Enumerative Combinatorics

Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds

Categorical Invariants of Graphs and Matroids

Graphs and matroids are two of the most important objects in combinatorics.We study invariants of graphs and matroids that behave well with respect to certain morphisms by realizing these invariants as functors from a category of graphs (resp. matroids). For graphs, we study invariants that respect deletions and contractions ofedges. For an integer $g > 0$, we define a category of $\mathcal{G}^{op}_g$ of graphs of genus at most g where morphisms correspond to deletions and contractions. We prove that this category is locally Noetherian and show that many graph invariants form finitely generated modules over the category $\mathcal{G}^{op}_g$. This fact allows us to exihibit many stabilization properties of these invariants. In particular we show that the torsion that can occur in the homologies of the unordered configuration space of n points in a graph and the matching complex of a graph are uniform over the entire family of graphs with genus $g$. For matroids, we study valuative invariants of matroids. Given a matroid,one can define a corresponding polytope called the base polytope. Often, the base polytope of a matroid can be decomposed into a cell complex made up of base polytopes of other matroids. A valuative invariant of matroids is an invariant that respects these polytope decompositions. We define a category $\mathcal{M}^{\wedge}_{id}$ of matroids whose morphisms correspond to containment of base polytopes. We then define the notion of a categorical matroid invariant which categorifies the notion of a valuative invariant. Finally, we prove that the functor sending a matroid to its Orlik-Solomon algebra is a categorical valuative invariant. This allows us to derive relations among the Orlik-Solomon algebras of a matroid and matroids that decompose its base polytope viewed as representations of any group $\Gamma$ whose action is compatible with the polytope decomposition. This dissertation includes previously unpublished co-authored material

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Scaling Limits in Models of Statistical Mechanics

The emphasis of the workshop was on the deep relations between, on the one hand, recent advances in probabilistic investigation of statistical mechanical models and spatial stochastic processes and, on the other hand, rigorous field-theoretic and analytic methods of mathematical physics. There were 52 participants, including 6 postdocs and graduate students, working in diverse intertwining areas of probability, statistical mechanics and field theory. Specific topics addressed during the 24 talks include: Universality and critical phenomena, disordered models, Gaussian free field (GFF), stochastic representation of classical and quantum-mechanical models and related random interchange and permutation processes, random planar graphs and unimodular planar maps, random walks on critical graphs and the Alexander-Orbach conjecture, reinforced random walks and non-linear -models, metastability, aging, equilibrium and dynamics for continuum particles with hard core interactions, non-equilibrium dynamics and Toom’s interfaces