Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block Göttsche, further extended Göttsche Schroeter in the case rational curves. In this paper, we study polynomial behavior coefficients these invariants. We prove that small codegree are polynomials Newto...

Current massively open online courses (MOOCs) are providing several technical challenges for educators. One of these challenges is automated generation of questions, along with the solutions, in order to deal with a large number of students. Geometry is an important part of the high school curriculum. Hence, in this paper, we have focused on the high school geometry domain. We have proposed a f...

Point configurations and convex polytopes play central roles in computational geometry and discrete geometry. For many problems, their combinatorial structures, i.e., the underlying oriented matroids up to isomorphism, are often more important than their metric structures. For example, the convexity, the face lattice of the convex hull and all possible triangulations of a given point configurat...

M.C. Escher is noted for his repeating patterns, usually with animal motifs. For a few motifs he created more than one pattern with different combinatorial characteristics, even in different geometries, leading to the concept of families of patterns with the same motif but different combinatorics. Hyperbolic geometry is useful in that it provides an infinite number of combinatorial possibilitie...

In 1958, the Dutch artist M.C. Escher became the first person to create artistic patterns in hyperbolic geometry. He used the Poincar é circle model of hyperbolic geometry. Slightly more than 20 years later, my students and I implemented a computer program that could draw repeating hyperbolic patterns in this model. The ...

This work is a combinatorial study of quantum gravity models related to Lorentzian quantum gravity. These models are discrete combinatorial objects called dynamical triangulations. They are related to classical combinatorial objects: Dyck paths, heaps, . . . This work is in collaboration with Xavier Viennot and is part of the speaker’s thesis [4]. Quantum gravity is a quantum description of the...

In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold that are analogues of objects in differential geometry. We study a cochain product and prove several statements about its convergence to the wedge product on differential forms. Also, for cochains with an inner product, we define a combinatorial Hodge star operator, and describe some appl...

My research is situated at the interface of algebraic combinatorics, representation theory, algebraic geometry, and algebraic topology. It focuses on developing new combinatorial structures and methods for computation, particularly related to the representation theory of Lie algebras and modern Schubert calculus on flag manifolds. Thus, my work is part of the current trend in mathematics which ...

In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of n definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semi-algebraic and semi-Pfaffian sets, and as a result vastly increases the applicability of results on combinat...

We present a combinatorial polynomial time algorithm to compute a maximum stable set of a t-perfect graph. The algorithm rests on an ε-approximation algorithm for general set covering and packing problems and is combinatorial in the sense that it does not use an explicit linear programming algorithm or methods from linear algebra or convex geometry. Instead our algorithm is based on basic arith...