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Hodge Theory in Combinatorics

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A Workshop on the Proof of Rota's Log-concavity Conjecture

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Georgia Tech, Atlanta GA

January 16, 2016 (Saturday)

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Speakers

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Schedule

All talks will be in Skiles 006 (ground floor).
**January 14 (Thu) 11am - noon. Mathematics Colloquium. **

Hodge Theory in Combinatorics, by Eric Katz.

We discuss applications of Hodge theory which is a part of algebraic geometry to problems in combinatorics, in particular to Rota's Log-concavity Conjecture. The conjecture was motivated by a question in enumerating proper colorings of a graph which are counted by the chromatic polynomial. This polynomial's coefficients were conjectured to form a unimodal sequence by Read in 1968. This conjecture was extended by Rota in his 1970 ICM address to assert the log-concavity of the characteristic polynomial of matroids which are the common combinatorial generalizations of graphs and linear subspaces. We discuss the resolution of this conjecture which is joint work with Karim Adiprasito and June Huh. The solution draws on ideas from the theory of algebraic varieties, specifically Hodge theory, showing how a question about graph theory leads to a solution involving Grothendieck's standard conjectures. This talk is a preview for the upcoming workshop at Georgia Tech.

**January 15 (Fri) 3-4pm. Combinatorics Seminar. **

Hard Lefschetz theorem and Hodge-Riemann relations in Combinatorics and Geometry, by June Huh.

I will give a broad overview of the Hard Lefschetz property and the Hodge-Riemann relations in the theory of polytopes, complex manifolds, invariants, algebraic varieties, and tropical varieties.

**January 16 (Sat).**

- 9:15-10:15. Proofs in the representable case, by Adiprasito, Huh, and Katz.
- Coffee
- 10:45-11:30. Geometry of Bergman fans and Stanley-Reisner rings, by Karim Adiprasito.
- 11:45-12:30. Matroidal intersection rings, duck theorem, alpha, beta classes, by June Huh.
- Lunch
- 2:30-3:15. Hodge theory in the abstract, by Eric Katz.
- Coffee
- 3:45-4:30. Flips and intermediate fans, by Karim Adiprasito.
- 4:45-5:50. Proof of Hodge-Riemann, by Eric Katz.
- Dinner

## Location

- All talks will be on the ground floor of the Skiles building on the Georgia Tech campus.
- On weekend street parking can be found in the neighborhood a few blocks south of the Skiles building (around Luckie St NW and Pine St NW, for example).

## References

## Poster

## Organizers

Matt Baker

Josephine Yu