Related papers: Rational curves on Grassmannians: systems theory, …
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn…
We survey a variety of cosmological problems where the issue of generality has arisen. This is aimed at providing a wider context for many claims and deductions made when philosophers of science choose cosmological problems for…
Any acceptable quantum gravity theory must allow us to recover the classical spacetime in the appropriate limit. Moreover, the spacetime geometrical notions should be intrinsically tied to the behavior of the matter that probes them. We…
We compute the Gromov-Witten invariants of the projective plane blown up in r general points. These are determined by associativity from r+1 intial values. Applications are given to the enumeration of rational plane curves with prescribed…
We describe an approach to the question of finding real solutions to problems of enumerative geometry, in particular the question of whether a problem of enumerative geometry can have all of its solutions be real. We give some methods to…
This paper shows that the well-known curve optimization problems which lead to the straight line, the catenary curve, the brachistochrone, and the circle, can all be handled using a unified formalism. Furthermore, from the general…
We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative…
This contribution summarizes recent work of the authors that combines methods from dynamical systems theory (discrete Painlev\'e equations) and asymptotic analysis of orthogonal polynomial recurrences, to address long-standing questions in…
We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{P}^3$, whose image lies in a $\mathbb{P}^2$, passing through $r$ lines and $s$ points, where $r + 2s = 3d+2$. This can be viewed as a family version of…
Quantum mechanics emerged as the result of a successful resolution of stringent empirical and profound conceptual conflicts within the development of atomic physics at the beginning of the last century. At first glance, it seems to be…
Generalized unitarity cut of a Feynman diagram generates an algebraic system of polynomial equations. At high-loop levels, these equations may define a complex curve or a (hyper-)surface with complicated topology. We study the curve cases,…
The existence of the Herman ring of a function adds interest and complexity to the dynamics of the function. We present a detailed and understandable summary of the core discoveries and recent developments on the Herman ring of rational and…
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
We introduce and study a new mathematical structure in the generalised (quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert calculus to a quantum integrable system known in the physics literature as the asymmetric…
A logical system derived from linear logic and called QMLL is introduced and shown able to capture all unitary quantum circuits. Conversely, any proof is shown to compute, through a concrete GoI interpretation, some quantum circuits. The…
In this paper, we prove three related results; (1) Extension of our result in [10] to all generic hypersurfaces. More precisely, the normal sheaf of a generic rational map $c_0$ to a generic hypersurface $X_0$ of $\mathbf P^n, n\geq 4$ has…
In this paper we consider a generalization of a well known result by Veronese about rational normal curves. More precisely, given a collection of linear spaces in $\PP^n$ we study the existence of rational normal curves intersecting each…
We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. We combine the methods and results from three different papers.
In this paper, I consider the issue of how two mathematical models of modern physics, the variational principles and the quantum path integral formalism, relate to reality. I assume that the observed phenomena are consistent with the…
We discuss first order systems of rational difference equations which have the property that lines through the origin are mapped into lines through the origin. We call such systems projective systems of rational difference equations and we…