Séminaires : Séminaire Dérivé

In 1994, Cohen-Jones-Segal proposed a program to understand the homotopy theory underlying Floer theory. They proposed that a Floer problem should give rise to a “Floer homotopy type”, refining the associated Floer homology. Moreover, they discussed how such a Floer homotopy type might be constructed via the use of flow categories and in particular sketch how to obtain a spectrum from a framed flow category. More recently, Abouzaid-Blumberg show that framed flow categories can be arranged into a stable infinity-category and show that this is equivalent to the infinity-category of spectra.

Far from all flow categories associated with Floer data are frameable, though. That some version of twisted stable homotopy theory is needed to deal with non-framed Floer homotopy theory has been known for a while. Twisted spectra were introduced by Douglas in his PhD thesis and recently recast in the infinity-categorical setting by Hedenlund-Moulinos. In this talk, we explain how these are indeed related to flow categories by exhibiting an equivalence between twisted spectra and flow categories structured by certain maps to U/O. This is joint work with Trygve Poppe Oldervoll.

In this talk, I will first survey well-known connections
between algebraic K-theory and algebraic cycles, including the
coniveau and Atiyah–Hirzebruch spectral sequences, as well as the
classical Riemann–Roch theorem. I will then explain the motivations
behind the current trend of enhancing these results using quadratic
(or more exactly symmetric bilinear) forms, and highlight a few
results in this direction.

A theorem of Deligne guarantees that under some finiteness assumptions, rigid tensor categories over an algebraically closed field admit a fiber functor and are therefore (super-)Tannakian. This, in turn, guarantees that they are relatively close to categories of modules over commutative rings. Beyond the Tannakian case, there is also a general feeling that rigid tensor categories behave "more" like categories of modules over commutative rings than arbitrary tensor categories.

In this talk, I will discuss a K-theoretic failure of this "feeling". More precisely, I will give examples to show that the K-theory of rigid tensor categories lacks one key structural property of the K-theory of commutative rings, by exhibiting failures of the so-called redshift principle (which holds for the K-theory of commutative rings). In the first half of the talk, I will focus on describing the context and discuss examples based on Deligne's category Rep(GL_t), and in the second half, I will discuss a general result that fully computes the K-theory of certain "algebraically closed" rigid tensor categories.

(Zero-dimensional) Donaldson-Thomas-invariants "count" things like ideal sheaves of a given length which have zero-dimensional support on a smooth projective complex threefold. Maulik, Nekrasov, Okounkov and Pandharipande have proven a formula for the generating series of these Donaldson-Thomas invariants in terms of the MacMahon function in the toric case.
We discuss a conjectural quadratically enriched analogue of this result for smooth projective real threefolds satisfying an orientation condition, using a quadratic version of Donaldson-Thomas invariants taking values in Witt rings which are constructed using work of Levine. We provide evidence for the conjecture coming from computations for  and .
This talk is based on my thesis and on joint work with Marc Levine.

The Yoga of motives is born with étale cohomology, as a conceptual way to explain elusive integral aspects of l-adic cohomologies. This has lead to deep independence of l problems that have found positive answers - from Deligne's proof of the Weil conjectures to the proof of Deligne's conjecture on companions over smooth algebraic varieties (Lafforgue, Drinfeld, Esnault and Kerz). The Tate conjecture, together with its variations due to Beilinson and Lichtenbaum remains open though. On the other hand, motives have become a full fledged theory that lead to the proof of the Bloch-Kato conjecture by Rost and Voevodsky. We will formulate categorified version of independence of l conjectures, in the language of motivic sheaves. To our knowledge, they are not equivalent to any of the classical conjectures, but interesting relations can be established. For instance, they follow from the Tate-Beilinson conjecture and imply Deligne's conjecture on companions in full generality (over normal algebraic varieties).

The Ravenel--Wilson Hopf ring is an algebraic object which describes the homology of the spaces comprising the spectrum MU. It is characterized by a universal property in a 1-category. I will report on joint work with M. Hopkins, in which we (a) show that (a slight variant of) the Ravenel--Wilson Hopf ring satisfies a related universal property in an oo-category, and (2) can be used to described the motivic homology of the spaces comprising MGL.

Quillen discovered a correspondence between cohomology theories and formal groups. While Quillen's correspondence is tight enough to be extremely successful in transporting phenomena back and forth, it is not one-to-one: some cohomology theories are missed, as are some formal groups. I will explain how to turn Quillen's correspondence into a one-to-one correspondence by changing the definition of a cohomology theory. From another perspective, this gives a functor-of-points description of Lurie's "derived moduli stack of formal groups", which he specified via charts. This is joint work with Robert Burklund and Ishan Levy.

The goal of this talk is to explain work joint with Scholze where we study a version of algebraic geometry which is built, not out of spectra of commutative rings, but out of spectra of symmetric monoidal higher categories. Unlike traditional algebraic geometry, where the category of affine schemes does not have well behaved gluings, our setup provides an (infinity) topos where every object is, in a sense, affine. This topos contains the usual category of qcqs schemes, but also provides a home to new and interesting objects which cannot be studied with more classical means. We will encounter some of these objects in this talk, with relevance in topological field theory, the theory of motives, and the geometric Langlands program.