A compact Lie group G and a faithful complex representation V determine the Sato-Tate measure μG,V on C, defined as the direct image of Haar measure with respect to the character map g 7→ tr(g|V ). We give a necessary and sufficient condition for a Sato-Tate measure to be an isolated point in the set of all Sato-Tate measures, regarded as a subset of the space of distributions on C. In particul...

entropy, and let Bn(X) be its set of words of length n. Define a random subset ω of Bn(X) by independently choosing each word from Bn(X) with some probability α. Let Xω be the (random) SFT built from the set ω. For each 0 ≤ α ≤ 1 and n tending to infinity, we compute the limit of the likelihood that Xω is empty, as well as the limiting distribution of entropy for Xω. For α near 1 and n tending ...

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K, and denote its Jacobian by J . Let d ≥ 1 be an integer and denote the d-th symmetric power of C by C(d). In this paper we adapt the classic Chabauty–Coleman method to study the K-rational points of C(d). Suppose that J(K) has Mordell–Weil rank at most g− d. We give an explicit and practical criterion...

Given a real closed field R, we define a real algebraic manifold as an irreducible nonsingular algebraic subset of some R. This paper is devoted to the notion of deformation of real algebraic manifolds. The main purpose is to prove rigorously the validity of the following informal principle, which is in sharp contrast with the compact complex case: The algebraic structure of every real algebrai...

We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg's deficiency zero theorem then implies that such a distribution exists ...

We prove the André–Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let n ≥ 0, and let Σ be a subset of Cn consisting of points all of whose coordinates are j-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreduc...

In this paper, we study the SL2(C) character variety of a hyperbolic link in S. We analyze a special smooth projective variety Y h arising from some 1-dimensional irreducible slices on the character variety. We prove that a natural symbol obtained from these 1-dimensional slices is a torsion in K2(C(Y )). By using the regulator map from K2 to the corresponding Deligne cohomology, we get some va...

For systems with a finite number of degrees of freedom, it is shown in [1] that first class constraints are Abelianizable if the FaddeevPopov determinant is not vanishing for some choice of subsidiary constraints. Here, for irreducible first class constraint systems with SO(3) or SO(4) gauge symmetries, including a subset of coordinates in the fundamental representation of the gauge group, we e...

The singular points of an irreducible plane cubic curve are quite limited: one knot/node, or one cusp. Our research starts originally with the Descartes Folium, which has a knot/node, and is able to have many group structures. The original results are concentrated in six directions: (i) special structures on affine algebraic varieties, (ii) theory of K-groups, (iii) isomorphisms of K-groups, (i...

Ulam asked in 1945 if there is an everywhere dense rational set, i.e. a point set in the plane with all its pairwise distances rational. Erdős conjectured that if a set S has a dense rational subset, then S should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erdős’ conjecture for algeb...