# Algebra and its Applications Spring 2017

Earlier sessions here.

- Apr 27
- Daniel Vallières (California State University at Chico)
*Numerical evidence for higher order Stark-type conjectures***Abstract.**Stark’s main conjecture has been refined by various authors especially in the abelian setting. Stark himself gave a more precise conjecture “over \(\mathbb{Z}\)” for imprimitive \(L\)-functions having precisely order of vanishing one at \(s=0\). Ever since this more refined conjecture was formulated, several authors provided numerical evidence in various different settings. A higher rank conjecture “over \(\mathbb{Z}\)” was formulated by Rubin in 1996 and Popescu gave another formulation in 2002 that is closely related to Rubin’s original conjecture, but very little numerical evidence has been provided for them. The goal of this talk will be to explain how one can verify numerically higher order Stark-type conjectures. This is joint work with Kevin McGown (California State University - Chico) and Jonathan Sands (University of Vermont).

- Apr 20
- Lydia Eldredge
*Invariant Harmonic Differential Forms on the Upper Half Space***Abstract.**We give a proof that there is no nonzero harmonic differential form on the upper half space that is invariant under \(\mathrm{SL}_2(\mathbb{C})\) that is of degree other than zero or the top degree. We will define the objects involved, such as the upper half space and harmonic differential forms.

- Apr 13
- Xiping Zhang
*Characteristic classes of Determinantal Varieties***Abstract.**Determinantal Varieties are quite interesting varieties. In this talk we give an algorithm to compute the Chern-Mather class, Chern-Schwartz-Macpherson class, and the local Euler obstruction. We also compute the equivariant version of the characteristic classes, with the natural action from the general linear group. From the arithmetic result one can find some interesting results that require explanations from geometry.

- Apr 6
- Grayson Jorgenson
*Duality defect in codimension 2 and 3***Abstract.**It is conjectured that the dual variety of any smooth subvariety of dimension \(> 2N/3\) in projective \(N\)-space is a hypersurface. This is weaker than Hartshorne’s complete intersection conjecture but is nevertheless unproven for the case of subvarieties of codimension \(> 2\). We will discuss a combinatorial approach to proving the conjecture in the codimension 2 case developed by Holme, and an algorithm based on this approach for proving the conjecture in the codimension 3 case for particular \(N\).

- Mar 30
- Wen Xu
*Reducible cases of the Appell F1 function***Abstract.**The Appell F1 function is a bivariate A-hypergeometric function, and satisfies a system of differential equations of rank 3. It is known for which parameters this system is reducible. We will show for all reducible cases that the rank 2 sub system or quotient system comes from a hypergeometric 2F1 function.

- Mar 23
- Xiping Zhang
*Hirzebruch–Riemann–Roch theorem on Singular Spaces***Abstract.**Hirzebruch–Riemann–Roch theorem is a fascinating result in Complex Manifold, and I will talk about how to tell the story on singular spaces.

- Mar 2
- Michael Niemeier
*Simplicial Abelian Groups and the W Bar Construction***Abstract.**A simplicial abelian group is a contravariant functor from the ordinal number category to the category of abelian groups and a bisimplicial abelian group is a contravariant functor from the ordinal number category to the category of simplicial abelian groups. A result of Duskin gives that the W bar construction on a simplicial abelian group factors as TN, where N is a functor from simplicial abelian groups to bisimplicial abelian groups taking the nerve level-wise and T is a functor from bisimplicial abelian groups to simplicial abelian groups which is the Artin-Mazur anti-diagonal functor. Using this result we will show that the W bar applied n times to a simplicial abelian group factors as TK(-,n), where K(-,n) is a certain functor from simplicial abelian groups to bisimplicial abelian groups that we will define in the talk.

- 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*