Schedule, Notes/Handouts & Videos

Stable infinity-categories provide a framework to study derived categories via universal properties, in contrast to the more traditional approach via the axioms of triangulated categories. In this series, we will give an introduction to this perspective, explain basic illustrative results, and conclude with a discussion of the enduring question as to what kind of categorical structure derived categories should form (motivated by an ongoing attempt to implement categorified variants of homological algebra).

Dg-categories have taken centre stage as models for spaces in noncommutative algebraic geometry through the work of Van den Bergh and others. From a homotopical viewpoint however, dg-categories are "too strict". For example, they possess a model structure and a monoidal structure, but these are not compatible. In higher category theory, a similar problem arises with simplicial categories, and this is resolved by turning to a Quillen-equivalent model of infinity-categories such as Joyal's quasi-categories.

In this talk I will introduce a candidate model for non-negatively graded dg-categories which is analogous to that of quasi-categories. We call them "quasi-categories in vectorspaces". Equipping these with some extra structure, they are (on the nose) equivalent to non-negatively graded dg-categories, and the comparison goes through a lift of the dg-nerve. At present, no analogue of Joyal's model structure exists for quasi-categories in vectorspaces, but I will outline some partial results in that direction.

The Rouquier dimension of a toric variety is shown to be achieved by the Frobenius pushforward of O via homological mirror symmetry. From the perspective of noncommutative geometry, this result leads to a geometric construction of the corresponding toric NCR of the invariant ring of the Cox ring with respect to a multi-grading which also gives the information about its global dimension. From the perspective of algebraic geometry, the same construction provides a universal ``wall skeleton'' capturing VGIT wall-crossings, which contains a window for each chamber as a full subcategory. From the perspective of commutative algebra, the same construction indicates the existence of virtual resolutions of the multigraded diagonal bimodule, which agrees with a recent result of Hanlon-Hicks-Larzarev constructing one such resolution explicitly. In this talk, I will explain how these fit together. This is a joint work in progress with D. Favero.

It has become accepted wisdom by now that when you localize a category with respect to a class of morphisms, what you get is not just a category but a category "with a homotopical enhancement".  Typically, the latter is made precise through the machinery of "infinity-categories", or "quasicategories", but this is quite heavy technically and not really optimal from the conceptual point of view.  I am going to sketch an alternative technique based on Grothendieck's idea of a "derivator".

Stable infinity-categories provide a framework to study derived categories via universal properties, in contrast to the more traditional approach via the axioms of triangulated categories. In this series, we will give an introduction to this perspective, explain basic illustrative results, and conclude with a discussion of the enduring question as to what kind of categorical structure derived categories should form (motivated by an ongoing attempt to implement categorified variants of homological algebra).

We define epsilon-stable LG quasimaps and their moduli spaces based the work of Fan--Jarvis--Ruan. We describe Favero--Kim's construction of general GLSM invariants via virtual matrix factorization; this generalizes Polischuk--Vaintrob's construction for affine LG models and Ciocan-Fontanine--Favero--Guéré--Kim--Shoemaker's construction for convex hybrid models.  

Stable infinity-categories provide a framework to study derived categories via universal properties, in contrast to the more traditional approach via the axioms of triangulated categories. In this series, we will give an introduction to this perspective, explain basic illustrative results, and conclude with a discussion of the enduring question as to what kind of categorical structure derived categories should form (motivated by an ongoing attempt to implement categorified variants of homological algebra).

One of the fundamental ways of understanding an algebraic structure is to find basic elements which can build all others through allowed operations, ie generators. For triangulated categories, where the allowed operation is taking a mapping cone, such generation was introduced by Bondal and Van den Bergh and expanded upon by Rouquier and others. 

Identifying generators, in particular ones that are of an elementary form and generate fast, can be difficult. In this talk, I will discuss how the action of Frobenius in positive characteristic can be used. As an application, for an F-finite Noetherian algebra $R$, $F_\ast^e R$ is a generator for $D^b(\operatorname{mod} R)$ for sufficiently large $e$. 

We will discuss joint work with Pat Lank, Srikanth Iyengar, Josh Pollitz, and Alapan Mukhopadhyay from arXiv:2303.18085.