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We use Naito-Sagaki's work [S. Naito & D. Sagaki, J. Algebra 245 (2001) 395--412, J. Algebra 251 (2002) 461--474] on Lakshmibai-Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We…

This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an…

This is a footnote to earlier joint work with Yu. Berest, which constructed a bijection between the space of ideal classes of the Weyl algebra and a union of Calogero-Moser varieties. A key property of this bijection is that it is…

We study the problem of sampling weighted partial triangulations of a convex polygon. We consider the distribution where each partial triangulation $\sigma$ is chosen with probability proportional to $\lambda^{|\sigma|}$, where $\lambda>0$…

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic…

We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.

An important breakthrough in understanding the geometry of Schubert varieties was the introduction of the notion of Frobenius split varieties and the result that the flag varieties G/P are Frobenius split. The aim of this article is to give…

A bijection is given between multi-edge trees and 3-coloured Motzkin paths.

We study the combinatorics of Gr\"obner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by tableaux induced by matching fields in the sense of…

Let $G$ be a finite group, and let $X$ be a smooth, orientable, connected, closed 4-dimensional $G$-manifold. Let $\mathcal{S}$ be a smooth, embedded, $G$-invariant surface in $X$. We introduce the concept of a $G$-equivariant trisection of…

We introduce a new isomorphism invariant for generalized Baumslag-Solitar (GBS) groups, which we call the limit angle. Unlike previously known invariants, which are primarily algebraic, the limit angle admits a dynamical interpretation,…

We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials…

Based on character decomposition, a dual transformation is introduced leading to two formulations of the theory which should allow for a removal/softening of the sign problem in the original version. Very preliminar numerical results are…

We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex…

We study Gr\"obner degenerations of Schubert varieties inside flag varieties. We consider toric degenerations of flag varieties induced by matching fields and semi-standard Young tableaux. We describe an analogue of matching field ideals…

Arnol'd proved in 1992 that Springer numbers enumerate the Snakes, which are type $B$ analogs of alternating permutations. Chen, Fan and Jia in 2011 introduced the labeled ballot paths and established a ``hard'' bijection with snakes.…

We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for…

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating…

We prove new bijections between different variants of Dyck paths and integer compositions, which give combinatorial explanations of their simple counting formula $4^{n-1}$. These give relations between different statistics, such as the…

For each positive integer $n$, we construct a bijection between the odd partitions and the distinct partitions of $n$ which extends Bressoud's bijection between the odd-and-distinct partitions of $n$ and the splitting partitions of $n$. We…