Nikolai Saveliev
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
Nikolai Saveliev has a total of 20 co-authors. Here are the top co-authors by number of shared publications:
- Daniel Ruberman
Department of Mathematics, Brandeis University, Waltham, MA, United States
17 shared publications - Dmitri Vassiliev
Department of Mathematics, University College London, London, UK
6 shared publications - Jianfeng Lin
Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
4 shared publications - Matteo Capoferri
Dipartimento di Matematica "Federigo Enriques" Università degli Studi di Milano via C. Saldini 50 20133 Milano Italy
3 shared publications - Anatoly Libgober
University of Illinois
2 shared publications - Demetre Kazaras
Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.
2 shared publications
Latest Publications
- Open Access
- 1 September 2026
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 314 (1) , 14
Instanton Floer homology and Milnor fibers
In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn–Hamm complete intersection singularities. The resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg–Witten theories. We use similar techniques to express the Heegaard Floer d -invariant of torus knots in terms of their Milnor fibers.
- 1 September 2026
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 314 (1) , 14
Open AccessInstanton Floer homology and Milnor fibers
In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn–Hamm complete intersection singularities. The resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg–Witten theories. We use similar techniques to express the Heegaard Floer d -invariant of torus knots in terms of their Milnor fibers.
- Open Access
Instanton Floer homology and Milnor Fibers
In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn-Hamm complete intersection singularities. The resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg-Witten theories. We use similar techniques to express the Heegaard Floer d-invariant of torus knots in terms of their Milnor fibers.
Open AccessInstanton Floer homology and Milnor Fibers
In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn-Hamm complete intersection singularities. The resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg-Witten theories. We use similar techniques to express the Heegaard Floer d-invariant of torus knots in terms of their Milnor fibers.
- Open Access
Topological obstructions to the diagonalisation of pseudodifferential systems
Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur.
Open AccessTopological obstructions to the diagonalisation of pseudodifferential systems
Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur.
- Open Access
- 27 December 2022
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society, Series B
- Vol. 9 (43) , 472-486
Topological obstructions to the diagonalisation of pseudodifferential systems
Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur.
- 27 December 2022
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society, Series B
- Vol. 9 (43) , 472-486
Open AccessTopological obstructions to the diagonalisation of pseudodifferential systems
Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur.
- 1 January 2022
- Published by International Press of Boston in Communications in Analysis and Geometry
- Vol. 30 (4) , 869-890
On positive scalar curvature cobordisms and the conformal Laplacian on end-periodic manifolds
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2022
- Published by International Press of Boston in Communications in Analysis and Geometry
- Vol. 30 (4) , 869-890
On positive scalar curvature cobordisms and the conformal Laplacian on end-periodic manifolds
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 31 December 2021
- Published by Mathematical Sciences Publishers in Geometry & Topology
- Vol. 25 (7) , 3591-3628
On the monopole Lefschetz number of finite-order diffeomorphisms
- 31 December 2021
- Published by Mathematical Sciences Publishers in Geometry & Topology
- Vol. 25 (7) , 3591-3628
On the monopole Lefschetz number of finite-order diffeomorphisms
- Open Access
On the spectral sets of Inoue surfaces
The Inoue surfaces are certain non-Kaehler complex surfaces that have the structure of a $T^3$ bundle over the circle. We study the Inoue surfaces $S_M$ with the Tricerri metric and the canonical spin$^c$ structure, and the corresponding chiral Dirac operators twisted by a flat $\mathbb C^*$--connection. The twisting connection is determined by $z \in \mathbb C^*$, and the points for which the twisted Dirac operators $\mathcal D^{\pm}_z$ are not invertible are called spectral points. We show that there are no spectral points inside the annulus $\alpha^{-1/4} < |z| < \alpha^{1/4}$, where $\alpha >1$ is the only real eigenvalue of the matrix $M$ that determines $S_M$, and find the spectral points on its boundary. Via Taubes' theory of end-periodic operators, this implies that the corresponding Dirac operators are Fredholm on any end-periodic manifold whose end is modeled on $S_M$.Open AccessOn the spectral sets of Inoue surfaces
The Inoue surfaces are certain non-Kaehler complex surfaces that have the structure of a $T^3$ bundle over the circle. We study the Inoue surfaces $S_M$ with the Tricerri metric and the canonical spin$^c$ structure, and the corresponding chiral Dirac operators twisted by a flat $\mathbb C^*$--connection. The twisting connection is determined by $z \in \mathbb C^*$, and the points for which the twisted Dirac operators $\mathcal D^{\pm}_z$ are not invertible are called spectral points. We show that there are no spectral points inside the annulus $\alpha^{-1/4} < |z| < \alpha^{1/4}$, where $\alpha >1$ is the only real eigenvalue of the matrix $M$ that determines $S_M$, and find the spectral points on its boundary. Via Taubes' theory of end-periodic operators, this implies that the corresponding Dirac operators are Fredholm on any end-periodic manifold whose end is modeled on $S_M$. - Open Access
On the monopole Lefschetz number of finite order diffeomorphisms
Let $K$ be a knot in an integral homology 3-sphere $Y$, and $\Sigma$ the corresponding $n$-fold cyclic branched cover. Assuming that $\Sigma$ is a rational homology sphere (which is always the case when $n$ is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of $\Sigma$. The proof relies on a careful analysis of the Seiberg--Witten equations on 3-orbifolds and of various $\eta$-invariants. We give several applications of our formula: (1) we calculate the Seiberg--Witten and Furuta--Ohta invariants for the mapping tori of all semi-free actions of $Z/n$ on integral homology 3-spheres; (2) we give a novel obstruction (in terms of the Jones polynomial) for the branched cover of a knot in $S^3$ being an $L$-space; (3) we give a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers.Open AccessOn the monopole Lefschetz number of finite order diffeomorphisms
Let $K$ be a knot in an integral homology 3-sphere $Y$, and $\Sigma$ the corresponding $n$-fold cyclic branched cover. Assuming that $\Sigma$ is a rational homology sphere (which is always the case when $n$ is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of $\Sigma$. The proof relies on a careful analysis of the Seiberg--Witten equations on 3-orbifolds and of various $\eta$-invariants. We give several applications of our formula: (1) we calculate the Seiberg--Witten and Furuta--Ohta invariants for the mapping tori of all semi-free actions of $Z/n$ on integral homology 3-spheres; (2) we give a novel obstruction (in terms of the Jones polynomial) for the branched cover of a knot in $S^3$ being an $L$-space; (3) we give a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers. - 5 March 2020
- Published by World Scientific Pub Co Pte Ltd in Reviews in Mathematical Physics
- Vol. 32 (09)
Classification of first order sesquilinear forms
A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial [ ]-bundle over a smooth [ ]-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our goal is to classify such forms up to [ ] gauge equivalence. We achieve this classification in the special case of [ ] and [ ] by means of geometric and topological invariants (e.g., Lorentzian metric, spin/spinc structure, electromagnetic covector potential) naturally contained within the sesquilinear form — a purely analytic object. Essential to our approach is the interplay of techniques from analysis, geometry, and topology.
- 5 March 2020
- Published by World Scientific Pub Co Pte Ltd in Reviews in Mathematical Physics
- Vol. 32 (09)
Classification of first order sesquilinear forms
A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial [ ]-bundle over a smooth [ ]-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our goal is to classify such forms up to [ ] gauge equivalence. We achieve this classification in the special case of [ ] and [ ] by means of geometric and topological invariants (e.g., Lorentzian metric, spin/spinc structure, electromagnetic covector potential) naturally contained within the sesquilinear form — a purely analytic object. Essential to our approach is the interplay of techniques from analysis, geometry, and topology.
- Open Access
Classification of first order sesquilinear forms
A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial $\mathbb{C}^n$-bundle over a smooth $m$-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our goal is to classify such forms up to $GL(n,\mathbb{C})$ gauge equivalence. We achieve this classification in the special case of $m=4$ and $n=2$ by means of geometric and topological invariants (e.g. Lorentzian metric, spin/spin$^c$ structure, electromagnetic covector potential) naturally contained within the sesquilinear form - a purely analytic object. Essential to our approach is the interplay of techniques from analysis, geometry, and topology.Open AccessClassification of first order sesquilinear forms
A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial $\mathbb{C}^n$-bundle over a smooth $m$-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our goal is to classify such forms up to $GL(n,\mathbb{C})$ gauge equivalence. We achieve this classification in the special case of $m=4$ and $n=2$ by means of geometric and topological invariants (e.g. Lorentzian metric, spin/spin$^c$ structure, electromagnetic covector potential) naturally contained within the sesquilinear form - a purely analytic object. Essential to our approach is the interplay of techniques from analysis, geometry, and topology.