Geometry and Combinatorics

Speaker: no talk this week

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Time: 14:30 - 15:30

Room: MC 108

Geometry and Topology

Speaker: Angelica Osorno (Reed College)

"Transfer systems and weak factorization systems"

Time: 15:30 - 16:30

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 - 16:30

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 - 15:30

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.

Geometry and Topology

Speaker: Steven Amelotte (Rochester)

"Cohomology operations for moment-angle complexes"

Time: 15:30 - 16:30

Room: online

Toric topology assigns to each simplicial complex a finite CW-complex, called the moment-angle complex, which comes with a natural torus action. Homotopy invariants of this space recover various homological invariants of Stanley-Reisner rings of interest in combinatorial commutative algebra. In this talk I will describe certain higher cohomology operations induced by the torus action on a moment-angle complex. Focusing on examples, I'll explain how these operations assemble into an explicit Hirsch-Brown model of the torus action and can be used to give a combinatorial description of the minimal free resolution of Stanley-Reisner rings. Time permitting, I will indicate the relevance of these operations to cohomological rigidity problems in toric topology. This talk is based on joint work with Benjamin Briggs.

Geometry and Topology

Speaker: Anna Marie Bohmann (Vanderbilt University)

"Algebraic K-theory for Lawvere theories: assembly and Morita invariance"

Time: 15:30 - 16:30

Room: Zoom Meeting ID: 958 6908 4555

Much like operads and monads, Lawvere theories are a way of encoding algebraic structures, such as those of modules over a ring or sets with a group action. In this talk, we discuss the algebraic K-theory of Lawvere theories, which contains information about automorphism groups of these structures. We'll discuss both particular examples and general constructions in the K-theory of Lawvere theories, including examples showing the limits of Morita invariance and the construction of assembly-style maps. This is joint work with Markus Szymik.

Algebra Seminar

Speaker: Tung T. Nguyen (University of Chicago)

"Heights and Tamagawa numbers of motives"

Time: 14:30 - 15:30

Room: Zoom

The class number formula is an inspiring pillar of number theory. By the work of many mathematicians, notably Deligne, Beilinson, Bloch, Kato, Fontaine, Perrin-Riou, Jannsen, and many others, we now have a quite general (conjectural) class number formulas for motives, i.e., the Tamagawa number conjecture of Bloch-Kato. Recently, Kato has proposed a new approach to this problem using heights of motives. In this talk, we will give an overview of this approach. In particular, we will show a precise relation between heights to Tamagawa numbers of motives. We also partially answer some of Kato's questions about the number of mixed motives of bounded heights in the case of mixed Tate motives.