Topology Seminar
Upcoming Talks
The seminar will meet at 4:30 on Monday in 2-131 unless otherwise noted.
Click here to add this seminar to your google calendar.
If you use a different calendar program, the ics file for this seminar is here:
http://math.mit.edu/topology/topology_seminar.ics
-
Sonja Farr (University of Nevada, Reno)
$\begingroup $In Higher Algebra, J. Lurie developed a theory of derived centers for algebras over $\infty$-operads. Similar to how the classical center of an associative $k$-algebra is a commutative $k$-algebra, the derived center of an $O$-algebra is an $\mathbb{E}_1$-algebra object in the category of $O$-algebras. In the case of $\mathbb{E}_1$-algebras, the Dunn Additivity Theorem thus promotes the derived center to an $\mathbb{E}_2$-algebra. By defining the Hochschild complex of an $\mathbb{E}_1$-algebra object as its derived center, we hence obtain a built-in solution of Deligne's conjecture on Hochschild cochains. We show that for an associative k-algebra, this definition recovers the classical Hochschild complex, including the correct Gerstenhaber algebra structure in cohomology. Globalizing to schemes, we show that the derived $\mathbb{E}_1$-center of the structure sheaf is indeed glued from the local centers, and that for a smooth scheme we recover the sheaf of polydifferential operators. The motivation for this work has its origin in Kontsevich's description of the action of the Grothendieck-Teichmuller group on the Hochschild cohomology of a smooth algebraic variety.
$\endgroup$ -
Adela Zhang (University of Copenhagen)
$\begingroup $We show that any $2D$ open field theory $F$ extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology of the $\mathbb{E}_1$-Frobenius algebra associated to $F$. In particular, the open-closed bordism category is obtained by formally adjoining iterated Hochchild homology to the open $2D$-bordism category. As a corollary, we identify the space of universal formal operations on the Hochschild homology of $\mathbb{E}_1$-Frobenius algebras to be moduli spaces of punctured surfaces. This is joint work with Barkan and Steinebrunner, and provides a space level refinement of work of Costello (over $\mathbb{Q}$) and Wahl (over $\mathbb{Z}$).
In dimension $\infty$, we establish analogous extensions of $\mathbb{E}_{\infty}$-Frobenius algebras, which are corepresented by the $\infty$-category Gr of graphs, to the $\infty$-category GrCob of graph cobordisms between spaces. In this case, we show that GrCob is obtained by formally adjoining to Gr the factorization homology of the universal $\mathbb{E}_{\infty}$-Frobenius algebra over any space $X$. Thus GrCob parametrizes universal operations among such factorization homologies. This is joint work with Andrea Bianchi.
$\endgroup$ -
Florian Riedel (University of Copenhagen)
-
Alexander Petrov (MIT)
$\begingroup $Given a complex algebraic variety, the profinite completion of its homotopy type can be recovered in purely algebraic terms as its etale homotopy type. In particular, if the variety was defined over a subfield F of complex numbers, the absolute Galois group of F naturally acts on this profinite completion. The Galois action on the fundamental group has many shared properties with the action on cohomology, in particular one expects (and in some case can) to associate a motive to the appropriately defined pro-algebraic completion of the fundamental group. On the other hand, the Galois action on higher etale homotopy groups turns out to be inconsistent with them arising from a motive: we will see that for some varieties over $\mathbb{Q}$ the (dual of) second etale homotopy group contains a finite-dimensional subrepresentaiton of the Galois group that cannot come from cohomology of an algebraic variety. This phenomenon arises from the difference between the cohomology of a discrete group and that of its pro-finite completion.
$\endgroup$ -
Natalia Pacheco-Tallaj (MIT)
-
Christian Kremer (MPIM)
Email Jeremy Hahn
for inquiries about the seminar.
The mailing list for this seminar is the MIT topology google group.
Email Mike Hopkins if you want to join the list.