Thin Distance-Regular Graphs with Classical Parameters $(D,q,q, \frac{q^{t}-1}{q-1}-1)$ with $t> D$ are the Grassmann Graphs

In the survey paper by Van Dam, Koolen and Tanaka (2016), they asked to classify the thin $Q$-polynomial distance-regular graphs. In this paper, we show that a thin distance-regular graph with the same intersection numbers as a Grassmann graph $J_q(n, D)~ (n \geqslant 2D)$ is the Grassmann graph if $D$ is large enough.

Counting Non-Crossing Permutations on Surfaces of any Genus

Given a surface with boundary and some points on the boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on the surface. If only bigons are allowed, then one obtains the notion of arc diagrams, whose enumeration is known to have a rich structure. We show that the count of polygon diagrams on surfaces with any genus and number of boundary components exhibits similar structure. In particular it is almost polynomial in the number of points on the boundary components, and the leading coefficients of those polynomials are intersection numbers on compactified moduli spaces of curves.

On Baer cones in $$\mathrm {PG}(3, q)$$

Recently, in Innamorati and Zuanni (J. Geom 111:45, 2020. 10.1007/s00022-020-00557-0) the authors give a characterization of a Baer cone of $$\mathrm {PG}(3, q^2)$$ PG ( 3 , q 2 ) , q a prime power, as a subset of points of the projective space intersected by any line in at least one point and by every plane in $$q^2+1$$ q 2 + 1 , $$q^2+q+1$$ q 2 + q + 1 or $$q^3+q^2+1$$ q 3 + q 2 + 1 points. In this paper, we show that a similar characterization holds even without assuming that the order of the projective space is a square, and weakening the assumptions on the three intersection numbers with respect to the planes.

Intersection numbers on $$ {\overline{M}}_{g,n} $$ and BKP hierarchy

In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar description for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.

On non-proper intersections and local intersection numbers

Given equidimensional (generalized) cycles $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 on a complex manifold Y we introduce a product $$\mu _1\diamond _{Y} \mu _2$$ μ 1 ⋄ Y μ 2 that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. If Y is projective, then given a very ample line bundle $$L\rightarrow Y$$ L → Y we define a product $$\mu _1{\bullet _L}\mu _2$$ μ 1 ∙ L μ 2 whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 are effective, this product satisfies a Bézout inequality. If $$i:Y\rightarrow {\mathbb P}^N$$ i : Y → P N is an embedding such that $$i^*\mathcal O(1)=L$$ i ∗ O ( 1 ) = L , then $$\mu _1{\bullet _L}\mu _2$$ μ 1 ∙ L μ 2 can be expressed as a mean value of Stückrad–Vogel cycles on $${\mathbb P}^N$$ P N . There are quite explicit relations between $${\diamond }_Y$$ ⋄ Y and $${\bullet _L}$$ ∙ L .

Intersection Numbers and the Statement of the Disc Embedding Theorem

‘Intersection Numbers and the Statement of the Disc Embedding Theorem’ provides detailed definitions of some of the notions involved in the statement of the disc embedding theorem, focusing specifically on intersection numbers. The chapter begins with a detailed analysis of immersions, regular homotopies, finger moves, and Whitney moves. Then it defines intersection and self-intersection numbers for families of discs and spheres, taking values in the group ring of the fundamental group of the ambient space, with the correct relations. Then it enumerates certain properties of intersection numbers, in particular relating them to the existence of Whitney discs. This work enables the disc embedding theorem to be stated carefully.

Notes on the Leonard System Classification

Around 2001 we classified the Leonard systems up to isomorphism. The proof was lengthy and involved considerable computation. In this paper we give a proof that is shorter and involves minimal computation. We also give a comprehensive description of the intersection numbers of a Leonard system.