About

Simion Filip is a professor in the Department of Mathematics at the University of Chicago. His research spans several areas in mathematics, including dynamics, geometry, and algebraic geometry, with a particular focus on the study of K3 surfaces, translation surfaces, and variations of Hodge structure. Filip's work often involves the interplay between dynamical systems and geometric structures, as evidenced by his contributions to measure rigidity for stationary measures of random walks generated by diffeomorphisms and actions of SL(2,R) on smooth manifolds. He has developed new techniques in the theory of normal forms for non-uniformly contracting dynamics and has explored the geometric and dynamical properties of hyperbolic manifolds and Calabi-Yau varieties.

Research topics

• Mathematical analysis
• Pure mathematics
• Mathematics

Selected publications

• Nonlinearizable embeddings of elliptic curves in rational surfaces

  arXiv (Cornell University) · 2026-05-05

  preprintOpen access1st authorCorresponding

  We show that for any smooth cubic in $\mathbb{P}^2$, there exists a dense $G_δ$ set of configurations of 9 distinct points such that blowing up $\mathbb{P}^2$ at these 9 points, the strict transform of the cubic is not linearizable and has nontorsion normal bundle. This answers a problem raised by Ogus in 1975.

  PublisherDOI

• Nonlinearizable embeddings of elliptic curves in rational surfaces

  ArXiv.org · 2026-05-05

  articleOpen access1st authorCorresponding

  We show that for any smooth cubic in $\mathbb{P}^2$, there exists a dense $G_δ$ set of configurations of 9 distinct points such that blowing up $\mathbb{P}^2$ at these 9 points, the strict transform of the cubic is not linearizable and has nontorsion normal bundle. This answers a problem raised by Ogus in 1975.

  PublisherOA PDF

• Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method

  ArXiv.org · 2025-02-19

  preprintOpen access

  We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the theorems are derived, applies also to the case of a single diffeomorphism or $1$-parameter flow and establishes extra invariance of a class of measures that we call ``generalized u-Gibbs states''.

  PublisherOA PDFDOI

• Measure Rigidity beyond Homogeneous Dynamics

  ArXiv.org · 2025-12-15

  preprintOpen access1st authorCorresponding

  We describe recent work that extends some of the measure and topological rigidity results in dynamical systems from situations homogeneous under a Lie group to quite general manifolds.

  PublisherOA PDFDOI

• The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds

  Journal of Algebraic Geometry · 2025-07-18

  article1st authorCorresponding

  We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain Calabi–Yau complete intersections known as Wehler <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -folds. We find that the volume function exhibits a pathological behavior when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we obtain examples of a pseudoeffective <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -divisor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the volume of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D plus s upper A"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> <mml:mi>s</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">D+sA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> small and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ample, oscillates between two powers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Superscript 0 Baseline left-parenthesis upper X comma left floor m upper D right floor plus upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mi>m</mml:mi> <mml:mi>D</mml:mi> <mml:mo>⌋</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h^0(X,\left \lfloor mD \right \rfloor +A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> displays a similar oscillatory behavior as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> increases, showing that several notions of numerical dimensions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> do not agree and disproving a conjecture of Fujino. We accomplish this by relating the behavior of the volume function along a segment to the visits of a corresponding hyperbolic geodesics to the cusps of a hyperbolic manifold.

  PublisherDOI

• A cyclotomic family of thin hypergeometric monodromy groups in $${\text {Sp}}_4({\mathbb {R}})$$

  Geometriae Dedicata · 2024-02-19

  article1st authorCorresponding
  PublisherDOI

• Translation surfaces: Dynamics and Hodge theory

  EMS Surveys in Mathematical Sciences · 2024-05-02 · 6 citations

  articleOpen access1st authorCorresponding

  A translation surface is a multifaceted object that can be studied with the tools of dynamics, analysis, or algebraic geometry. Moduli spaces of translation surfaces exhibit equally rich features. This survey provides an introduction to the subject and describes some developments that make use of Hodge theory to establish algebraization and finiteness statements in moduli spaces of translation surfaces.

  PublisherOA PDFDOI

• Finiteness of totally geodesic hypersurfaces

  arXiv (Cornell University) · 2024-08-06

  preprintOpen access1st authorCorresponding

  We prove that a closed negatively curved analytic Riemannian manifold that contains infinitely many totally geodesic hypersurfaces is isometric to an arithmetic hyperbolic manifold. Equivalently, any closed analytic Riemannian manifold with negative sectional curvature has only finitely many totally geodesic hypersurfaces, unless it has constant curvature.

  PublisherOA PDFDOI

• Gaps in the Support of Canonical Currents on Projective K3 Surfaces

  Journal of Geometric Analysis · 2024-01-19 · 2 citations

  article1st author
  PublisherDOI

• Canonical currents and heights for K3 surfaces

  Cambridge Journal of Mathematics · 2023-01-01 · 6 citations

  articleOpen access1st authorCorresponding

  We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the ample cone. Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.

  PublisherOA PDFDOI

Recent grants

• Geometry and Dynamics of K3 Surfaces

  NSF · $216k · 2020–2024

Frequent coauthors

• Valentino Tosatti

  16 shared

• Alexander I. Bufetov

  7 shared

• Carlos Matheus

  Heilbronn Institute for Mathematical Research

  5 shared

• Charles Fougeron

  Université Sorbonne Paris Nord

  5 shared

• Giovanni Forni

  Istituti Clinici Scientifici Maugeri

  3 shared

• Alex Eskin

  University of Chicago

  3 shared

• Alex Wright

  University of Michigan–Ann Arbor

  3 shared

• Fabian Haiden

  2 shared

Education

• Ph.D.

  University of Chicago

  2016

Awards & honors

• 2020 EMS Prize