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The F\'elix-Tanr\'e rational model for the polyhedral product of a fibre inclusion is considered. In particular, we investigate the rational model for the polyhedral product of a pair of Lie groups corresponding to arbitrary simplicial…
We define the unipotent tropical fundamental group of a polyhedral complex in $\mathbb{R}^n$ as the Tannakian fundamental group of the category of unipotent tropical vector bundles with integrable connection. We show that it is computable…
In this paper, we make the case that Clifford algebra is the natural framework for root systems and reflection groups, as well as related groups such as the conformal and modular groups: The metric that exists on these spaces can always be…
We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a…
We consider Noether symmetries within Hamiltonian setting as transformations that preserve Poincar\'e-Cartan form, i.e., as symmetries of characteristic line bundles of nondegenerate 1-forms. In the case when the Poincar\'e-Cartan form is…
The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold (C-space) consists not…
In this article we give an expression of the motivic Milnor fiber at infinity and the motivic nearby cycles at infinity of a polynomial $f$ in two variables with coefficients in an algebraic closed field of characteristic zero. This…
We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$. We study its interplay with the general presentation for any finite-dimensional basic algebra. We give an interpretation of evaluating $f_M$ at a weight vector.…
We demonstrate the common bihamiltonian nature of several integrable systems. The first one is an elliptic rotator that is an integrable Euler-Arnold top on the complex group GL(N) for any $N$, whose inertia ellipsiod is related to a choice…
Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton--Sturmfels, and…
For each quadratic form Q in Quad(V) over a given vector space over a field R we have the Clifford algebra Cl(V,Q) defined as the quotient T(V)/I(Q) of the tensor algebra T(V) over the two-sided ideal generated by expressions of the form $x…
The topological framework of circuit topology has recently been introduced to complement knot theory and to help in understanding the physics of molecular folding. Naturally evolved linear molecular chains, such as proteins and nucleic…
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley-Menger ideal for $n$ points in 2D. We…
The Tolman-Bondi (TB) model is defined up to some transformation of a co-moving coordinate but the transformation is not fixed. The use of an arbitrary co-moving system of coordinates leads to the solution dependent on three functions $f,…
We study the interplay between noncommutative tori and noncommutative elliptic curves through a category of equivariant differential modules on $\mathbb{C}^*$. We functorially relate this category to the category of holomorphic vector…
We consider Milnor's "tower algorithm" in the space of piecewise monotone maps, an iterative algorithm on the space of metrics which unifies, on the one hand, Thurston's iterative scheme which converges to holomorphic models, and, on the…
We classify projective toric manifolds whose dual variety is not a hypersurface in the dual projective space. Under the standard dictionary between toric geometry and convex geometry, they correspond to certain convex Delzant integer…
Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are…
Full binary trees naturally represent commutative non-associative products. There are many important examples of these products: finite-precision floating-point addition and NAND gates, among others. Balance in such a tree is highly…
R. P. Feynman was quite fond of inventing new physics. It is shown that some of his physical ideas can be supported by the mathematical instruments available from the Lorentz group. As a consequence, it is possible to construct a…