Weiyan Chen (Minnesota), Ronno Das (UChicago), Rachel Davis (Wisconsin), Federico Scavia (UBC), Abishek Shukla (UBC), Padmavathi Srinivasan (Georgia Tech), Rebecca Winarski (University of Michigan), and Lynnelle Ye (Harvard).

Lightning Session

Weiyan Chen (University of Minnesota, Twin Cities)
Title: Homological stability of the space of complex irreducible polynomials in several variables
Abstact: The space of complex irreducible polynomials of degree d in n variables satisfies two forms of homological stability: first, its cohomology stabilizes as d increases, and second, its compactly supported cohomology stabilizes as n increases. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.

Ronno Das (University of Chicago)
Title: Points and lines on cubic surfaces
Abstract: The Cayley-Salmon theorem states that every smooth cubic surface in CP^3 has exactly 27 lines; marking a line on each cubic surface produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces. Similarly, marking a point produces a 'universal family' of cubic surfaces over M. I will describe the rational cohomology of these spaces. These purely topological computations have purely arithmetic consequences: the typical smooth cubic surface over a finite field F_q contains 1 line and q^2 + q + 1 points.

Rachel Davis (University of Wisconsin-Madison)
Title: Irreducible sextic polynomials and their absolute resolvents
Abstract: I will outline a 2015 paper by Awtrey, French, Jakes, and Russell with this name. They give an algorithm to determine the Galois group of an irreducible degree 6 polynomial using a degree 30 resolvent polynomial and the discriminant of the original polynomial (a degree 2 resolvent). Moreover, they show that at least 2 resolvent polynomials are necessary to completely determine the Galois group of the sextic.

Federico Scavia (University of British Columbia)
Title: Essential dimension of representations of algebras
Abstract: The essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. I will explain how the representation type of a finitely-generated algebra (finite, tame, wild) is determined by the essential dimension of the functors of its n-dimensional representations and I will introduce new numerical invariants for algebras. I will then explicitely determine the invariants in the case of quiver algebras.

Abishek Shukla (University of British Columbia)
Title: Minimal number of generators of an \'etale algebra
Abstract: O.Forster proved that over a ring R of Krull dimension d a finite module M of rank at most n can be generated by n+d elements. Generalizing this in great measure Z.Reichstein and U.First showed that any finite R-algebra A can be generated by n+d elements if each A?_Rk(p), for p?MaxSpec(R), is generated by n elements. It is natural to ask if the upper bounds can be improved. For modules over rings R.Swan produced examples to match the upper bound. Recently the second author obtained weaker lower bounds in the context of Azumaya algebras. In this paper we investigate this question for ´etale algebras. We show that the upper bound is indeed sharp. Our main result is a construction of universal varieties for degree-2 ´etale algebras equipped with a set of r generators and explicit examples realizing the upper bound of First & Reichstein.

Padmavathi Srinivasan (Georgia Tech)
Title: Conductors and minimal discriminants of hyperelliptic curves
Abstract: Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.
Presentation Slides

Rebecca Winarski (University of Michigan)
Title: Polynomials, dynamics, and twisted rabbits
Abstract: Motivated by problems in complex dynamics, we study branched covers from the sphere to itself. Thurston proves that a branched self-cover of the sphere that satisfies certain finiteness conditions is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines which polynomial a topological branched cover is equivalent to. Our algorithm provides a new solution to Hubbard's twisted rabbit problem, a celebrated theorem from complex dynamics, and allows us to solve a generalization of the twisted rabbit problem. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.

Lynnelle Ye (Harvard University)
Title: Slopes in eigenvarieties for definite unitary groups
Abstract: An eigenvariety is a geometric object parametrizing p-adic automorphic Hecke eigenforms on a given group. We will give bounds on the p-adic valuations of the Hecke eigenvalues appearing in eigenvarieties for definite unitary groups of rank n. (These bounds generalize ones of Liu-Wan-Xiao for definite unitary groups of rank 2, which formed the core of their proof of the Coleman-Mazur-Buzzard-Kilford conjecture in that setting.)

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