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Deformations of topological open string theories are described, with an emphasis on their algebraic structure. They are encoded in the mixed bulk-boundary correlators. They constitute the Hochschild complex of the open string algebra -- the…
We generalize Coincidence theorem due to Walsh for symmetric and linear polynomial in n complex variables, that is linear in each of them having total degre n. We discuss case when total degree is smaller then n. This case has been already…
We consider harmonic immersions in $\R^{\N}$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. We prove that such surfaces have finite total…
Both Morse theory and Lusternik-Schnirelmann theory link algebra, topology and analysis in a geometric setting. The two theories can be formulated in finite geometries like graph theory or within finite abstract simplicial complexes. We…
Geometric discretisation draws analogies between discrete objects and operations on a complex with continuum ones on a manifold. We generalise the theory to the cubic case and incorporate metric, by adding volume factors to our discrete…
Let X,Y be finite sets and T a set of functions from X -> Y which we will call "tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such "tableau complexes" have many nice…
Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on…
For the ordered set $[n]$ of $n$ elements, we consider the class $\Bscr_n$ of bases $B$ of tropical Pl\"ucker functions on $2^{[n]}$ such that $B$ can be obtained by a series of mutations (flips) from the basis formed by the intervals in…
Area and orientation preserving diffeomorphisms of the standard 2-disc, referred to as symplectomorphisms of $\mathbb{D}^{2}$, allow decompositions in terms of positive twist diffeomorphisms. Using the latter decomposition we utilize the…
We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of…
We prove that the minimum number of critical points of a Weinstein Morse function on a Weinstein domain of dimension at least six is at most two more than the minimum number of critical points of a smooth Morse function on that domain; if…
This paper shows that discrete Morse-Bott theory can be developed as a natural extension of R. Forman's discrete Morse theory by improving the definition of the discrete Morse-Bott function introduced by S. Yaptieu. To this end, we…
We study discrete complex analysis and potential theory on a large family of planar graphs, the so-called isoradial ones. Along with discrete analogues of several classical results, we prove uniform convergence of discrete harmonic…
Studying crepant blow-ups of (compound) du Val singularities, we classify complexes of coherent sheaves which admit no negative self-extensions -- such a complex, up to flops and mutation equivalences, must either be (1) a module over a…
We present a Morse-theoretic characterization of collapsibility for 2-dimensional acyclic simplicial complexes by means of the values of normalized optimal combinatorial Morse functions.
In the paper the foundation of the $k$-orbit theory is developed. The theory opens a new simple way to the investigation of groups and multidimensional symmetries. The relations between combinatorial symmetry properties of a $k$-orbit and…
Bierstone and Parusi\'nski studied the desingularization of $d$-dimensional closed subanalytic sets and in particular of $d$-dimensional closed semialgebraic sets. Their main tools are Hironaka's desingularization of real algebraic sets (to…
This is a continuation of developing mutation theory in exact WKB analysis using the framework of cluster algebras. Here we study the Schrodinger equation on a compact Riemann surface with turning points of simple-pole type. We show that…
We develop the notion of a "filtered cospan" as an algebraic object that stands in the same relation to interlevel persistence modules as filtered chain complexes stand with respect to sublevel persistence modules. This relation is…
We use discrete Morse theory to provide another proof of Bernini, Ferrari, and Steingrimson's formula for the Mobius function of the consecutive pattern poset. In addition, we are able to determine the homotopy type of this poset. Earlier,…