Sergey Fomin
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
Sergey Fomin has a total of 59 co-authors. Here are the top co-authors by number of shared publications:
- Andrei Zelevinsky
Northeastern University
9 shared publications Andrei Zelevinsky
7 shared publicationsCurtis Greene
4 shared publications- Jonah Blasiak
Dept. of Mathematics, Drexel University, Philadelphia, PA
4 shared publications Anatol N. Kirillov
University of Tokyo
3 shared publications- Dylan P. Thurston
Indiana University, Bloomington, IN
3 shared publications
Latest Publications
- Top Cited PapersOpen Access
- 29 October 2025
- Published by Springer Nature in Selecta Mathematica
- Vol. 31 (5) , 1-44
Long mutation cycles
A mutation cycle is a cycle in a graph whose vertices are labeled by the quivers in a given mutation class and whose edges correspond to single mutations. For any fixed $$n\ge 4$$ , we describe arbitrarily long mutation cycles involving n -vertex quivers. Each of these mutation cycles allows for an arbitrary choice of $$\left( {\begin{array}{c}n\\ 2\end{array}}\right) $$ positive integer parameters. None of the mutation cycles we construct can be paved by short mutation cycles.- 29 October 2025
- Published by Springer Nature in Selecta Mathematica
- Vol. 31 (5) , 1-44
Top Cited PapersOpen AccessLong mutation cycles
A mutation cycle is a cycle in a graph whose vertices are labeled by the quivers in a given mutation class and whose edges correspond to single mutations. For any fixed $$n\ge 4$$ , we describe arbitrarily long mutation cycles involving n -vertex quivers. Each of these mutation cycles allows for an arbitrary choice of $$\left( {\begin{array}{c}n\\ 2\end{array}}\right) $$ positive integer parameters. None of the mutation cycles we construct can be paved by short mutation cycles. - Open Access
- 28 August 2023
- Published by American Mathematical Society (AMS) in Communications of the American Mathematical Society
- Vol. 3 (10) , 669-743
Expressive curves
We initiate the study of a class of real plane algebraic curves which we call expressive . These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve C C is expressive if (a) each irreducible component of C C can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of C C in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of C C in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.
- 28 August 2023
- Published by American Mathematical Society (AMS) in Communications of the American Mathematical Society
- Vol. 3 (10) , 669-743
Open AccessExpressive curves
We initiate the study of a class of real plane algebraic curves which we call expressive . These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve C C is expressive if (a) each irreducible component of C C can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of C C in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of C C in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.
- Open Access
Expressive curves
We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve $C$ is expressive if (a) each irreducible component of $C$ can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of $C$ in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of $C$ in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)-(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.Open AccessExpressive curves
We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve $C$ is expressive if (a) each irreducible component of $C$ can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of $C$ in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of $C$ in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)-(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more. - Open Access
Incidences and tilings
We show that various classical theorems of real/complex linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a single "master theorem" that involves an arbitrary tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones.
Open AccessIncidences and tilings
We show that various classical theorems of real/complex linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a single "master theorem" that involves an arbitrary tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones.
- 5 January 2022
- Published by American Mathematical Society (AMS) in Representation Theory
- Vol. 26 (1) , 1-16
Coordinate rings and birational charts
Let G G be a semisimple simply connected complex algebraic group. Let U U be the unipotent radical of a Borel subgroup in G G . We describe the coordinate rings of U U (resp., G / U G/U , G G ) in terms of two (resp., four, eight) birational charts introduced by Lusztig [ Total positivity in reductive groups , Birkhäuser Boston, Boston, MA, 1994; Bull. Inst. Math. Sin. (N.S.) 14 (2019), pp. 403–459] in connection with the study of total positivity.
- 5 January 2022
- Published by American Mathematical Society (AMS) in Representation Theory
- Vol. 26 (1) , 1-16
Coordinate rings and birational charts
Let G G be a semisimple simply connected complex algebraic group. Let U U be the unipotent radical of a Borel subgroup in G G . We describe the coordinate rings of U U (resp., G / U G/U , G G ) in terms of two (resp., four, eight) birational charts introduced by Lusztig [ Total positivity in reductive groups , Birkhäuser Boston, Boston, MA, 1994; Bull. Inst. Math. Sin. (N.S.) 14 (2019), pp. 403–459] in connection with the study of total positivity.
- Open Access
Morsifications and mutations
We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications.
Open AccessMorsifications and mutations
We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications.
- Open Access
Universal quivers
We show that for any positive integer $n$, there exists a quiver $Q$ with $O(n^2)$ vertices and $O(n^2)$ edges such that any quiver on $n$ vertices is a full subquiver of a quiver mutation equivalent to $Q$. We generalize this statement to skew-symmetrizable matrices and obtain other related results. In particular, we show that any quiver is a full subquiver of a quiver mutation equivalent to a quiver of a plabic graph.Open AccessUniversal quivers
We show that for any positive integer $n$, there exists a quiver $Q$ with $O(n^2)$ vertices and $O(n^2)$ edges such that any quiver on $n$ vertices is a full subquiver of a quiver mutation equivalent to $Q$. We generalize this statement to skew-symmetrizable matrices and obtain other related results. In particular, we show that any quiver is a full subquiver of a quiver mutation equivalent to a quiver of a plabic graph. - 11 May 2020
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2021 (1) , 648-694
Heronian Friezes
Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type $A$, we introduce and study Heronian friezes, the Euclidean analogues of Coxeter’s frieze patterns. We prove that a generic Heronian frieze possesses the glide symmetry (hence is periodic) and establish the appropriate version of the Laurent phenomenon. For a closely related family of Cayley–Menger friezes, we identify an algebraic condition of coherence, which all friezes of geometric origin satisfy. This yields an unambiguous propagation rule for coherent Cayley–Menger friezes, as well as the corresponding periodicity results.- 11 May 2020
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2021 (1) , 648-694
Heronian Friezes
Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type $A$, we introduce and study Heronian friezes, the Euclidean analogues of Coxeter’s frieze patterns. We prove that a generic Heronian frieze possesses the glide symmetry (hence is periodic) and establish the appropriate version of the Laurent phenomenon. For a closely related family of Cayley–Menger friezes, we identify an algebraic condition of coherence, which all friezes of geometric origin satisfy. This yields an unambiguous propagation rule for coherent Cayley–Menger friezes, as well as the corresponding periodicity results.- Open Access
Cluster algebras and triangulated surfaces. Part II: Lambda lengths
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.
Open AccessCluster algebras and triangulated surfaces. Part II: Lambda lengths
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.
- 4 June 2018
- Published by Springer Nature in computational complexity
- Vol. 27 (4) , 595-616
On semiring complexity of Schur polynomials
- 4 June 2018
- Published by Springer Nature in computational complexity
- Vol. 27 (4) , 595-616
On semiring complexity of Schur polynomials