Schedule for: 22w5088 - Rank Conjectures in Algebraic Topology and Commutative Algebra

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We will review basic facts about transformation groups and the resulting restrictions on groups that can act freely on a finite CW-complex. Cohomological methods will be introduced and geometric examples will be provided. We will consider the relationship between the rank of the group and the cohomology of a space on which it acts freely. We will review the work of Carlsson, Browder and others on this problem, motivated by the case of a product of spheres.

Rational homotopy theory. Sullivan-de Rham theorem, small cochain models via Postnikov decomposition, examples, cochain models for torus actions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We will cover the following topics: 1. Motivation and Basic Definitions of Hilbert functions, Free resolutions, and Betti numbers. 2. Formulas. 3. Upper Bounds. 4. Lower Bounds. 5. The Buchsbaum–Eisenbud–Horrocks Conjecture. 6. A related conjecture about Ext. 7. Other problems about resolutions in the artinian case. 8. Conjectures about Hilbert functions in the artinian case.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

This is an online working group, for participants signing in from Europe and further east.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we will consider the problem of constructing group actions with prescribed isotropy on a finite complex with a fixed homotopy type. This will involve methods from representation theory and homotopy theory. For rank one groups this builds on the classical characterization of groups acting freely on spheres, leading to a generalized notion of cohomological periodicity with a corresponding geometric characterization. We will discuss extensions to groups of higher rank.

Tame homotopy theory. Cenkl-Porter theorem, tame Hirsch lemma, cochain models for p-torus actions, the stable free rank of symmetry of products of spheres.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

This is an online working group, for participants signing in from Europe and further east.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will first discuss the ``Total Rank Conjecture" in commutative algebra, which is a weak form of the famed Buchsbaum-Eisenbud-Horrocks Conjecture regarding lower bounds on the Betti numbers of modules of finite projective dimension. The discussion will include both the proof of the Total Rank Conjecture in some cases (due to myself) and counter-examples to a generalized version of it (due to S. Iyengar and myself). I will explain the connection of these results with the Toral Rank Conjecture in topology, and, in particular, I will discuss why the positive results in algebra fail to prove the Toral Rank Conjecture, and why the counter-examples do not disprove it. The work of Iyengar and myself is also related to a conjecture of Carlsson regarding free actions of elementary abelian p-groups on topological spaces; specifically, we produce counter-examples to a purely algebraic generalization of this conjecture. I will discuss these examples and why, as before, they don't disprove Carlsson's original conjecture.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A long-standing Rank Conjecture states that if an elementary abelian $p$-group acts freely on a product of spheres, then the rank of the group is at most the number of spheres in the product. In this talk, we will discuss the algebraic version of the Rank Conjecture given by Carlsson for a differential graded module $M$ over a polynomial ring. Then we consider the varieties of square-zero upper triangular matrices corresponding to the differentials of such modules. By stratifying these varieties via Borel orbits and imposing conditions coming from the algebraic conjecture, we state a stronger conjecture about varieties of matrices. Using the corresponding free flag construction, we show that $(\mathbb{Z}/2\mathbb{Z})^4$ cannot act freely on a product of $3$ spheres of any dimensions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Syzygies are a notion from commutative algebra that interpolates between torsion-free, reflexive and free modules. We discuss an application of syzygies to equivariant cohomology. Let $X$ be a ``nice'' space with an action of a torus $T$ of rank $n$, and take cohomology with rational coefficients. By a result of Chang-Skjelbred, the sequence $$0 \to H_T^*(X) \to H_T^*(X^T) \to H_T^{*+1}(X_1,X^{T})$$ is exact if $H_T^*(X)$ is a free module over $R=H^*(BT)$. Here $X_1$ denotes the orbits of dimension at most 1. This significantly simplifies the computation of $H_T^*(X)$ and forms the basis of the so-called GKM method. The CS sequence above can be extended to the Atiyah-Bredon sequence $$\begin{multline*} 0 \to H_T^*(X) \to H_T^*(X_{0}) \to H_T^{*+1}(X_{1},X_{0}) \to H_T^{*+2}(X_{2},X_{1}) \to \\ \cdots \to H_T^{*+n-1}(X_{n-1},X_{n-2}) \to H_T^{*+n}(X_{n},X_{n-1}) \to 0, \end{multline*}$$ where $X_k\subset X$ denotes the subset of orbits of dimension at most $k$. It turns out that the AB sequence is exact at the first $k$ terms if and only if $H_T^*(X)$ is a $k$-th syzygy over $R$. In particular, the CS sequence is exact if and only if $H_T^*(X)$ is a reflexive $R$-module. If $X$ satisfies Poincar\'e duality, then this is also equivalent to the perfection of the equivariant Poincar\'e pairing. (This is joint work with Chris Allday and Volker Puppe.)

I’ll discuss the Buchweitz-Greuel-Schreyer Conjecture on the minimal rank of a matrix factorization, and my recent proof that the conjecture holds for generic polynomials.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

This is an online working group, for participants signing in from Europe and further east.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will report on recent results with Erg\"un Yalcin and Marc Stephan providing obstructions to the existence of free $A_4$ actions on a product of two spheres. For such an action we can look at the map in cohomology induced by the classifying map. Its kernel is a Steenrod closed parameter ideal in $H^*(BA_4)$. We provide a full classification of these ideals. It turns out that these ideals are sparse. While the question of which ideals can be realized this way is interesting, much less is known there.

Work on infinite resolutions beyond the cases of complete intersections and Golod rings has tended to focus on the sequence of Betti numbers. Hai Long Dao and I have recently begun to study a question of a different kind, and I will report on this joint work: Let R = S/I be an artinian quotient of a regular local ring S, with residue field k. When does it happen that k is a direct summand of a syzygy module in the R-free resolution of k, or indeed in the R-free resolution of every module? We were surprised by what we found experimentally, and were able to prove a little of what we observed.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.