WARTHOG* 2015

Positivity in Combinatorial Algebraic Geometry

10 August - 14 August 2015
University of Oregon
Eugene, OR

* Workshop on Algebra and Representation Theory Held on Oregonian Grounds

The goal of this workshop will be to understand some of the recent work of June Huh, using Hodge theory, tropical geometry, and intersection theory of toric varieties to prove various positivity statements in combinatorics.

The workshop will be led by June Huh, and will consist of a combination of lectures and problem sessions.
It will be organized by Nicholas Proudfoot and Ben Elias.

Registration

If you are interested in participating, email Nick or Ben. Please include your school, advisor, and a brief description of your research interests. Funding for accommodations in Eugene (but not for travel) will be available to students and postdocs as long as space and funds remain.

Participants

If you are on this list, then you are registered, and you can expect us to arrange and pay for your accommodations in Eugene for the duration of the workshop. We will contact each of you to confirm your participation later in the year.

History

This is the sixth of a series of summer workshops that have been funded by an NSF CAREER grant. You can read about the previous ones here:

2010: Operator Algebras and Conformal Field Theory (with André Henriques)
2011: Cluster Algebras and Lusztig's Semicanonical Basis (with David Speyer)
2012: Categorical Representation Theory (with David Ben-Zvi)
2013: Quantum/Affine Schubert Calculus (with Allen Knutson)
2014: Kazhdan-Lusztig theory and Soergel bimodules (with Ben Elias)

Practical information

Participants will be staying in a dormitory on campus. When you arrive, you need to check in at the Area Desk of the Living Learning Center, which is on 15th Avenue between University Street and Agate Street; there you will receive a key to your room. The confusing thing is that the building is disconnected (there is a South component and a North component). The component on 15th Avenue is the South component, but the component that contains the Area Desk is the North component (which is not bordered by any street at all). Also, if you arrive after 8pm, you will have to call this number: (541) 346-5686.

There is no public transportation from the airport, so you'll have to take a cab, which should take about half an hour and cost about $30. Since a lot of you will be arriving around the same time, you might consider trying to find each other at the airport and sharing a cab. To help coordinate this, you can use this page.

Lectures will begin on Monday at 9:00 in HEDCO 146, with coffee and pastries available starting around 8:30.

Program

Below is a more detailed discussion of the topics that will be covered.

1. Positivity of Chern classes of Schubert cells and varieties

MacPherson's functorial Chern class is defined for noncompact or singular varieties, and in particular for Schubert cells and varieties in G/P. Aluffi and Mihalcea conjectured that Chern classes of Schubert cells and varieties in Grassmannians are positive. This is a purely combinatorial statement. I proved the conjecture, but the proof is not combinatorial, and only works for Grassmannians. The true source of the positivity remains unclear, and we do not know whether the same positivity holds for other G/P. A reasonable goal of the week is to decide whether we should try to prove or disprove the general case.

References:

Paolo Aluffi, Characteristic classes of singular varieties
Paolo Aluffi and Leonardo Mihalcea, Chern classes of Schubert cells and varieties
Paolo Aluffi and Leonardo Mihalcea, Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds
June Huh, Positivity of Chern classes of Schubert cells and varieties

2. Log-concavity conjectures and positivity of algebraic cycles in permutohedral varieties

Rota conjectured that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. This is a purely combinatorial statement. Eric Katz and I proved the conjecture for representable matroids, but one specific part of the proof is not combinatorial, and breaks down in the non-representable case. The log-concavity comes from the Hodge-Riemann bilinear relation, and for non-representable matroids there is no object to apply the Hodge theory to. This ties in nicely with the theme of the previous year's workshop led by Ben Elias. In characteristic 0, MacPherson's Chern class for hyperplane arrangement complement gives stronger log-concavity, but it is unclear whether the same can be achieved for matroids representable only over characteristic p, let alone non-representable matroids.

In covering this material, we will discuss toric varieties, tropical algebraic geometry, and matroid theory, among other topics.

References:

Eric Katz, Matroid theory for algebraic geometers
June Huh and Eric Katz, Log-concavity characteristic polynomials and the Bergman fan of matroids
June Huh, Rota's conjecture and positivity of algebraic cycles in permutohedral varieties
June Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomials of graph
June Huh, The maximum likelihood degree of a very affine variety
June Huh, h-vectors of matroids and logarithmic concavity

3. Tropical currents and the Hodge conjecture for positive currents

Demailly showed that the Hodge conjecture is equivalent to the statement that any closed current of Hodge type can be approximated by linear combinations of currents associated with subvarieties, and asked whether any positive closed current of Hodge type can be approximated by positive linear combinations of currents associated with subvarieties. In joint work with Farhad Babaee, we found a counterexample to the latter statement by combining Babaee's thesis with ideas coming from topic (2) above. We will go through details of the construction of the example, and try to see how far we can generalize the method of construction.

Reference:

Farhad Babaee and June Huh, A tropical approach to the strongly positive Hodge conjecture
Farhad Babee, Complex tropical currents

Schedule

Here is a tentative schedule of lecture topics. The first four days will be held in HEDCO 146, and the last day will be held in HEDCO 142. Lectures will begin at 9:00 each day, and pastries and coffee will be available starting at 8:30.

Monday

9:00 Overview (June Huh)

9:45 Hodge Theory for Combinatorial Geometries 1 (June Huh)

11:00 Hodge Theory for Combinatorial Geometries 2 (June Huh) Slides

2:00 Introduction to Matroids and Bergman fans 1 (Federico Ardila) Slides

3:15 Introduction to Matroids and Bergman fans 2 (Federico Ardila) Slides Exercises

4:30 Discussion/Exercises

Tuesday

9:00 Triage (June Huh) Exercises

9:45 A Tropical Approach to Hodge Conjecture for Positive Currents 1 (June Huh)

11:00 A Tropical Approach to Hodge Conjecture for Positive Currents 2 (June Huh)

2:00 Flag Varieties and Matroids 1 (Alex Fink)

3:15 Flag Varieties and Matroids 2 (Alex Fink) Exercises

4:30 Discussion/Exercises

Wednesday

9:00 Chern classes of Schubert Cells and Varieties (June Huh)

10:15 Introduction to Schubert Varieties and Their Singularities 1 (Alex Woo)

11:30 Introduction to Schubert Varieties and Their Singularities 2 (Alex Woo) Exercises

1:30 Hike

6:30 Picnic: University Park (5 blocks south of campus on University Avenue).

Thursday

9:00 Triage (June Huh)

9:45 The Kazhdan-Lusztig Polynomial of a Matroid 1 (Nicholas Proudfoot)

11:00 The Kazhdan-Lusztig Polynomial of a Matroid 2 (Nicholas Proudfoot) Exercises

2:00 Hodge Theory for Combinatorial Geometries 3 (June Huh)

3:15 Hodge Theory for Combinatorial Geometries 4 (Eric Katz)

4:30 Discussion/Exercises

Friday

9:00 Triage (June Huh)

9:45 Hodge Theory for Combinatorial Geometries 5 (Eric Katz)

11:00 Hodge Theory for Combinatorial Geometries 6 (Eric Katz)

12:15 Final Discussion