Related papers: On a conjecture of Varchenko
In bounding the homology of a manifold, Forman's Discrete Morse theory recovers the full precision of classical Morse theory: Given a PL triangulation of a manifold that admits a Morse function with c_i critical points of index i, we show…
We prove the convergence case of Khintchine's theorem, with general approximation functions that are not necessarily monotonic, for analytic nonplanar manifolds over local fields of positive characteristic. Our approach is based on the…
A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of…
We prove that any compact complex surface with positive first Chern class admits an Einstein metric which is conformally related to a Kaehler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex…
We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex…
We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected)…
We compute by Monte Carlo numerical simulations the critical exponents of two-dimensional scalar field theories at the $\lambda\phi^6$ tricritical point. The results are in agreement with the Zamolodchikov conjecture based on conformal…
We explore systems of polynomial equations where we seek complex solutions with absolute value 1. Geometrically, this amounts to understanding intersections of algebraic varieties with tori -- Cartesian powers of the unit circle. We study…
The classic Schneider-Lang theorem in transcendence theory asserts that there are only finitely many points at which algebraically independent complex meromorphic functions of finite order of growth can simultaneously take values in a…
We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on…
To a transitive pseudo-Anosov flow $\varphi$ on a $3$-manifold $M$ and a representation $\rho$ of $\pi_1(M)$, we associate a zeta function $\zeta_{\varphi,\rho}(s)$ defined for $\Re s \gg 1$, generalizing the Anosov case. For a class of…
We give results on the asymptotic in Waring's problem over function fields that are stronger than the results obtained over the integers using the main conjecture in Vinogradov's mean value theorem. Similar estimates apply to Manin's…
A variational formula is derived by combining the Gaussian volume of the epigraph of a convex function $\varphi$ and the perturbation of $\varphi$ via the infimal convolution. This formula naturally leads to a Borel measure on…
In this paper, making use of Theorem 2 of [5], we establish a new four critical points theorem which can be regarded as a companion to Theorem 1 of [4]. We also present an application to the Dirichlet problem for a class of quasilinear…
An Eulerian orientation is an orientation of the edges of a graph such that every vertex is balanced: its in-degree equals its out-degree. Counting Eulerian orientations corresponds to the crucial partition function in so-called ``ice-type…
In the geometric theory of defects, media with a spin structure, for example, ferromagnet, is considered as a manifold with given Riemann--Cartan geometry. We consider the case with the Euclidean metric corresponding to the absence of…
The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational normal curve at real points. This sparked…
Given a sequence of uniformly convex norms $ \phi_h $ on $ \mathbf{R}^{n+1} $ converging to an arbitrary norm $ \phi $, we prove rigidity of $ L^1 $-accumulation points of sequences of sets $ E_h \subseteq \mathbf{R}^{n+1} $ of finite…
We give a new proof of Witten asymptotic conjecture for Seifert manifolds with non vanishing Euler class and one exceptional fiber. Our method is based on semiclassical analysis on a two dimensional phase space torus. We prove that the…
We propose that geometric quantization of symplectic manifolds is the arrow part of a functor, whose object part is deformation quantization of Poisson manifolds. The `quantization commutes with reduction' conjecture of Guillemin and…