Asymptotics of local genus distributions and the genus distribution of the complete graph

We are interested in $2$-cell embeddings of graphs on orientable surfaces. The distribution of genus across all the embeddings of a graph is only known explicitly for a few graphs. In order to study these genus distributions, [Gross et al., European J. Combin. 2016] introduced local genus distributions. These describe how the faces are distributed around a single vertex across all the embeddings of a graph. Not much is known about the local or non-local genus distributions for general graphs. [F{\'e}ray, 2015] noted that it is possible to study local genus distributions by studying the product of conjugacy classes $C_n C_\lambda$. We therefore start by showing a central limit theorem on this product. We show that the difference between $C_n C_\lambda$ and the uniform distribution on all even/odd permutations is at most the number of fixed points in $\lambda$ plus one, divided by $\sqrt{n-1}$. This can be thought of as an analogue of a result of [Chmutov and Pittel, Adv. Appl. Math. 2016], who show a similar result for the product $C_{2^n} C_\lambda$. We use this to show that any graph with large vertex degrees and a small average number of faces has an asymptotically uniform local genus distribution at each of its vertices. Then, we study the whole genus distribution of the complete graph. We show that a portion of the complete graph of size $(1-o(1))|K_n|$ has the same genus distribution as the set of all permutations, up to parity

Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications

Lattice polytopes arise naturally in many different branches of pure and applied mathematics such as number theory, commutative algebra, combinatorics, toric geometry, optimization, and mirror symmetry. The miniworkshop on “Lattice polytopes: methods, advances, applications” focused on two current hot topics: the classification of lattice polytopes with few lattice points and unimodality questions for Ehrhart polynomials. The workshop consisted of morning talks on recent breakthroughs and new methods, and afternoon discussion groups where participants from a variety of different backgrounds explored further applications, identified open questions and future research directions, discussed specific examples and conjectures, and collaboratively tackled open research problems

Quantum Barnes function as the partition function of the resolved conifold

We suggest a new strategy for proving large $N$ duality by interpreting Gromov-Witten, Donaldson-Thomas and Chern-Simons invariants of a Calabi-Yau threefold as different characterizations of the same holomorphic function. For the resolved conifold this function turns out to be the quantum Barnes function, a natural $q$-deformation of the classical one that in its turn generalizes Euler's gamma function. Our reasoning is based on a new formula for this function that expresses it as a graded product of $q$-shifted multifactorials.Comment: 47 pages, 7 figure

Tautological classes of matroids

We introduce certain torus-equivariant classes on permutohedral varieties which we call "tautological classes of matroids" as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between K-theory and Chow theory.Comment: 69 pages; comments welcome. v2: minor edits, addition of subsection 10.

Integration on complex Grassmannians, deformed monotone Hurwitz numbers, and interlacing phenomena

We introduce a family of polynomials, which arise in three distinct ways: in the large $N$ expansion of a matrix integral, as a weighted enumeration of factorisations of permutations, and via the topological recursion. More explicitly, we interpret the complex Grassmannian $\mathrm{Gr}(M,N)$ as the space of $N \times N$ idempotent Hermitian matrices of rank $M$ and develop a Weingarten calculus to integrate products of matrix elements over it. In the regime of large $N$ and fixed ratio $\frac{M}{N}$, such integrals have expansions whose coefficients count factorisations of permutations into monotone sequences of transpositions, with each sequence weighted by a monomial in $t = 1 - \frac{N}{M}$. This gives rise to the desired polynomials, which specialise to the monotone Hurwitz numbers when $t = 1$. These so-called deformed monotone Hurwitz numbers satisfy a cut-and-join recursion, a one-point recursion, and the topological recursion. Furthermore, we conjecture on the basis of overwhelming empirical evidence that the deformed monotone Hurwitz numbers are real-rooted polynomials whose roots satisfy remarkable interlacing phenomena. An outcome of our work is the viewpoint that the topological recursion can be used to "topologise" sequences of polynomials, and we claim that the resulting families of polynomials may possess interesting properties. As a further case study, we consider a weighted enumeration of dessins d'enfant and conjecture that the resulting polynomials are also real-rooted and satisfy analogous interlacing properties.Comment: 30 pages, 3 figures. Comments welcome