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Two methods of constructing 2D Toda $\tau$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group…
We introduce the notion of "color rules" for computing class functions of $Z_k \wr S_n$, where $Z_k$ is the cyclic group of order $k$ and $S_n$ is the symmetric group on $n$ letters. Using a general sign-reversing involution and a map of…
Sunada's work on crystallography emphasizes the role of the "maximal abelian cover" of a graph $X$. This is a covering space of $X$ for which the group of deck transformations is the first homology group $H_1(X,\mathbb{Z})$. An embedding of…
Knots and knotted fields enrich physical phenomena ranging from DNA and molecular chemistry to the vortices of fluid flows and textures of ordered media. Liquid crystals provide an ideal setting for exploring such topological phenomena…
In this paper we calculate certain chiral quantities from the cyclic permutation orbifold of a general completely rational net. We determine the fusion of a fundamental soliton, and by suitably modified arguments of A. Coste , T. Gannon and…
We consider the role of deformations in graphene heterostructures with hexagonal crystals (including strain, wrinkles and dislocations) on the geometrical properties of moire patterns characteristic for a pair of two incommensurate…
We propose the use of two-dimensional photonic crystals with engineered defects for the generation of an arbitrary-profile beam from a focused input beam. The cylindrical harmonics expansion of complex-source beams is derived and used to…
Patterned two-dimensional (2D) magnetic nanostructures constitute geometry-engineered spin systems in which exchange, anisotropy, dipolar coupling, and finite-size effects operate on comparable energy scales. Spatial modulation of…
We study the relation between $\mathcal{W}_{1+\infty}$ algebra and Arbesfeld-Schiffmann-Tsymbaliuk Yangian using the Maulik-Okounkov R-matrix. The central object linking these two pictures is the Miura transformation. Using the results of…
A systematic method for constructing trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two affinizable representations of a quantum algebra or superalgebra has been developed by the Brisbane group and…
In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are…
We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all $n_1+m$ sources are separated from all $n_2+m$ sinks, we can construct a cluster-algebra realization of elements of an affine…
For the flip action of $\mathbb{Z}_2$ on an $n$-dimensional noncommutative torus $A_\theta,$ using an exact sequence by Natsume, we compute the K-theory groups of $A_\theta \rtimes \mathbb{Z}_2$. The novelty of our method is that it also…
Here, we propose a comprehensive first-principle atomistic approach to predict the Wulff-Kaischew equilibrium shape of crystals heterogeneously integrated on a dissimilar material. This method uses both reconstructed surface and interface…
We introduce ``virtual'' crystals of the affine types $g=D_{n+1}^{(2)}$, $A_{2n}^{(2)}$ and $C_n^{(1)}$ by naturally extending embeddings of crystals of types $B_n$ and $C_n$ into crystals of type $A_{2n-1}$. Conjecturally, these virtual…
The subject matter of this paper is the geometry of the affine group over the integers, $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$. Turing-computable complete $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-orbit invariants are…
By combining two distinct renormalization group transformations, opposing scale transformations, we obtain a composite transformation which does not rescale the system, and drives it to a "geometrical" fixed point, controlling the effective…
We construct finite $R$-matrices for the first fundamental representation $V$ of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ for classical $\mathfrak{g}$, both through the decomposition of $V\otimes V$ into irreducibles…
We define an integral form of the deformed W-algebra of type gl_r, and construct its action on the K-theory groups of moduli spaces of rank r stable sheaves on a smooth projective surface S, under certain assumptions. Our construction…
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…