Geometry and Combinatorics

Speaker: Alex Suciu (Northeastern University)

"Sigma-invariants and tropical geometry"

Time: 14:30

Room: Zoom

The Bieri--Neumann--Strebel--Renz invariants $\Sigma^q(X)$ of a connected, finite-type CW-complex $X$ are the vanishing loci for the Novikov--Sikorav homology of $X$ in degrees up to $q$. These invariants live in the unit sphere inside $H^1(X,\mathbb{R})$; this sphere can be thought of as parametrizing all free abelian covers of $X$, while the $\Sigma$-invariants keep track of the geometric finiteness properties of those covers. On the other hand, the characteristic varieties $V^q(X) \subset H^1(X,\mathbb{C}^{*})$ are the non-vanishing loci in degree $q$ for homology with coefficients in rank $1$ local systems. After explaining these notions and providing motivation, I will describe a rather surprising connection between these objects, to wit: each BNSR invariant $\Sigma^q(X)$ is contained in the complement of the tropicalization of $V^{\le q}(X)$. I will conclude with some examples and applications pertaining to complex geometry, group theory, and low-dimensional topology.

Geometry and Topology

Speaker: Elden Elmanto (Harvard University)

"A Dundas-Goodwillie-McCarthy Theorem for Algebraic Stacks"

Time: 15:30

Room: Zoom Meeting ID: 958 6908 4555

The Dundas-Goodwillie-McCarthy (DGM) theorem asserts that the difference between the K-theory of a ring and its thickening is the same as the difference in topological cyclic homology (TC). This has had spectacular applications in computations for K-theory as well as inspiring recent developments in p-adic Hodge theory. According to Bondal and van den Bergh, the category of perfect complexes on a qcqs scheme is equivalent to the category of perfect complexes on a $A_{\infty}$-ring. Therefore, the DGM theorem is applicable in this geometric context.

However, the category of perfect complexes on algebraic stacks do not enjoy this "monogenic generation" property. In joint work with Vova Sosnilo. we proved a version of the DGM theorem where this is applicable in many cases. I will explain how the proof works, whose new input comes from Bondarko's theory of weights (aka co-t-structures), and also some applications.

Algebra Seminar

Speaker: Pal Zsamboki (Renyi Institute)

"A homotopical Skolem-Noether theorem"

Time: 13:30

Room: Zoom: 998 5635 1219

Joint work with Ajneet Dhillon. See arXiv:2007.14327 [math.AG]. The classical Skolem--Noether Theorem by Giraud shows us (1) how we can assign to an Azumaya algebra A on a scheme X a cohomological Brauer class in H^2(X,G_m) and (2) how Azumaya algebras correspond to twisted vector bundles. The Derived Skolem--Noether Theorem by Lieblich generalizes this result to weak algebras in the derived 1-category locally quasi-isomorphic to derived endomorphism algebras of perfect complexes. We show that in general for a co-family of presentable monoidal quasi-categories with descent over a quasi-category with a Grothendieck topology, there is a fibre sequence giving in particular the above correspondences. For a totally supported perfect complex E over a quasi-compact and quasi-separated scheme X, the long exact sequence on homotopy group sheaves splits giving equalities pi_i(Aut Perf E,id_E)=pi_i(Aut_Alg Perf REnd E, id_REnd E) for i >= 1. Further applications include complexes in Derived Algebraic Geometry, module spectra in Spectral Algebraic Geometry and ind-coherent sheaves and crystals in Derived Algebraic Geometry in characteristic 0.

Geometry and Combinatorics

Speaker: Michael Kaminski (Purdue University)

"A Syzygetic Approach to Resonance Varieties"

Time: 14:30

Room: Zoom

For a complex hyperplane arrangement, the cohomology ring of the complement depends only on the combinatorics of the arrangement, and this cohomology ring has an explicit expression as the quotient of an exterior algebra E. One may study such rings by studying their resonance varieties, collections of points corresponding to (nontrivial) zero divisors. Viewing these cohomology rings as modules over E, the Chen Ranks Theorem expresses some of the graded Betti numbers of the ring in terms of the first resonance variety. Inspired by this result, I will define a collection of varieties consisting of the points in the resonance varieties that "contribute to the Betti numbers," and state a theorem I proved computing these for all square-free E-modules. I will conclude with an application of the result to cohomology rings of hyperplane arrangements.

Geometry and Topology

Speaker: Joel Villatoro (KU Leuven)

"Diffeological Vector Bundles vs Sheaves of Modules"

Time: 11:30

Room: Zoom Meeting ID: 958 6908 4555

In this talk I will give a quick overview on both diffeology, and sheaves of modules on smooth manifolds. Both diffeological vector bundles and sheaves of modules, can be thought of as generalizations of the notion of a vector bundle on a manifold. At first glance, the two approaches seem quite different, but I will show that one can construct a natural functor from the category of sheaves of modules to the category of diffeological vector bundles. We will then see that, under some fairly mild assumptions, this functor is actually an equivalence of categories. In the last part of the talk I will give some examples where understanding this correspondence has proven to be useful.

Algebra Seminar

Speaker: Ahmed Ashraf (Western)

"Chromatic symmetric function and star basis"

Time: 13:30

Room: Zoom

We introduce Stanley's chromatic symmetric function (CSF) for a simple graph, which is a generalization of the chromatic polynomial. Stanley gave the expansion of CSF in various bases of space of symmetric functions, and asked whether a tree is determined by its respective CSF. We describe a four term relation satisfied by CSF and give a recursive formula to expand it in star bases.

Geometry and Topology

Speaker: Yuki Maehara (Macquarie University)

"Triangulating weak $\omega$-categories into weak $\omega$-categories"

Time: 19:00

Room: Zoom Meeting ID: 958 6908 4555

Tim Campion, Chris Kapulkin and I recently proposed certain (marked) cubical sets as a model for weak $\omega$-categories (a.k.a. $(\infty,\infty)$-categories). This model is a sort of cubical adaptation of Verity's complicial sets (which are simplicial), and we made the analogy more precise (than a mere resemblance of definitions) by proving the triangulation functor to be left Quillen and strong monoidal with respect to both lax and pseudo Gray tensor products. In this talk, I will define these models, explain the intuition behind the definitions, and give a flavour of the sort of combinatorics that went into the proofs.

Geometry and Topology

Speaker: Karol Szumilo (University of Leeds)

"$\infty$-groupoids in lextensive categories"

Time: 11:30

Room: Zoom Meeting ID: 958 6908 4555

I will discuss a construction of a new model structure on simplicial objects in a countably extensive category (i.e., a category with well behaved finite limits and countable coproducts). This builds on previous work on a constructive model structure on simplicial sets, originally motivated by modelling Homotopy Type Theory, but now applicable in a much wider context. This is joint work with Nicola Gambino, Simon Henry and Christian Sattler.

Geometry and Combinatorics

Speaker: Ivan Limonchenko (University of Toronto)

"On Koszul homology of face rings and toric topology"

Time: 14:30

Room: MC 108

In this talk we will discuss the relation between homological properties of Stanley--Reisner rings of simplicial complexes and topology of polyhedral products. A key result in this direction is the characterization of Golod rings over rationals in terms of their Poincaré series and loop homology of the corresponding moment-angle-complexes. Much more can be said if only flag simplicial complexes are considered. We will see how the methods and objects of toric topology allow us to interpret the results on Poincaré series and Koszul homology of face rings as well as to get new results.

Geometry and Topology

Speaker: Bob Lutz (MSRI Berkeley)

"Teaching a new dog old tricks: Classical topology theorems in the discrete setting"

Time: 15:30

Room: Zoom Meeting ID: 958 6908 4555

An exciting theme in combinatorics is degenerating a continuous theory into a discrete one and asking which features of the original are preserved. This talk will focus on our effort to replicate classical theorems of topology in the setting of discrete homotopy and singular homology theories for graphs. These combinatorial theories have a distinct cubical flavor, with the roles of spheres and simplices played by grids and hypercube graphs. A major goal has been to connect the two by way of a discrete Hurewicz theorem. Our first result marks progress toward this goal: We will describe a natural map from discrete homotopy to discrete homology, and show that it is surjective in a large number of cases. As a corollary, we prove the existence of nontrivial higher discrete homotopy groups. Our second result is a discrete version of a theorem of P. A. Smith, which says that the fundamental group of a nontrivial symmetric product of $X$ is isomorphic to the first homology group of $X$.

Algebra Seminar

Speaker: Alexander Neshitov (Western)

"Torsion in codimension 2 Chow groups of classifying spaces of algebraic tori"

Time: 13:30

Room: Zoom: 998 5635 1219

Chow groups of classifying spaces of tori arise in the theory of cohomological invariants. In this talk we will give an overview of a computer-assisted proof that \({\rm CH}^2\) groups of the classifying space \(BT\) are torsion free when \({\rm dim}(T) <= 5\)