# Algebra & Number Theory Seminar

**Winter 2021**

**ANT Zoominar**

**Fridays @ 3pm California Time**

**9045829721**--- password: seminar

For further information please contact

**Schedule** *(click dates for title and abstract)*

2/19 | Libby Taylor |
Stanford University |

2/26 | Leo Herr | University of Utah |

3/12 | Hatice Mutlu Akaturk | UCSC |

**Libby Taylor, Stanford University**

**Derived equivalences of gerbey curves**

We study derived equivalences of certain stacks over genus 1 curves, which arise as connected components of the Picard stack of a genus 1 curve. To this end, we develop a theory of integral transforms for these algebraic stacks. We use this theory to answer the question of when two stacky genus 1 curves are derived equivalent. We use integral transforms and intersection theory on stacks to answer the following questions: if C'=Pic^d (C), is C=Pic^f (C') for some integer f? If C'=Pic^d (C) and C''=Pic^f (C'), then is C''=Pic^g (C) for some integer g?

**Leo Herr, University of Utah**

**Log intersection theory and the product formula**

(joint with Jonathan Wise, Y.P. Lee, and You-Cheng Chou)

Log structures stratify a space and manage mild singularities. Cohomological invariants are defined via an inverse limit of blowups akin to a rigid analytic or Berkovich space. When schemes are enriched with log structure, intersections and other fiber products are different from the intersections of underlying schemes. I define a "log intersection product," or log Gysin map that takes place on the log intersection and mimics Fulton's intersection theory. This can be defined in log versions of Chow or K theory groups, similar to Shokurov's bChow groups in birational geometry. As an application, I prove a log version of the product formula in Gromov-Witten theory.

**Hatice Mutlu Akaturk, UCSC**

** Monomial posets and their Lefschetz invariants**(joint with Serge Bouc and Robert Boltje)

The Euler-Poincaré characteristic of a given poset X is defined as the alternating sum of the orders of the sets of chains Sd_{n}(X) with cardinality n + 1 over the natural numbers n. Given a finite group G, Thévenaz extended this definition to G-posets and defined the Lefschetz invariant of a G-poset X as the alternating sum of the G-sets of chains Sd_{n}(X) with cardinality n+1 over the natural numbers n which is an element of Burnside ring B(G). Let A be an abelian group. We will introduce the notions of A-monomial G-posets and A-monomial G-sets, and state some of their categorical properties. The category of A-monomial G-sets gives a new description of the A-monomial Burnside ring B_{A}(G). We will also introduce Lefschetz invariants of A-monomial G-posets, which are elements of B_{A}(G). An application of the Lefschetz invariants of A-monomial G-posets is the A-monomial tensor induction. Another application is a work in progress that aims to give a reformulation of the canonical induction formula for ordinary characters via A-monomial G-posets and their Lefschetz invariants. For this reformulation we will introduce A-monomial G-simplicial complexes and utilize the smooth G-manifolds and complex G-equivariant line bundles on them.