Fall 2024 @ Dartmouth College - November 8, 2024
The BATMOBILE is a vehicle for bringing together the algebraic and tropical geometry communities of Brown and surrounding institutions for a biannual day of talks.
Greg Smith (Queen's University) |
Cohomology of toric vector bundles
A toric vector bundle is a vector bundle on a toric variety
equipped with a torus action that is compatible with canonical
action on the underlying variety. Klyachko proves that toric
vector bundles are classified by finite-dimensional vector spaces
with a suitable family of filtrations. Building on this
equivalence of categories, we construct a complex of modules over
the Cox ring which simultaneously encodes the cohomology of a
toric vector bundle and many of its twists by line bundles.
Beyond the improved computational efficiency, this approach leads
to new insights into virtual resolutions and vanishing theorems.
This talk is based on joint work with Michael Perlman.
|
The morning events will be in Kemeny Hall and Haldeman Center (two sides of the same building). The morning talks will be in Haldeman 041 and the evening talk (co-organized with AGNES) will be in Kemeny 008.
Make it an algebraic geometry weekend! After BATMOBILE, stay for AGNES @ Dartmouth Fall 2024, including the co-organized GEMS of Algebraic Geometry Mixer and Diane Maclagan's Friday evening talk!
For transportation and parking information, see the AGNES transportation site.
| Time | Event | Location |
|---|---|---|
| 10:00 - 11:00 | Coffee and pastries | Kemeny Hall 300 |
| 11:00 - 12:00 |
Mario Sanchez (Cornell)
Derived categories of permutahedral varieties through matroids |
Haldeman 041 |
| 12:00 - 2:00 | Lunch | Hanover |
| 2:00 - 3:00 |
Greg Smith (Queen's University)
Cohomology of toric vector bundles |
Haldeman 041 |
| The following events are organized jointly with AGNES, so stick around! | ||
| 3:00 - 4:30 | GEMS of Algebraic Geometry mixer | Kemeny Hall 300 |
| 4:30 - 6:00 |
Diane
Maclagan (University of Warwick)
Tropical vector bundles |
Kemeny 008 |
Abstracts
The derived category of a variety is an important invariant that is difficult to compute. One way to describe this category is to hope that it contains a nice set of sheaves known as a full strongly exceptional collection. I discuss a convex-geometric and combinatorial approach to finding these collections for toric varieties through the study of polytopal subdivisions and the homology of set difference of polytopes. I will focus on the toric variety of the permutahedron, also known as the Losev-Manin space, which has played an important role in many recent developments in matroid theory. In this context, the exceptional collection comes from a special collection of matroids.
Greg Smith (Queen's University) -- Cohomology of toric vector bundles
A toric vector bundle is a vector bundle on a toric variety equipped with a torus action that is compatible with canonical action on the underlying variety. Klyachko proves that toric vector bundles are classified by finite-dimensional vector spaces with a suitable family of filtrations. Building on this equivalence of categories, we construct a complex of modules over the Cox ring which simultaneously encodes the cohomology of a toric vector bundle and many of its twists by line bundles. Beyond the improved computational efficiency, this approach leads to new insights into virtual resolutions and vanishing theorems. This talk is based on joint work with Michael Perlman.
Diane Maclagan (University of Warwick) -- Toric vector bundles
Tropicalization replaces a variety by a combinatorial shadow that preserves some of its invariants. When the variety is a subspace of projective space the tropical variety is determined by a (valuated) matroid. I will review this, and discuss a resulting definition for a tropical vector bundle in the context of tropical scheme theory. This is joint work with Bivas Khan.
Organizers:
Asher Auel (Dartmouth),
Juliette Bruce (Dartmouth),
Melody Chan (Brown),
Sarah Frei (Dartmouth),
Andrew Hanlon (Dartmouth),
Eric Larson (Brown),
Nathan Pflueger (Amherst),
and
Isabel Vogt (Brown).
Previous BATMOBILE/BATMOBYLE models:
Fall 2022, Spring 2022, Fall 2020, Fall 2019, Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, and Fall 2014.