Geometry and Topology

Speaker: Brandon Doherty (Western)

"Cubical models of $(\infty,1)$-categories (part 1)"

Time: 15:30 - 16:30

Room: Zoom Meeting ID: 958 6908 4555

We introduce the categories of cubical sets and marked cubical sets, and discuss the construction of a new model structure on the category of marked cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to open boxes with certain edges marked. This talk is based on joint work with Chris Kapulkin, Zachery Lindsey, and Christian Sattler.

Geometry and Topology

Speaker: Brandon Doherty (Western)

"Cubical models of $(\infty,1)$-categories (part 2)"

Time: 15:30 - 16:30

Room: Zoom Meeting ID: 958 6908 4555

We introduce a model structure on the category of cubical sets which is constructed using our model structure on marked cubical sets from part 1, and whose fibrant objects are defined by the right lifting property with respect to open boxes with certain edges degenerate. We discuss the proof that this model structure is Quillen-equivalent to the Joyal model structure on simplicial sets. We will also introduce the homotopy category and mapping spaces of a fibrant cubical set, and characterize weak equivalences between fibrant cubical sets in terms of these concepts.

Algebra Seminar

Speaker: Andrew Herring (Western)

"Dynamics of Rational Quadratic Polynomials: Uniform Boundedness, Hilbert Irreducibility, and Finite Exceptional Sets"

Time: 13:30 - 14:30

Room: Zoom: 998 5635 1219

Arithmetic dynamics asks number theoretic questions about dynamical systems (for example, a polynomial over Q which is iteratively composed with itself). One of the "holy grails" of the discipline is the Dynamical Uniform Boundedness Conjecture (DUBC) which purports a uniform bound on the number of points with finite orbit. The simplest case of the DUBC concerns quadratic polynomials g(z) over Q, and Flynn, Poonen, and Schaefer conjecture that no periodic point of g(z) can have period greater than 3. Poonen went on to show that if he, Flynn, and Schaefer are correct, then the largest possible g(z)-orbit has size 9, and the first full case of the DUBC is proved. The "nth dynatomic polynomial," $\Phi_n$, has as its zeros the points of period n for g(z), so studying G_n, the Galois group of $\Phi_n$, may shed light on Flynn, Poonen, and Schaer's conjecture. By considering Phi_n over Q(t), we have an expectation for each G_n, and Hilbert's Irreducibility Theorem says that our expectation is almost always correct. For periods n=1,2,3, there are infinitely many g(z) with smaller than expected G_n, but for n=4,5,6,7,9 it's known that there are only finitely many such exceptions, and it's conjectured that n=1,2,3 are the only values exhibiting infinitely many exceptions. We will discuss how giving lower bounds on genus of fixed fields of maximal subgroups of the expected G_n can be used to prove that there are only finitely many exceptions.

Geometry and Combinatorics

Speaker: Alex Suciu (Northeastern University)

"Sigma-invariants and tropical geometry"

Time: 14:30 - 15:30

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 - 16:30

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.