Geometry and Combinatorics

Speaker: no talk this week

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Time: 14:30

Room: MC 108

Geometry and Topology

Speaker: Angelica Osorno (Reed College)

"Transfer systems and weak factorization systems"

Time: 15:30

Room: Zoom Meeting ID: 958 6908 4555

$N_\infty$-operads over a group $G$ encode homotopy commutative operations together with a class of equivariant transfer (or norm) maps. Their homotopy theory is given by transfer systems, which are certain discrete objects that have a rich combinatorial structure defined in terms of the subgroup lattice of $G$. In this talk, we will show that when $G$ is finite Abelian, transfer systems are in bijection with weak factorization systems on the poset category of subgroups of $G$. This leads to an involution on the lattice of transfer systems, generalizing the work of Balchin-Bearup-Pech-Roitzheim for cyclic groups of squarefree order. We will conclude with an enumeration of saturated transfer systems and comments on the Rubin and Blumberg-Hill saturation conjecture. This is joint work with Evan Franchere, Usman Hafeez, Peter Marcus, Kyle Ormsby, Weihang Qin, and Riley Waugh.

Colloquium

Speaker: Kiumars Kaveh (University of Pittsburgh)

" Vector bundles, valuations and piecewise linear functions"

Time: 15:30

Room: Online via zoom

The talk explores an interplay between three concepts from different areas of algebra and geometry: vector bundles from geometry and topology, valuations from commutative algebra and piecewise linear functions from convex geometry. A "vector bundle" over a geometric space X (such as a manifold) is, roughly speaking, an assignment of vector spaces to each point in X. Vector bundles are a central object of study in geometry and topology. We introduce the notion of a valuation with values in piecewise linear functions and see that these are the right gadgets to classify (equivariant) vector bundles on so-called "toric varieties". Examples include classification of all (equivariant) vector bundles on a projective space. This can be regarded as a reformulation of Klyachko's famous classification of toric vector bundles. This point of view leads to far reaching extensions which I will touch on if there is time. This is joint work with Chris Manon.

Algebra Seminar

Speaker: Owen Barrett (University of Chicago)

"The derived category of the abelian category of constructible sheaves"

Time: 14:30

Room: Zoom

Nori proved in 2002 that given a complex algebraic variety $X$, the bounded derived category of the abelian category of constructible sheaves on $X$ is equivalent to the usual triangulated category $D(X)$ of bounded constructible complexes on $X$. He moreover showed that given any constructible sheaf ${\mathcal F}$ on ${\mathbb A}^n$, there is an injection ${\mathcal F}\hookrightarrow {\mathcal G}$ with ${\mathcal G}$ constructible and ${\rm H}^i({\mathbb A}^n, {\mathcal G})=0$ for $i>0$.

In this talk, I'll discuss how to extend Nori's theorem to the case of a variety over an algebraically closed field of positive characteristic, with Betti constructible sheaves replaced by $\ell$-adic sheaves. This is the case $p=0$ of the general problem which asks whether the bounded derived category of $p$-perverse sheaves is equivalent to $D(X)$,resolved affirmatively for the middle perversity by Beilinson.