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.

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.

Geometry and Topology

Speaker: Steven Amelotte (Rochester)

"Cohomology operations for moment-angle complexes"

Time: 15:30

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

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

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.

Geometry and Combinatorics

Speaker: Botong Wang (University of Wisconsin (Madison))

"The Hodge theory of hyperplane arrangements and matroids"

Time: 15:30

Room: Zoom

Given a hyperplane arrangement, we associate two projective varieties: the wonderful compactification and the matroid Schubert variety. Adiprasito, Huh and Katz used the Hodge-Riemann relations of the wonderful compactification (and their combinatorial generalizations) to prove that the coefficients of their characteristic polynomials form a log-concave sequence. In a joint work with Huh, we proved Dowling and Wilson's Top-heavy conjecture for realizable matroids by applying the hard Lefschetz theorem to the matroid Schubert varieties. In a more recent work with Braden, Huh, Matherne and Proudfoot, we proved the Top-heavy conjecture to arbitrary matroids. In this talk, I will go over some of the key ideas about the proof of the Top-heavy conjectures. If time permits, I will also mention some on-going works with my students Colin Crowley and Connor Simpson towards generalizations to type A Coxeter matroids.

Geometry and Topology

Speaker: Viktoriya Ozornova (Ruhr-Universitaet Bochum)

"Pasting diagrams in $(\infty,2)$-categories"

Time: 11:30

Room: Zoom Meeting ID: 958 6908 4555

In the world of $(\infty,1)$-categories, it is well-known that the composition of specified morphisms is well-defined up to a contractible choice. The situation for $(\infty,2)$-categories is more subtle, as there are many potential ways of composing 2-cells. In a joint work in progress with Hackney, Riehl, Rovelli, we prove a uniqueness statement for the composition of so-called pasting diagrams, which I will explain during the talk.

Algebra Seminar

Speaker: Tung T. Nguyen (University of Chicago)

"Power sums and special values of L-functions"

Time: 14:30

Room: Zoom

The zeta functions are a pillar of number theory. Zeta functions have been objects of great interest for number theorists due to their beauty, mystery, and power. I will discuss this study in the simplest case: the Hurwitz zeta functions. Recently, some surprising direct connections between the special values of Hurwitz zeta functions and power sums were found. In my talk, I will introduce these discoveries. In particular, we will see that special values of Hurwitz zeta functions have some nice integral representations. This is joint work with Jan Minac and Nguyen Duy Tan.