The UC Berkeley Combinatorics Seminar

DATE	SPEAKER	TITLE (click to show abstract)
January 29th	Annie Raymond, UMass Amherst	The generalized Pitman-Stanley polytope The eponymous Pitman-Stanley polytope introduced in 1999 is related to plane partitions of skew shape with entries 0 and 1. It has been well studied because of its connections to probability, parking functions, generalized permutahedra, and flow polytopes. We consider a generalization of this polytope related to plane partitions with entries 0, 1, ... , m. We show that this polytope can also be realized as a flow polytope of a grid graph. We give multiple characterizations of its vertices in terms of plane partitions of skew shape and integer flows. We also study formulas for the volume as well as for the number of lattice points and vertices of this polytope. This is joint work with William Dugan, Maura Hegarty and Alejandro Morales.
February 5th	Eugene Gorsky, UC Davis	Skewing formulas for Delta Conjecture Delta Conjecture of Haglund, Remmel and Wilson is the identity describing the action of Macdonald operators on elementary symmetric functions. The conjecture was proved independently by Blasiak-Haiman-Morse-Pun-Seelinger, and D'Adderio-Mellit. In this talk, I will explain yet another proof of Delta Conjecture by applying a Schur skewing operator to both sides of another identity known as Rational Shuffle Theorem. I will also describe geometric models for Delta conjecture and Rational Shuffle Theorem using affine Springer fibers. This is a joint work with Sean Griffin and Maria Gillespie.
February 12th	Stephanie van Willigenburg, UBC	The $e$-positivity of chromatic symmetric functions We will meet the chromatic symmetric function, dating from 1995, which is a generalization of the chromatic polynomial. We will also hear about a famed problem regarding it, called the Stanley-Stembridge (3+1)-free problem. This has been the focus of much research lately including resolving another problem of Stanley of whether the (3+1)-free problem can be widened. The resulting paper on the latter problem was recently awarded the 2023 MAA David P. Robbins Prize, and we will hear this story too. This talk requires no prior knowledge and will be suitable for a broad audience. Pre-talk Title: Noncommutative chromatic symmetric functions revisited Pre-talk Abstract: The chromatic polynomial of a graph dates from 1912, when Birkhoff created it while trying to solve the 4-colour problem. In 1995 Stanley generalized it to the chromatic symmetric function, which stored more information about the graph. This was then generalized again to noncommuting variables, by Gebhard-Sagan in 2001. In this talk we will meet these newest chromatic functions, the space they live in, and discover new results for them. No background is needed for this talk and it is suitable for students.
February 19th	Daniel Kráľ, Masaryk University -> Leipzig University	Extremal problems with quasirandom constraints A combinatorial structure is said to be quasirandom if it resembles a random structure in a certain robust sense. Classical work of Rödl, Thomason, Chung, Graham and Wilson from the 1980s led to the notion of quasirandom graphs, which is nowadays considered to be well-understood. In this talk, we first review classical and recent results on quasirandom combinatorial structures, and we then focus on problems from extremal combinatorics with additional quasirandom constraints. The study of such extremal problems was initiated by Erdős and Sós in the early 1980s, however, substantial progress appeared only recently with use of the hypergraph regularity method, which was independently developed by Kohayakawa, Nagle, Rödl, Schacht and Skokan, and Gowers. We will present some of recent results, e.g. a solution of a 40-year-old problem of Erdős and Sós concerning the uniform Turán densities of K_4^3-, introduce methods developed that have been developed to tackle extremal problems with quasirandom constraints, and discuss some of many open problems concerning extremal problems with quasirandom constraints. The talk will include results obtained jointly with various collaborators, particularly, with Matija Bucić, Jacob W. Cooper, Frederik Garbe, Daniel Iľkovič, Filip Kučerák, Ander Lamaison, Samuel Mohr, David Munhá Correia and Gábor Tardos.
February 26th	Talia Blum, Stanford
March 5th	Warut Thawinrak, UC Davis
March 12th	Peter Winkler, Dartmouth
March 19th	Sara Billey, University of Washington
March 26th	No seminar - spring break
April 2nd	Matthew Nicoletti, UC Berkeley
April 9th	Greta Panova, USC
April 16th	Kayla Wright, Oregon
April 23rd	Stephan Pfannerer, Waterloo
April 30th	Jessica Striker, NDSU

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