes. These spectra can be also realised as the suspension spectrum of a certain homotopy colimit, and in this talk we will show that this homotopy colimit can sometimes be computed as the realisation of a semi-simplicial set. This is joint work with Marithania Silvero.

Monday 4 June 2018, 12:00, IMUB

Abstract: Orientable manifolds have even Euler characteristic unless the dimension is a multiple of 4. I give a generalisation of this theorem: k-orientable manifolds have even Euler characteristic (and in fact vanishing top Wu class), unless their dimension is 2^(k+1)m for some integer m. Here we call a manifold k-orientable if the i-th Stiefel–Whitney class vanishes for 0 < i < 2^k. This theorem is strict for k = 0, 1, 2, 3, but whether there exist 4-orientable manifolds with an odd Euler characteristic is a new open question. Such manifolds would have dimensions that are multiples of 32. I discuss manifolds of dimension high powers of 2 and present the results of calculations on the cohomology of the second Rosenfeld plane, a special 64-dimensional manifold with odd Euler characteristic.

Monday 11 June 2018, 12:00, IMUB

Abstract: Intersection cohomology has become an important tool in algebraic geometry and geometric representation theory. This theory has its roots in algebraic topology has a powerful gadget to restore Poincaré duality for spaces with singularities and extend the theory of characteritic classes from manifolds to singular spaces. A series of foundational questions remain open since its introduction in the beginning of the 1980's by M. Goresky and R. MacPherson. In particular, they asked for a homotopical treatment of this theory. In this talk I plan to present a homotopical treatment based on joint works with M. Saralegui (Lens) and D. Tanré (Lille) and of recent results of S. Douteau (Amiens). As a byproduct, I will explain that intersection cohomology is representable by a spectrum in a stable homotopy category of stratified simplicial sets, a kind of stratified Eilenberg-Mac Lane spectrum. This answers positively to a problem of Goresky and MacPherson on the stratified homotopic representability of intersection cohomology, and could lead in the future to the construction of generalized intersection cohomology theories.

Abstract: We give a simple combinatorial model for plethystic substitution. Precisely, the plethystic bialgebra is realised as the homotopy cardinality of the incidence bialgebra of a certain simplicial groupoid, obtained from surjections by a construction reminiscent of the Waldhausen S and the Quillen Q-construction.

Abstract: I will outline the history of the Kauffman Bracket Skein Algebras from Summer 1985 and L. Kauffman's discovery of his bracket, via my definition of the Kauffman bracket skein module in April 1987, my work with J. Hoste, D. Bullock, and A. Sikora (coordinate ring of SL₂(C) character variety) till the recent work of R. Bakshi, S. Mukherjee, M. Silvero, and X. Wang on product to sum formulas. I will discuss positivity conjectures by D. Thurston, H. Queffelec, and P. Wedrich (motivated by E. Witten). As an example I will show that in a thickened four-holed sphere the curve (fraction) products, in Chebyshev basis, 1/n and 1/0 and 0/n with 2/0 are positive. We give explicit formulas for structure constants in these cases.