Algebra and its Applications

Welcome to the Algebra and its Applications seminar. We meet on Thursday, at 3:35 pm in 104 LOV.

Seminar organizing duties rotate. Schedules for currently running seminars, including this one, are always posted to the Department’s events page. Earlier sessions I have organized are here.

Spring 2017

• Feb 23
• Kyounghee Kim
• Rational Surface Automorphism: Homology actions of real mappings
• Abstract Let $f:X \to X$ be a quadratic rational surface automorphism fixing a cuspidal cubic. Let $X_R$ be the closure of the $\mathbb{RP}^2$ inside $X$. When the multiplier at the invariant cuspidal cubic is real, $f$ induces the automorphism $f_R$ of $X_R$. In this talk, we will discuss the homology action of $f_R$ and and the real mappings with maximal entropy.
• Feb 16
• Ivan Martino (Northeastern)
• Syzygies of the Veronese and Pinched Veronese modules
• Abstract There has been a lot of effort to find the graded Betti numbers of the Veronese ring. I am going to define the (pinched) Veronese modules and present a sample of literature results, like the work of Ein and Lazarsfeld, Ottaviani and Paoletti and, Bruns, Conca and Römer. Then, I will show a combinatorial approach to the question and I will discuss new results about the linearity of the resolution of the Veronese modules and the pinched Veronese modules.
• Feb 9
• Yaineli Valdes
• A multifunctor from Waldhausen Categories to the 1-type of their $K$-Theory spectra
• Abstract Muro and Tonks constructed an algebraic model for the stable 1-type of the $K$-Theory spectrum of any Waldhausen by ways of stable quadratic modules. Stable 1-types are classified by Picard groupoids, so they constructed a 1-functor from the category of Waldhausen categories to the category of Picard groupoids. Zakharevich proved the category of Waldhausen categories is a closed symmetric multicategory and there is a multifunctor from the category of Waldhuasen categories to the category of spectra by assigning to any Waldhausen category its algebraic $K$-Theory spectrum. Symmetric monoidal categories are themselves symmetric multicategories and it is known that the category of Picard groupoids is a symmetric monoidal category and is closed as well. We want to show the 1-functor defined by Muro and Tonks extends to a multifunctor of closed symmetric multicategories. This is useful because it will describe the algebraic structures on the 1-type of the $K$-Theory spectra induced by the multiexactness pairings on the level of Waldhausen categories.
• Feb 2
• Ettore Aldrovandi
• Extensions of Lie Algebroids and generalized differential operators
• Abstract A Lie algebroid, or Lie-Rinehart algebra, is a module over a commutative $k$-algebra $A$ and at the same time it is a $k$-Lie Algebra. The prototypical example is the algebra of derivations of a $k$-algebra $A$ itself, whose envelope consists of differential operators. I will recall the notions of Lie Algebroid and generalized differential operator, and discuss some examples. I will then discuss some aspects of the classification of sheaves of Lie algebroids on a scheme, in particular that of their extensions (Joint with U. Bruzzo, Trieste).
• Jan 26
• Ezra Miller (Duke University)
• Algebraic data structures for topological summaries
• Abstract. This talk introduces a combinatorial algebraic framework to encode, compute, and analyze topological summaries of geometric data. The motivating problem from evolutionary biology involves statistics on a dataset comprising images of fruit fly wing veins. The algebraic structures take their cue from graded polynomial rings and their modules, but the theory is complicated by the passage from integer exponent vectors to real exponent vectors. The path to effective methods is built on appropriate finitness conditions, to replace the usual ones from commutative algebra, and on an understanding of how datasets of this nature interact with moduli of modules. I will introduce the biology, algebra, and topology from first principles. Joint work with David Houle (Biology, Florida State), Ashleigh Thomas (grad student, Duke Math), and Justin Curry (postdoc, Duke Math).
• Jan 19
• Organizational Meeting

Fall 2016

• Dec. 8
• Erdal Imamoglu
• Integral Bases for Differential Operators
• Abstract. The goal of this talk is to find a transformation that reduces a complicated differential operator $L$ (with numerous apparent singularities) to a simpler one. To find such transformation, we present a fast algorithm to compute an integral basis of a differential operator $L$ with rational function coefficients, and an algorithm to normalize it at infinity. Examples show that this often reduces $L$ to an operator that is easier to solve.
• Nov. 17
• Xiping Zhang
• G.A.G.A.
• Abstract. I will talk about the great theorem by Serre on connecting Analytic spaces and Algebraic Schemes.
• Nov. 10
• Jay Leach
• The A-polynomial
• Abstract. I’ll be discussing the A-polynomial and some of its properties.
• Nov. 3
• Sarah Algee
• Introduction to Motivic Integration
• Abstract. Motivic integration is a notion in algebraic geometry that was introduced by Kontsevich in 1995 and was developed by Denef and Loeser. Motivic integration can be quite useful in various branches of algebraic geometry and it resembles $p$-adic integration. Essentially, motivic integration assigns to subsets of the arc space a measure living in the (completed) Grothendieck ring of algebraic varieties.
• Oct. 20 & 27
• Corey Harris
• Tritangents of algebraic and tropical space sextics
• Abstract. The theory of the tritangent planes to a canonically embedded sextic in $\mathbb{C}P^3$ is classical. The number of such planes was known to Klein and their structure worked out by Coble. In this talk we’ll quickly review some of this classical theory and see what happens when you tropicalize.
• Oct. 6 & 13
• Kyounghee Kim
• No smooth Julia sets for polynomial diffeomorphisms
• Abstract. The Fatou set of a holomorphic mapping $f$ is the set where the iterates $f^n$ are locally equicontinuous. The Julia set is defined as the complement of the Fatou set. The Julia set is where any chaotic behavior occurs. For one dimensional case, there are mappings with smooth Julia sets and these mappings play important roles. We will show that there is no polynomial diffeomorphism of $\mathbb{C}^2$ with the $C^1$-smooth Julia set. This is a joint work with Eric Bedford.
• Sept. 22 & 29
• Paolo Aluffi
• Chern classes of Schubert varieties
• Abstract. We compute the Chern-Schwartz-MacPherson classes of Schubert varieties in flag manifolds. These classes are obtained by constructing a representation of the Weil group, by means of certain Demazure-Lusztig type operators. The construction extends to the equivariant setting. Based on explicit computations in low dimension, we conjecture that these classes are Schubert-positive; the analogous conjecture for Schubert varieties of Grassmannians was recently proven by June Huh. This is joint work with Leonardo Mihalcea.
• Sept. 15
• Ettore Aldrovandi
• The Heisenberg group and a geometric approach to cup products
• Abstract. The Heisenberg Group is a functor that to any pair of abelian groups $A$ and $B$, assigns a nilpotent central extension of $A\times B$ by $A\otimes B$. I will show that the universal cup product map for degrees $1+1=2$ is equal to the class of this extension. (Joint work with N. Ramachandran.)