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Rustam Sadykov

Publications and source records attributed to Rustam Sadykov.

At least 19 recordsLinked to original sources
arXiv2025

On the Mather stability theorem for smooth maps

In [MaII] Mather proved that a smooth proper infinitesimally stable map is stable. This result is the key component of the Mather stability theorem [MaV], which can be reformulated as follows: a smooth proper map $f: M\to N$ is stable if and only if it is infinitesimally stable if and only if it satisfies the Mather normal crossing condition. The latter condition, roughly speaking, means that all map germs of $f$ are stable and $f$ maps the singular strata of $f$ to $N$ in a mutually transversal manner. In this note we adapt a short argument from the book by Golubitsky and Guillemin to derive the Mather stability theorem presented in [MaV] from the theorem in [MaII].

math.GT↗
arXiv2025

Cache-a-lot: Pushing the Limits of Unsatisfiable Core Reuse in SMT-Based Program Analysis

Satisfiability Modulo Theories (SMT) solvers are integral to program analysis techniques like concolic and symbolic execution, where they help assess the satisfiability of logical formulae to explore execution paths of the program under test. However, frequent solver invocations are still the main performance bottleneck of these techniques. One way to mitigate this challenge is through optimizations such as caching and reusing solver results. While current methods typically focus on reusing results from fully equivalent or closely related formulas, they often miss broader opportunities for reuse. In this paper, we propose a novel approach, Cache-a-lot, that extends the reuse of unsatisfiable (unsat) results by systematically considering all possible variable substitutions. This enables more extensive reuse of results, thereby reducing the number of SMT solver invocations and improving the overall efficiency of concolic and symbolic execution. Our evaluation, conducted against the state-of-the-art Utopia solution using two benchmark sets, shows significant improvements, particularly with more complex formulas. Our method achieves up to 74% unsat core reuse, compared to Utopia's 41%, and significant increase in the time savings. These results demonstrate that, despite the additional computational complexity, the broader reuse of unsat results significantly enhances performance, offering valuable advancements for formal verification and program analysis.

cs.SE↗
arXiv2024

Manturov Projection for Virtual Legendrian Knots in $ST^*F$

Kauffman virtual knots are knots in thickened surfaces $F\times R$ considered up to isotopy, stabilizations and destabilizations, and diffeomorphisms of $F\times R$ induced by orientation preserving diffeomorphisms of $F$. Similarly, virtual Legendrian knots, introduced by Cahn and Levi~\cite{CahnLevi}, are Legendrian knots in $ST^*F$ with the natural contact structure. Virtual Legendrian knots are considered up to isotopy, stabilization and destabilization of the surface away from the front projection of the Legendrian knot, as well as up to contact isomorphisms of $ST^*F$ induced by orientation preserving diffeomorphisms of $F$. We show that there is a projection operation $proj$ from the set of virtual isotopy classes of Legendrian knots to the set of isotopy classes of Legendrian knots in $ST^*S^2$. This projection is obtained by substituting some of the classical crossings of the front diagram for a virtual crossing. It restricts to the identity map on the set of virtual isotopy classes of classical Legendrian knots. In particular, the projection $proj$ extends invariants of Legendrian knots to invariants of virtual Legendrian knots. Using the projection $proj$, we show that the virtual crossing number of every classical Legendrian knot equals its crossing number. We also prove that the virtual canonical genus of a Legendrian knot is equal to the canonical genus. The construction of $proj$ is inspired by the work of Manturov.

math.SG↗
arXiv2023

Conjectures about virtual Legendrian knots and links

We formulate conjectures generalizing some known results to the category of virtual Legendrian knots. This includes statements relating virtual Legendrian knots to ordinary Legendrian knots, non-existence of positive virtual Legendrian self isotopy for the class of the fiber of $ST^*M$ and the conjectural relation of virtual Legendrian isotopy to causality in generalized spacetimes. We prove the conjectures in the case of $2$-dimensional $M$ and $(2+1)$-dimensional spacetimes. We also formulate and prove the version of the Arnold's $4$ cusp conjecture for virtual isotopies.

math.GT↗
arXiv2022

The minimal number of critical points of a smooth function on a closed manifold and the ball category

Introduced by Seifert and Threlfall, cylindrical neighborhoods of isolated critical points of smooth functions is an essential tool in the Lusternik- Schnirelmann theory. We conjecture that every isolated critical point of a smooth function admits a cylindrical ball neighborhood. We show that the conjecture is true for cone-like critical points, Cornea reasonable critical points, and critical points that satisfy the Rothe H hypothesis. In particular, the conjecture holds true at least for those critical points that are not infinitely degenerate. If, contrary to the assertion of the conjecture, there are isolated critical points that do not admit cylindrical ball neighborhoods, then we say that such critical points are exotic. We prove a Lusternik-Schnirelmann type theorem asserting that the minimal number of critical points of smooth functions without exotic critical points on a closed manifold of dimension at least 6 is the same as the minimal number of elements in a Singhof-Takens filling of M by smooth balls with corners.

math.GT↗
arXiv2022

A homotopy invariant of stable maps to oriented surfaces

The singular set of a generic map $f: M\to F$ of a manifold $M$ of dimension $m\ge 2$ to an oriented surface $F$ is a closed smooth curve $\Sigma(f)$. We study the parity of the number of components of $\Sigma(f)$. The image $f(\Sigma)$ of the singular set inherits canonical local orientations via so-called chessboard functions. Such a local orientation gives rise to the cumulative winding number $\omega(f)\in \frac{1}{2}\mathbb{Z}$ of $\Sigma(f)$. When the dimension of the manifold $M$ is even we also define an invariant $I(f)$ which is the residue class modulo $4$ of the sum of the number of components of $\Sigma(f)$, the number of cusps, and twice the number of self-intersection points of $f(\Sigma)$. Using the cumulative winding number and the invariant $I(f)$, we show that the parity of the number of connected components of $\Sigma(f)$ does not change under homotopy of $f$ provided that one of the following conditions is satisfied: (i) the dimension of $M$ is even, (ii) the singular set of the homotopy is an orientable manifold, or (iii) the image of the singular set of the homotopy does not have triple self-intersection points.

math.GT↗
arXiv2022

Isolated singularities of hypersurfaces

Introduced by Seifert and Threlfall, cylindrical neighborhoods is an essential tool in the Lusternik-Schnirelmann theory. We conjecture that every isolated critical point of a smooth function admits a cylindrical ball neighborhood. We show that the conjecture is true for cone-like critical points, Cornea reasonable critical points, and critical points that satisfy the Rothe H hypothesis. In particular, the conjecture holds true at least for those critical points that are not infinitely degenerate.

math.GT↗
arXiv2021

The weak b-principle: Mumford conjecture

In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b-principle and in dimension 2 approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension 2 to a closed simply connected manifold is bordant to a submersion. We show that the Madsen-Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b-principle. Namely, a version of the b-principle for oriented colored broken submersions together with the Harer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem.

math.AT↗
arXiv2018

On non-orientable surfaces in 4-manifolds

We find conditions under which a non-orientable closed surface S embedded into an orientable closed 4-manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem and the Auckly-Kim-Melvin-Ruberman-Schwartz "one is enough" theorem to the case of non-orientable surfaces.

math.GT↗
arXiv2017

Minimizing intersection points of curves under virtual homotopy

A flat virtual link is a finite collection of oriented closed curves $\mathfrak L$ on an oriented surface $M$ considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves $(L_1,L_2)$, we show that the minimal number of intersection points of curves in the virtual homotopy class of $(L_1, L_2)$ equals to the number of terms of a generalization of the Anderson--Mattes--Reshetikhin Poisson bracket. Furthermore, considering a single curve, we show that the minimal number of self-intersections of a curve in its virtual homotopy class can be counted by a generalization of the Cahn cobracket.

math.GT↗
arXiv2017

On LS-category and topological complexity of connected sum

The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\cat(M\# N)=\max\{\cat M,\cat N\}$. For topological complexity we prove the inequality $\TC (M\# N)\ge\max\{\TC M,\TC N\}$ for simply connected manifolds.

math.AT↗
arXiv2014

The number of framings of a knot in a 3-manifold

In view of the self-linking invariant, the number $|K|$ of framed knots in $S^3$ with given underlying knot $K$ is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that $|K|$ is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of $S^1\times S^2$; the knot $K$ need not be zero-homologous and the manifold is not required to be compact. We show that when $M$ is orientable, the number $|K|$ is infinite unless $K$ intersects a non-separating sphere at exactly one point, in which case $|K|=2$; the existence of a non-separating sphere implies that $M$ contains a connected sum factor of $S^1\times S^2$. For knots in nonorientable manifolds we show that if $|K|$ is finite, then $K$ is disorienting, or there is an isotopy from the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded $S^2$ or $\mathbb R P^2$ at exactly one point, or it intersects some embedded $S^2$ at exactly two points in such a way that a closed curve consisting of an arc in $K$ between the intersection points and an arc in $S^2$ is disorienting.

math.GT↗
arXiv2012

The space of non-degenerate closed curves in a Riemannian manifold

Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space Omega FTM on the bundle of frames in the tangent bundle of M. We show that Omega FTM is the group completion of LM, and prove that it is obtained by localizing LM with respect to adding a "small twist".

math.DG↗
arXiv2012

Approximations of periodic functions to R^n by curvatures of closed curves

We show that for any n real periodic functions f_1,..., f_n with the same period, such that f_i>0 for i 0, there is a closed curve in R^{n+1} with curvatures k_1, ..., k_n such that |k_i(t)-f_i(t)| < e for all i and t. This neither holds for closed curves in the hyperbolic space H^{n+1}, nor for parametric families of closed curves in R^{n+1}.

math.DG↗
arXiv2010

Topological properties of manifolds admitting a $Y^x$-Riemannian metric

A complete Riemannian manifold $(M, g)$ is a $Y^x_l$-manifold if every unit speed geodesic $\gamma(t)$ originating at $\gamma(0)=x\in M$ satisfies $\gamma(l)=x$ for $0\neq l\in \R$. B\'erard-Bergery proved that if $(M^m,g), m>1$ is a $Y^x_l$-manifold, then $M$ is a closed manifold with finite fundamental group, and the cohomology ring $H^*(M, \Q)$ is generated by one element. We say that $(M,g)$ is a $Y^x$-manifold if for every $\epsilon >0$ there exists $l>\epsilon$ such that for every unit speed geodesic $\gamma(t)$ originating at $x$, the point $\gamma(l)$ is $\epsilon$-close to $x$. We use Low's notion of refocussing Lorentzian space-times to show that if $(M^m, g), m>1$ is a $Y^x$-manifold, then $M$ is a closed manifold with finite fundamental group. As a corollary we get that a Riemannian covering of a $Y^x$-manifold is a $Y^x$-manifold. Another corollary is that if $(M^m,g), m=2,3$ is a $Y^x$-manifold, then $(M, h)$ is a $Y^x_l$-manifold for some metric $h.$

math.DG↗