Related papers: Crystals, regularisation and the Mullineux map
Partition functions of certain classes of "spin glass" models in statistical physics show strong connections to combinatorial graph invariants. Also known as homomorphism functions they allow for the representation of many such invariants,…
Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…
Periodic boundary condition (PBC) is a standard approximation for calculating crystalline materials properties. However, a PBC crystal is not the same as the real macroscopic crystal, therefore, if applied indiscriminately, it can lead to…
In this paper, we announce results from our thesis, which studies for the first time the categorification of the theory of generalized permutohedra. The vector spaces in the categorification are tightly constrained by certain continuity…
Motivated by a conjecture of Lian and Yau concerning the mirror map in string theory, we determine when the mirror map q-series of certain elliptic curve and K3 surface families are Hauptmoduln (genus zero modular functions). Our geometric…
Graph clustering aims at discovering a natural grouping of the nodes such that similar nodes are assigned to a common cluster. Many different algorithms have been proposed in the literature: for simple graphs, for graphs with attributes…
The problem of membrane topology in the matrix model of M-theory is considered. The matrix regularization procedure, which makes a correspondence between finite-sized matrices and functions defined on a two-dimensional base space, is…
We establish a new simple explicit description of combinatorial wall-crossing for the rational Cherednik algebra applied to the trivial representation. In this way we recover a theorem of P. Dimakis and G. Yue. We also present two…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear…
The Poincare series of a multi-index filtration on the ring of germs of functions can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring. Here this integral is considered with…
We establish a Poincar\'e-Dulac theorem for sequences (G_n)_n of holomorphic contractions whose differentials d_0 G_n split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of…
This paper establishes combinatorial characterisations of forced-symmetric and forced-periodic rigidity (under a fixed lattice) of bar-joint frameworks in non-Euclidean normed planes. In $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$,…
In this paper we introduce a new clustering technique called Regularity Clustering. This new technique is based on the practical variants of the two constructive versions of the Regularity Lemma, a very useful tool in graph theory. The…
In this paper, we study and build the Hamiltonian system attached to any $\mathfrak{gl}_2(\mathbb{C})$ meromorphic connection with an arbitrary number of non-ramified poles of arbitrary degrees. In particular, we propose the Lax pairs and…
The symmetric group on a set acts transitively on its subsets of a given size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from…
We employ projective Fra\"iss\'e theory to define the "generic combinatorial $n$-simplex" as the pro-finite, simplicial complex that is canonically associated with a family of simply defined selection maps between finite triangulations of…
We introduce a type $A$ crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing…
The crystalline period map is a tool for linearizing $p$-divisible groups. It has been applied to study the Langlands correspondences, and has possible applications to the homotopy groups of spheres. The original construction of the period…
Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more…