Boston Algebraic Geometry Day

Registration: Please fill out the Registration Form (headcount for food, dietary restrictions, and parking passes) NOTE: Registration ends at 5:00PM on Wednesday, October 7.

Location: University Hall Y02-2110, UMass Boston. See Visitor Resources for directions.

Parking: After parking in a ticketed lot (e.g. the Campus Center Garage), we give you a parking pass, and you use this in combination with your ticket to exit the lot.

Speakers: Xin Jin, Eric Jovinelly, Daniel Litt, Eyal Markman

Schedule: 

9:30-10  Check-in and refreshments

10-11  Xin Jin: Around the multiplicative universal centralizer

11-11:30  Coffee break

11:30-12:30  Eric Jovinelly: Rational chain connectivity of globally F-regular varieties

12:30-2  Lunch (not provided; lunch is available for purchase at the Dining Commons in Residence Hall East)

2-3  Daniel Litt: Tuples of matrices (after Charlie Wu)

3-3:30  Coffee break

3:30-4:30  Eyal Markman: Secant sheaves and Weil classes

Abstracts:

Xin Jin (Boston College): Around the multiplicative universal centralizer

The multiplicative universal centralizer $J$ of a complex reductive group $G$ plays an important role in geometric representation theory. For $G=GL_n$, $J$ is a transverse Hilbert scheme of $n$ points on $\mathbb{C}^\times\times \mathbb{C}^\times$. For $G$ of ADE type, $J$ is the K-theoretic Coulomb branch of pure gauge theory. I will present joint work with Ben Webster (to appear) establishing two remarkable structures on $J$: a (toric-like) Bruhat stratification and a cluster structure. Time permitting, I will discuss applications of these structures, including a partial resolution of a conjecture of Cautis–Williams, applications to the cohomology of $J$, and the determination of the Donaldson–Thomas transformation on $J$.

Eric Jovinelly (Brown University): Rational chain connectivity of globally F-regular varieties

Rational connectivity is a fundamental property of complex Fano varieties.  While it has long been conjectured that Fano varieties in positive characteristic share this property, the behavior of curves in this setting is much harder to understand.  In this talk, we study curves on globally F-regular varieties, which are the analogue of complex klt Fano varieties in positive characteristic.  By analyzing properties of their tangent bundles, we show globally F-regular varieties are rationally chain connected whenever the characteristic is large compared to their dimension.  

Daniel Litt (University of Toronto): Tuples of matrices (after Charlie Wu)

Consider the space of n-tuples of matrices whose product is the identity, up to simultaneous conjugation, and where the matrices are constrained to lie in some collection of fixed conjugacy classes. This is an example of a "relative character variety," in this case of the projective line minus n points. I'll discuss work of Charlie Wu (in his PhD thesis) describing some of the topology of these spaces, as well as relative character varieties of higher genus curves, and in particular a classification of the compact components of the real points of these spaces. I'll also discuss joint work in progress that further studies the rich geometry of these compact components and their mapping class group dynamics. 

Eyal Markman (UMass Amherst): Secant sheaves and Weil classes

We will describe a general strategy for proving the algebraicity of the Hodge Weil classes on abelian varieties of Weil type. The latter are even dimensional abelian varieties admitting a suitable embedding of a CM number field in their rational endomorphism ring. We will describe the implementation of the strategy for abelian varieties of dimension 4 and 6, and why it implies the Hodge conjecture for abelian varieties of dimension at most 5. The cohomology of an abelian variety X is naturally the spin representation of Spin(V), where V is the first cohomology of the cartesian product X x Pic^0(X). We will emphasize the central role played by semi-regular coherent sheaves F on an abelian varieties X of dimension equal to half that of the abelian varieties of Weil-type and with a Chern character ch(F) lying on a line secant to the even spinorial variety (semi-regular secant sheaves).