Related papers: On the totally positive grassmannian
We introduce totally nonnegative Grassmannians over finite fields where an element of a finite field is nonnegative if it is a square of an element of the finite field. Explicit point counts are given in some special cases where we find new…
We notice new Hermitian counterpart of Swanson's Hamiltonian.
We settle in the affirmative the Graham-Sloane conjecture.
These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014. In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a…
We provide a short proof of a classical result of Kasteleyn, and prove several variants thereof. One of these results has become key in the parametrization of positroid varieties, and thus deserves the short direct proof which we provide.
We introduce a new factorial function which agrees with the usual Euler gamma function at both the positive integers and at all half-integers, but which is also entire. We describe the basic features of this function.
We explain how a slight variant in the use of our recursive algorithm leads to improve the known lower bounds for the absolute trace of a totally positive algebraic integer. We also link the absolute trace of a totally positive algebraic…
We give a new proof of a classical theorem on approximation of continuous functions on totally real sets
We provide a proof and a counterexample to two conjectures made by N. Kuznetsov.
We consider two {seemingly} different definitions of infinite words which contain {the} utmost number of palindromes. We show that these two definitions coincide. {The keynote of the proof is a meticulous inspection of properties of…
Cattani's theorem for graded Artinian Gorenstein algebras states that the ordinary Hodge-Riemann relations imply the mixed Hodge-Riemann relations under certain conditions. We give a new proof of this result for codimension two algebras.…
The main aim of this paper is to establish a deep link between the totally nonnegative grassmannian and the quantum grassmannian. More precisely, under the assumption that the deformation parameter $q$ is transcendental, we show that…
We present a question which implies a complete positive answer for the Bass-Quillen Conjecture.
We introduce a new criterion which if satisfied implies the Riemann hypothesis.
We consider the Pfaffian-Grassmannian equivalence from the motivic point of view. The main result is that under certain numerical conditions, both sides of the equivalence are related on the level of Chow motives. The consequences include a…
We define a word in two positive definite (complex Hermitian) matrices $A$ and $B$ as a finite product of real powers of $A$ and $B$. The question of which words have only positive eigenvalues is addressed. This question was raised some…
The theory of total positivity for reductive groups is here extended to the case of symmetric spaces.
We revisit the notion of one-sided recognizability of morphisms and its relation to two-sided recognizability.
We show the Graceful Tree Conjecture holds.
A problem of completing a linear map on C*-algebras to a completely positive map is analyzed. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some…