Geometry and Combinatorics

Speaker: Chris Eur (Stanford University)

"Tautological bundles of matroids"

Time: 14:30

Room: internet

Matroid theory has seen fruitful developments arising from different algebro-geometric approaches, such as the K-theory of Grassmannians and Chow rings of wonderful compactifications. However, these developments have remained somewhat disjoint. We introduce "tautological bundles of matroids" as a new geometric framework for studying matroids. We show that it unifies, recovers, and extends much of these recent developments including log-concavity statements, as well as answering some open conjectures. This is an on-going work with Andrew Berget, Hunter Spink, and Dennis Tseng.

Geometry and Topology

Speaker: Dan Christensen (Western)

"No set of spaces detects isomorphisms in the homotopy category"

Time: 15:30

Room: Zoom Meeting ID: 958 6908 4555

Whitehead's theorem says that a map of pointed, connected CW complexes is a homotopy equivalence if and only if it induces an isomorphism on homotopy groups.

In the unpointed setting, one can ask whether there is a set $\mathcal{S}$ of spaces such that a map $f : X \to Y$ between connected CW complexes is a homotopy equivalence if and only if it induces bijections $[A, X] \to [A, Y]$ for all $A$ in $\mathcal{S}$. Heller claimed that there is no such set $\mathcal{S}$, but his argument relied on an "obvious" statement about weak colimits in the homotopy category of spaces. We show that this obvious statement is false, thus reopening the question above. We then show that Heller was in fact correct that no such set $\mathcal{S}$ exists, using a different, more direct method.

This is joint work with Kevin Arlin; see arXiv:1910.04141.

Algebra Seminar

Speaker: Zinovy Reichstein (UBC)

" On the minimal number of generators of an algebra over a commutative ring."

Time: 14:30

Room: Zoom: 978 8611 6423 (passcode required, check email)

Let R be a commutative ring of Krull dimension d. A 1964 theorem of Forster asserts that every projective R-module of rank n can be generated by d+n elements. Chase and Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than d+n elements. We view projective R-modules as R-forms of the non-unital R-algebra where the product of any two elements is 0. A few years ago Uriya First and I generalized Forster's theorem to forms of other algebras (not necessarily commutative, associative or unital). For example, every etale algebra over R can be generated by d + 1 elements, every Azumaya algebra can be generated by d + 2 elements, every octonion algebra by d + 3 elements. Abhishek Shukla and Ben Williams then showed that this generalized Forster bound is optimal for etale algebras. In this talk, based on joint work with First and Williams, I will address the following question: Is the Forster bound optimal for other types of algebras?