Geometry and Combinatorics

Speaker: (no talk this week) (Western)

"~"

Time: 14:30

Room: MC 108

Geometry and Topology

Speaker: Anibal Medina-Mardones (MPI Bonn)

"Chain level Steenrod operations"

Time: 11:30

Room: online

Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-i products; a family of coherent homotopies derived from the broken symmetry of Alexander-Whitney's chain approximation to the diagonal. Later, following a viewpoint developed by Adem, Steenrod defined his homonymous operations for all primes using the homology of symmetric groups. This viewpoint enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at 2. In recent years, thanks to the development of new applications of cohomology -- most notably in Applied Topology and Quantum Field Theory -- having a definition of Steenrod operations that can be effectively computed in specific examples has become a key issue. In this talk, I will review Steenrod's definition of the operations and describe an effective construction of them at every prime.

Algebra Seminar

Speaker: Federico Scavia (UBC)

"Steenrod operations on the de Rham cohomology of algebraic stacks"

Time: 14:30

Room: Zoom : 978 8611 6423 (passcode needed)

Let k be a field. Totaro studied the de Rham cohomology of algebraic stacks over k, and computed it for classifying stacks of linear algebraic k-groups in many cases. Combining previous work of Drury, May and Epstein, I define and study Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field k of characteristic p>0. These operations share many properties with their topological analogues, but there are also important differences. I then determine the Steenrod operations on the de Rham cohomology of linear algebraic k-groups computed by Totaro.

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?

Geometry and Combinatorics

Speaker: (no talk this week) (Western)

"~"

Time: 14:30

Room: MC 108

Geometry and Topology

Speaker: Homotopy Theory Group (Western)

"Special Event"

Time: 15:30

Room: Zoom Meeting ID: 958 6908 4555

Algebra Seminar

Speaker: Olaf Schnuerer (Universitaet Paderborn)

"The inclusion of small modules into big ones: Is it fully faithful on unbounded derived categories"

Time: 14:30

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

Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring R to the unbounded derived category of all modules. We answer the natural question whether this functor defines an equivalence onto the full subcategory of complexes with finitely generated cohomology modules in two special cases. If R is a quasi-Frobenius ring of infinite global dimension, then this functor is not full. If R has finite left global dimension, this functor is an equivalence. We also prove variants of the latter assertion for left coherent rings, for noetherian schemes and for locally noetherian Grothendieck categories.

This is joint work with Leonid Positselski

Geometry and Topology

Speaker: Daniel Schaeppi (Universitaet Regensburg)

"Flat replacements of homology theories"

Time: 11:30

Room: Zoom Meeting ID: 958 6908 4555

A flat homology theory naturally takes values in comodules over a flat Hopf algebroid. In this talk, we will "reverse" this. Starting with a non-flat homology theory, we will construct a new homology theory with values in an abelian category C. Under some conditions, one can show that C is equivalent to the category of comodules of a flat Hopf algebroid. By composing with the forgetful functor to modules, we obtain a new homology theory which is always flat; this is the flat replacement mentioned in the title. The motivating example (due to Piotr Pstragowski) is that complex cobordism is a flat replacement of singular homology with integer coefficients.

Algebra Seminar

Speaker: Luuk Verhoeven (Western)

"A brief introduction to KK-theory."

Time: 15:30

Room: Zoom

Kasparov's KK-theory is a very interesting theory with various applications, such as extensions of $C^*$-algebras and the Novikov conjecture. We will discuss the basic definitions of KK-theory and see how it allows us to recover various properties of K-theory and K-homology such as Bott periodicity, as well as a nice formulation of the Atiyah-Singer index theorem. If time permits, I will also briefly discuss my own interest in KK-theory via the unbounded picture.