Posets (partially ordered sets) are fundamental objects in algebraic combinatorics. On the one hand, this course will cover important partial orders on objects of interest in algebra, combinatorics, and geometry; such objects include permutations, integer partitions, set partitions, rooted plane trees, elements of Coxeter groups, faces of polytopes, and regions of hyperplane arrangements. On the other hand, we will discuss posets in their own right in topological, dynamical, and algebraic contexts. Particular topics we will discuss include order complexes, poset dynamics, combinatorial billiards, lattice theory, and polytopality.
Meeting times: T & TH 12:00-1:15 PM
Contact Information:
Office: SC 235
Email: colindefant@gmail.com
Office Hours: TH 1:30-2:30 PM or by appointment (this might change)
Potential Sources for Final Projects
(You are allowed to choose something that is not from this list.)
Lattices of acyclic pipe dreams https://arxiv.org/abs/2303.11025
Rowmotion in slow motion https://arxiv.org/abs/1712.10123
Acyclic reorientation lattices and their lattice quotients https://arxiv.org/abs/2111.12387
The canonical complex of the weak order https://arxiv.org/abs/2111.11553
Permutrees https://arxiv.org/abs/1606.09643
Toric partial orders https://arxiv.org/abs/1211.4247
Two poset polytopes https://link.springer.com/content/pdf/10.1007/BF02187680.pdf
Upho lattices I: examples and non-examples of cores https://arxiv.org/abs/2407.08013
Affine extended weak order is a lattice https://arxiv.org/abs/2311.05737
Mobius functions of lattices https://arxiv.org/abs/math/9801009
Posets of shuffles https://www.sciencedirect.com/science/article/pii/0097316588900180
Differential posets https://math.mit.edu/~rstan/pubs/pubfiles/77.pdf
Effective poset inequalities https://arxiv.org/abs/2205.02798
Permutation statistics and linear extensions of posets https://www.sciencedirect.com/science/article/pii/009731659190075R
Promotion of Kreweras words https://arxiv.org/abs/2005.14031
A unifying framework for the $\nu$-Tamari lattice and principal order ideals in Young's lattice https://arxiv.org/abs/2101.10425
The core label order of a congruence-uniform lattice https://arxiv.org/abs/1708.02104