Graduate student, Cornell University Department of Mathematics
annab at math dot cornell dot edu
Advisor: Allen Knutson
Here are my CV, research statement and teaching statement .
Slides from my talk on the equations defining unions of matrix Schubert varieties at the Cornell Syzygies Workshop are here, also available is a draft of the associated paper.
Research interests:
I'm interested in algebraic combinatorics and algebraic geometry. Lately, I've been working on a few different problems.
The first problem is in Schubert calculus. It is a fact (from the geometry) that when we take the product of two Schubert classes in the cohomology ring of the full flag manifold and re-expand in terms of the basis of Schubert classes, then the structure constants are positive. However, no known combinatorial, or even manifestly positive (no negative signs) formula exists. I have been using a non-commutative algebra, the Fomin-Kirillov algebra to try to write manifestly combinatorial formulae for multiplying by some Schubert classes.
I have also recently been working on two collaborative projects in quantum Schubert calculus. Classical Schubert calculus asks questions (some though not all answered--see above!) about the cohomology of flag varieties. These cohomology computations often turn out to be governed by extraordinarily beautiful combinatorial rules. Unlike the product in usual cohomology of a flag manifold, which asks about Schubert varieties overlapping, the product in quantum cohomology asks for given pairs of schemes, which other schemes are connected by a curve with three marked points, one in each subspace. More formally, the structure constants are the Gromov-Witten invariants. These rings similarly have many nice combinatorial properties.
With Tara Holm and Kaisa Taipale, I have been working on a version of the Kirwan map for quantum cohomology to calculate the quantum cohomology of spaces that appear as GIT quotients, such as polygon space as the GIT quotient of the Grassmannian of 2 planes in complex n-space by the torus.
With Elizabeth Beazley and Kaisa Taipale, we are working to prove an equivariant version of the "rim hook rule" of Bertram, Ciocan-Fontanine and Fulton, which gives the product for QH*(Gr(k,n)) in terms of the product in H*(Gr(k,2n-k)). We are working on proving a conjecture for a similar map using equivariant versions of these two rings, with success so far in the Pieri case.