Alexander Grigor'Yan
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Alexander Grigor'Yan has a total of 47 co-authors. Here are the top co-authors by number of shared publications:
- Yury Vladimirovich Muranov
University of Warmia and Mazury in Olsztyn, Olsztyn, Poland
5 shared publications - Alexander Bendikov
Wroclaw University
4 shared publications - Jiaxin Hu
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
4 shared publications - Shing-Tung Yau
Beijing Institute of Mathematical Sciences and Applications, Beijing, China
4 shared publications - Hong-Wei Zhang
Ghent University
3 shared publications - Jun Cao
School of Mathematical Sciences, Zhejiang University of Technology, Hangzhou, People’s Republic of China
3 shared publications
Latest Publications
- 17 July 2026
- Published by Springer Nature in Mathematische Annalen
- Vol. 396 (1) , 4
Construction of heat kernels on metric measure spaces via multiresolution analysis
- 17 July 2026
- Published by Springer Nature in Mathematische Annalen
- Vol. 396 (1) , 4
Construction of heat kernels on metric measure spaces via multiresolution analysis
- 14 January 2026
- Published by Walter de Gruyter GmbH in Forum Mathematicum
A long exact sequence of path homology of digraphs
In this paper, we develop a long exact sequence for the path homology of digraphs, providing a useful tool for computing the path homology of digraphs. One application of this result is the proof of a conjecture proposed by S. Chowdhury, which was initially observed through extensive computational experiments. Another interesting application demonstrates that the path homology of n -dimensional grid-like digraphs is concentrated in dimension ≤ n - 1 {\leq n-1} .
- 14 January 2026
- Published by Walter de Gruyter GmbH in Forum Mathematicum
A long exact sequence of path homology of digraphs
In this paper, we develop a long exact sequence for the path homology of digraphs, providing a useful tool for computing the path homology of digraphs. One application of this result is the proof of a conjecture proposed by S. Chowdhury, which was initially observed through extensive computational experiments. Another interesting application demonstrates that the path homology of n -dimensional grid-like digraphs is concentrated in dimension ≤ n - 1 {\leq n-1} .
- 12 January 2026
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- p. 1-41
Rigidity for the heat equation with density on Riemannian manifolds through a conformal change
We investigate uniqueness of solution to the heat equation with a density $\rho$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_{\phi}$ . We distinguish between the cases $p \gt 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $\rho$ . We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $\rho$ .- 12 January 2026
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- p. 1-41
Rigidity for the heat equation with density on Riemannian manifolds through a conformal change
We investigate uniqueness of solution to the heat equation with a density $\rho$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_{\phi}$ . We distinguish between the cases $p \gt 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $\rho$ . We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $\rho$ .- Open Access
Rigidity for the heat equation with density on Riemannian manifolds through a conformal change
We investigate uniqueness of solution to the heat equation with a density $ρ$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_φ$. We distinguish between the cases $p > 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $ρ$. We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $ρ$.Open AccessRigidity for the heat equation with density on Riemannian manifolds through a conformal change
We investigate uniqueness of solution to the heat equation with a density $ρ$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_φ$. We distinguish between the cases $p > 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $ρ$. We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $ρ$. - 1 January 2024
- Published by International Press of Boston in Pure and Applied Mathematics Quarterly
- Vol. 20 (2) , 703-755
Analytic and Reidemeister torsions of digraphs and path complexes
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2024
- Published by International Press of Boston in Pure and Applied Mathematics Quarterly
- Vol. 20 (2) , 703-755
Analytic and Reidemeister torsions of digraphs and path complexes
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 December 2023
- Published by Pleiades Publishing Ltd in Proceedings of the Steklov Institute of Mathematics
- Vol. 323 (1) , 12-46
Hierarchical Schrödinger Operators with Singular Potentials
We consider the operator \(H=L+V\) that is a perturbation of the Taibleson–Vladimirov operator \(L=\mathfrak{D}^\alpha\) by a potential \(V(x)=b\|x\|^{-\alpha}\), where \(\alpha>0\) and \(b\geq b_*\). We prove that the operator \(H\) is closable and its minimal closure is a nonnegative definite self-adjoint operator (where the critical value \(b_*\) depends on \(\alpha\)). While the operator \(H\) is nonnegative definite, the potential \(V(x)\) may well take negative values as \(b_*<0\) for all \(0<\alpha<1\). The equation \(Hu=v\) admits a Green function \(g_H(x,y)\), that is, the integral kernel of the operator \(H^{-1}\). We obtain sharp lower and upper bounds on the ratio of the Green functions \(g_H(x,y)\) and \(g_L(x,y)\).- 1 December 2023
- Published by Pleiades Publishing Ltd in Proceedings of the Steklov Institute of Mathematics
- Vol. 323 (1) , 12-46
Hierarchical Schrödinger Operators with Singular Potentials
We consider the operator \(H=L+V\) that is a perturbation of the Taibleson–Vladimirov operator \(L=\mathfrak{D}^\alpha\) by a potential \(V(x)=b\|x\|^{-\alpha}\), where \(\alpha>0\) and \(b\geq b_*\). We prove that the operator \(H\) is closable and its minimal closure is a nonnegative definite self-adjoint operator (where the critical value \(b_*\) depends on \(\alpha\)). While the operator \(H\) is nonnegative definite, the potential \(V(x)\) may well take negative values as \(b_*<0\) for all \(0<\alpha<1\). The equation \(Hu=v\) admits a Green function \(g_H(x,y)\), that is, the integral kernel of the operator \(H^{-1}\). We obtain sharp lower and upper bounds on the ratio of the Green functions \(g_H(x,y)\) and \(g_L(x,y)\).- 26 August 2023
- Published by Springer Nature in Mathematische Annalen
- Vol. 389 (3) , 2411-2467
Parabolic mean value inequality and on-diagonal upper bound of the heat kernel on doubling spaces
- 26 August 2023
- Published by Springer Nature in Mathematische Annalen
- Vol. 389 (3) , 2411-2467
Parabolic mean value inequality and on-diagonal upper bound of the heat kernel on doubling spaces
- 26 June 2023
- Published by Springer Nature
- p. 165-179
Asymptotic Behavior of the Heat Semigroup on Certain Riemannian Manifolds
- 26 June 2023
- Published by Springer Nature
- p. 165-179
Asymptotic Behavior of the Heat Semigroup on Certain Riemannian Manifolds
- Open Access
- 3 May 2023
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 322 (1) , 39-58
On homology theories of cubical digraphs
We prove the equivalence of the singular cubical homology and the path homology on the category of cubical digraphs. As a corollary we obtain a new relation between the singular cubical homology of digraphs and simplicial homology. * and the path homology H * on the category of digraphs using [1; 2; 3; 5]. In Section 3, we recall the definition of cubical digraph from [2] and prove the contractibility of the image of a digraph cube in a cubical digraph for any digraph map. In Section 4, we prove the main result of the paper: Theorem 1.1. On the category of cubical digraphs, the singular cubical homology theory is equivalent to the path homology theory.
- 3 May 2023
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 322 (1) , 39-58
Open AccessOn homology theories of cubical digraphs
We prove the equivalence of the singular cubical homology and the path homology on the category of cubical digraphs. As a corollary we obtain a new relation between the singular cubical homology of digraphs and simplicial homology. * and the path homology H * on the category of digraphs using [1; 2; 3; 5]. In Section 3, we recall the definition of cubical digraph from [2] and prove the contractibility of the image of a digraph cube in a cubical digraph for any digraph map. In Section 4, we prove the main result of the paper: Theorem 1.1. On the category of cubical digraphs, the singular cubical homology theory is equivalent to the path homology theory.
- Open Access
Poincaré constant on manifolds with ends
We obtain optimal estimates of the Poincar\'e constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincar\'e constant is determined by the second largest end. The proof is based on the argument by Kusuoka-Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends.
Open AccessPoincaré constant on manifolds with ends
We obtain optimal estimates of the Poincar\'e constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincar\'e constant is determined by the second largest end. The proof is based on the argument by Kusuoka-Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends.