KOALA 2024:
Workshop of the Kentucky-Ohio ALgebra Alliance
Registration:
All participants, including invited speakers, are asked to register
online. The deadline for registration is April 30. Please
fill out the following registration
form.
Discord Server:
The KOALA discord can be found here: https://discord.gg/N7bKwTPTDn
About KOALA:
The Mathematics Departments at both The
Ohio State University (Columbus, OH) and the University of Kentucky (Lexington,
KY) have a critical mass of faculty, postdocs and graduate students in
Combinatorial Algebraic Geometry. Faculty at both institutions often
have research collaborations and meet each other at conferences.
Younger members of both groups do not have as many opportunities to
interact.
KOALA workshops (regular 2-day meetings held once a year, and
alternating between both institutions) provide a venue to further
expand collaborations between these two groups. Their main goals are:
- to establish a community of mathematicians in Kentucky and Ohio
working in Combinatorial Algebraic Geometry;
- to expose graduate students and junior researchers in both
institutions to the many outstanding problems in Combinatorial
Algebraic Geometry, and to the tools currently being used to attack
them; and
- to survey recent developments in Combinatorial Algebraic
Geometry (including moduli of curves, toric and tropical geometry, and
the related geometry and combinatorics of homogeneous spaces)
Speakers:
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Schedule:
Time
|
Location
|
Speaker/Event
|
Title
|
|
Tuesday |
|
|
|
1:00 - 1:50 PM
|
CP 103
|
Chris Eur
|
Matroids as Vector Bundles
|
2:00 - 3:00 PM
|
CP 111
|
Coffee Break
|
|
3:00 - 3:50 PM
|
CP 103
|
Juliette Bruce
|
The Top-Weight Cohomology of A_g
|
4:00 - 4:30 PM
|
CP 111
|
Coffee Break
|
|
4:30 - 5:30 PM
|
CP 103
|
Lightning Session
|
|
6:00 - 8:00 PM
|
Math House (654 Maxwelton Ct)
|
Dinner
|
|
8:00 PM - ???
|
Kentucky Native Cafe
|
Drinks
|
|
|
|
|
|
Wednesday
|
|
|
|
8:30 - 9:30 AM
|
CP 111
|
Breakfast
|
|
9:30 - 10:20 AM
|
CP 103
|
Austin Alderete
|
Toric Matroid Bundles
|
10:30 - 11:00 AM
|
CP 111
|
Coffee Break
|
|
11:00 - 11:50 AM
|
CP 103
|
Madeline Brandt
|
Topology of (Tropical) Moduli Spaces: Hyperelliptic Curves
|
12:00 PM - ???
|
Wherever you like
|
Lunch
|
|
Titles and Abstracts:
Austin Alderete: Toric Matroid Bundles
The
Kaveh-Manon classification of torus-equivariant principal G-bundles in
terms of piecewise-linear maps allows one to define a toric matroid
bundle. We discuss these new objects and their connection to the
tautological classes of matroids, ending with an example of a toric
matroid bundle over the projective line which does not split.
Madeline Brandt: Topology of (Tropical) Moduli Spaces: Hyperelliptic Curves
Moduli spaces offer a geometric solution to geometric classification
problems by parameterizing all objects of some type. Tropical versions
of these spaces explain the combinatorics of their compactifications.
Moreover, these tropical moduli spaces can be used to compute a
piece of the cohomology of the corresponding classical moduli space. In
joint work with Melody Chan and Siddarth Kannan, we study the topology
of the moduli space of hyperelliptic curves using these techniques.
Juliette Bruce: The Top-Weight Cohomology of A_g
I will discuss recent work calculating the top weight cohomology of the
moduli space A_g of principally polarized abelian varieties of
dimension g for small values of g. The key idea is that this piece of
cohomology is encoded combinatorially via the relationship between the
boundary complex of a compactification of A_g and the moduli space of
tropical abelian varieties. This is joint work with Madeline Brandt,
Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.
Chris Eur: Matroids as Vector Bundles
Matroids are combinatorial abstractions of linear
subspaces in a coordinatized vector space. Could these objects
further be interpreted as some sort of a vector bundle? If so,
what good does it do? We discuss some past results, developing
ideas, and possible future directions.
5 Minute Talks:
Ishan Banerjee: An Aysmptotic Lefschetz Theorem for Rational Fibrations
Let Y, Z be smooth projective
varieties. Let e>0. Let X be a degree d hypersurface in
Y. Let RatFib_{\le e}(Y,Z), Ratfib_{\le e}(X,Z)
denote the set of dominant rational maps from Y to Z and X to Z of
degree $\le e$ respectively. Then for d sufficiently large, we
prove that there is a restriction map from RatFib_{\le e}(Y,Z) to RatFib_{\le
e}(X,Z) and that furthermore this map is a bijection. I will
comment briefly on some motivation for this result and about the
proof. This is joint work with David Stapleton.
Kyle Binder: Higher Tor Groups for Matroids
For a loopless matroid, the Chow group is an important invariant which
has been used to resolve log-concavity conjectures in matroid
theory. Geometrically, this group is the Chow group of the toric
variety attached to the Bergman fan of the matroid. In this talk,
I will mention how to generalize the Chow group of matroids via higher
Tor groups, their relation to the cohomology groups of toric varieties,
and the case of uniform matroids.
Eric Burkholder: Searching for a Tropical Basepoint Free Pencil Trick
Classically, the Basepoint Free Pencil Trick provides a lower bound on
the number of linearly independent functions in the product of two
linear series, in which one of the linear series is rank 1 and
basepoint free. To pursue a tropicalized version of this result,
we will define a notion of tropical independence of functions and
provide our most recent result for tropical linear series on the metric
graph of a loop.
Hugh Dennin: Cauchy Identities for Schubert Polynomials via Pipe Dreams
Schubert polynomials are a family of polynomials indexed by partitions
which represent Schubert cycles in the cohomology of the complete flag
variety. They can be interpreted combinatorially
as the generating series of a class of diagrams called pipe
dreams. We introduce a new "push" bijection on pipe dreams which
gives a new proof of certain Cauchy identities relating Schubert
polynomials to double Schubert polynomials.
Casey Hill: The Algebra of SL_4 Conformal Blocks
Conformal blocks are objects from mathematical physics which appear
naturally as the spaces of global sections of line bundles on the
moduli of vector bundles on smooth curves. In this talk, we will
discuss how to determine a presentation for the algebra of SL_4
conformal blocks.
William Newman: Moduli Spaces and Higher Chow Groups
In this talk, I will explain how to use higher groups and the
localization exact sequence to compute Chow groups of moduli spaces.
Jordan Sawdy: A Bicategorical Grothendieck-Riemann-Roch
In 2020, Hoyois, Safronov, Scherotzke, and Sibilla established a GRR-type theorem in the context of
symmetric monoidal infinity-categories. I'll discuss some work
I've been doing on a similar result, but in the context of symmetric
monoidal 2-categories.
