• Speaker: Nick Rekuski, Michigan State
• Title: Splitting Criteria for Vector Bundles on $\mathbb{P}^n$
• Date: 09/23/2019
• Time: 4:30 PM - 5:30 PM
• Place: C304 Wells Hall

Grothendieck's Theorem says that any vector bundle on $\mathbb{P}^1$ can be decomposed as a finite sum of line bundles. In this talk, we will discuss a generalization of this theorem: Horrocks Splitting Criterion. This criterion completely describes when a vector bundle on $\mathbb{P}^n$ splits as a sum of line bundles. We will then discuss an open conjecture of Hartshorne. If time permits, we will also consider the similar question of classifying when a vector bundle on $\mathbb{P}^n$ decompose as line bundles and twists of the tangent bundle.

• Speaker: Honghao Gao, MSU
• Title: More on Morse homology
• Date: 09/24/2019
• Time: 12:00 PM - 1:00 PM
• Place: C117 Wells Hall

No abstract available.

• Speaker: Honghao Gao, MSU
• Title: Legendrian knots and augmentation varieties
• Date: 09/24/2019
• Time: 3:00 PM - 4:00 PM
• Place: C204A Wells Hall

We begin with a gentle introduction to Legendrian knot and its invariant theory. We will define the Chekanov-Eliashberg different graded algebra and augmentations associated to the dga. We also present an example where the augmentation variety is a cluster variety.

• Speaker: Bruce Sagan, MSU
• Title: Lucas atoms
• Date: 09/25/2019
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall

We introduce a powerful algebraic method for proving that Lucas analogues are polynomials with nonnegative coefficients. In particular, we factor a Lucas polynomial as $\{n\}=\prod_{d|n} P_d(s,t)$, where we call the polynomials $P_d(s,t)$ Lucas atoms. This permits us to show that the Lucas analogues of the Fuss-Catalan and Fuss-Narayana numbers for all irreducible Coxeter groups are polynomials in $s,t$. Using gamma expansions, a technique which has recently become popular in combinatorics and geometry, one can show that the Lucas atoms have a close relationship with cyclotomic polynomials $\Phi_d(q)$. Certain results about the $\Phi_d(q)$ can then be lifted to Lucas atoms. In particular, one can prove analogues of theorems of Gauss and Lucas, deduce reduction formulas, and evaluate the $P_d(s,t)$ at various specific values of the variables. This is joint work with Jordan Tirrell based on an idea of Richard Stanley.

• Speaker: Chuangtian Guan, MSU
• Title: Almost Mathematics
• Date: 09/26/2019
• Time: 10:00 AM - 12:00 PM
• Place: C329 Wells Hall

No abstract available.

• Speaker: Corey Jones, The Ohio State University
• Title: The higher dimensional algebra of matrix product operators and quantum spin chains
• Date: 09/26/2019
• Time: 11:30 AM - 12:30 PM
• Place: C304 Wells Hall

In the context of 1D quantum spin chains, matrix product operators provide a way to study non-local operators such as translation in terms of quasi-local information. They have been used to describe a generalized form of symmetry for 1D systems on the boundary of 2D topological phases. In this talk, we will introduce some concepts of higher dimensional algebra, and a broad hypotheses about higher categories and spatially extended quantum systems. We will then explain how the collection of matrix product operators assembles into a higher (symmetric monoidal 2-) category, and discuss some implications of this. Based on joint work with David Penneys.

• Speaker: A. Volberg/P. Mozolyako
• Title: Unexpected combinatorial properties of all planar measures
• Date: 09/26/2019
• Time: 1:00 PM - 1:50 PM
• Place: C517 Wells Hall

We will start with paraproducts--operators used in PDE to prove Leibniz rule with fractional derivatives. Then we move to bi-parameter paraproducts and prove the property from the title.

• Speaker: David Boozer, UCLA
• Title: Holonomy perturbations of the Chern-Simons functional for lens spaces
• Date: 09/26/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

We describe a scheme for constructing generating sets for Kronheimer and Mrowka's singular instanton knot homology for the case of knots in lens spaces. The scheme involves Heegaard-splitting a lens space containing a knot into two solid tori. One solid torus contains a portion of the knot consisting of an unknotted arc, as well as holonomy perturbations of the Chern-Simons functional used to define the homology theory. The other solid torus contains the remainder of the knot. The Heegaard splitting yields a pair of Lagrangians in the traceless $SU(2)$-character variety of the twice-punctured torus, and the intersection points of these Lagrangians comprise the generating set that we seek. We illustrate the scheme by constructing generating sets for several example knots. Our scheme is a direct generalization of a scheme introduced by Hedden, Herald, and Kirk for describing generating sets for knots in $S^3$ in terms of Lagrangian intersections in the traceless $SU(2)$-character variety for the 2-sphere with four punctures.