# Algebra and its Applications Fall 2016

Earlier sessions are here.

• 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.)