Recent Publications and Preprints
Resolving surface singularities in positive characteristic (with S. Perlega), Publications RIMS 60, 4 (2024), 44 pp, to appear. PDF
Phylogenetic trees and the moduli of n points on the projective line (with J. Qi and J. Schicho), forthcoming Preprint 2024, 50 pp. PDF
On Abel’s problem about logarithmic integrals in positive characteristic (with F. Fürnsinn, H. Kawanoue), Preprint 2024, 28 pp. PDF
Fuchs' theorem on ordinary linear differential equations in arbitrary characteristic (with F. Fürnsinn), Preprint 2023, 40 pp. PDF
A normal form theorem for Fuchsian differential equations over fields of characteristic zero. Manuscript 2022. PDF
How regular are regular singularities? In: Proceedings of Transient Transcendence in Transylvania, Brasov 2019 (eds. A. Bostan, K. Raschel). Springer 2021 PDF
Arquile varieties - varieties consisting of power series in a single variable (with S. Woblistin). Forum Math. Sigma 9 (2021), 1-39. PDF
Algebraic power series. Lecture Notes 2018, 60 pp.
Algebraic Aspects of Fuchsian Differential Equations. Monograph 2020, 160 pp.
Isosingular loci of algebraic varieties (with C. Chiu). J. Pure Applied Algebra 226 (2022), art. 107131. PDF
On the formal neighborhood of degenerate arcs (with C. Chiu), Manuscript, 9 pp. PDF
Resolution of Singularities
Cycles of singularities appearing in the resolution problem in positive characteristic (with S. Perlega), J. Algebraic Geometry 28 (2019), 391-403. PDF
Characterizing the increase of the residual order under blowup in positive characteristic (with S. Perlega), Publications RIMS 55 (2019), 835-857. PDF
Blowups and resolution, in: The resolution of singular algebraic varieties, Clay Mathematics Institute Summer School 2012, Obergurgl. CMI series, Amer. Math. Soc. 2014, pp. 1-80. PDF
Alternative invariants for the embedded resolution of purely inseparable surface singularities (with D. Wagner), L'Enseign Math. 60 (2014), 177–224. PDF
A game for the resolution of singularities (with J. Schicho), Proc. London Math. Soc. 105 (2012), 1149-1182. PDF
Forty Questions on Singularities (with J. Schicho), Asian J. Math. 15 (2011), 417-436. PDF
Today's Menu: Geometry and Resolution of Singular Algebraic Surfaces (with E. Faber), Bull. Amer. Math. Soc. 47 (2010), 373-417. PDF
On the Problem of Resolution of Singularities in Positive Characteristic (Or: A proof we are still waiting for), Bull. Amer. Math. Soc. 47 (2010), 1-30. PDF
Platonic Stars (with A. Fritz), Math. Intelligencer 32 (2010), 22-36. PDF
Wild Singularities and Kangaroo Points for the Resolution in Positive Characteristic, Manuscript, 2010, 31 pp. PDF
Resolution Principles. In: Proceedings of Clay Research Conference (2009). PDF
Plain Varieties (with G. Bodnár, J. Schicho, O. Villamayor), Bull. London Math. Soc. 40 (2008), 965-971. PDF
Desingularization of ideals and varieties, In: Singularity Theory, Marseille 2005 (ed. Anne Pichon), pp. 33-69. World Sci. Publ. 2007. PDF
Seven short stories on blowups and resolution. Proceedings Gökova Conference on Geometry and Topology 2005 (ed. Selman Akbulut), International Press 2006, 1-48. PDF
Explizite Auflösung von ebenen Kurvensingularitäten in beliebiger Charakteristik (with G. Regensburger). L'Enseign. Math. 50 (2004), 305-353. PDF
Why the characteristic zero proof of resolution of singularities fails in positive characteristic. Manuscript 2003. 51 pp. PDF
The Hironaka Theorem on resolution of singularities (Or: A proof that we always wanted to understand). Bull. Amer. Math. Soc. 40 (2003), 323-403. PDF
Strong resolution of singularities in characteristic zero (with S. Encinas). Commentarii Math. Helvetici 77 (2002), 421-445. PDF
Excellent surfaces and their taut resolution. In: Resolution of Singularities (with H.Hauser, J.Lipman, F.Oort, A.Quirós), Educational Progress in Math. 181, Birkhäuser (2000). PDF
Resolution of singularities 1860-1999. In: Resolution of Singularities (with H.Hauser, J.Lipman, F.Oort, A.Quirós), Educational Progress in Math. 181, Birkhäuser (2000). PDF
Seventeen obstacles for resolution of singularities. In: The Brieskorn anniversary volume 1996 (ed. V.I. Arnol'd, G.-M. Greuel, J. Steenbrink). Progress in Math. 162, Birkhäuser (1997). PDF
Blowup-Algebra. Contribution to the Encyclopaedia of Mathematics. Kluwer (1997).
Commutative Algebra and Infinite Dimensional Geometry
Encoding algebraic power series (with M.E. Alonso, F.J. Castro-Jiménez), Found. Comp. Math. 18 (2018), 789-833. PDF
Echelons of power series and Gabrielov’s counterexample to nested linear Artin approximation (with M.E Alonso, F.J. Castro-Jiménez, C. Koutschan), Bull. London Math. Soc. (2018), 635-648. PDF
The classical Artin approximation theorems, Bull. Amer. Math. Soc. 54 (2017), 595-633. PDF
Corrigendum to "The classical Artin approximation theorems", Bull. Amer. Math. Soc. 55 (2018), 289-293. PDF
Artin approximation (with G. Rond), Preprint 2015, 74 pp. PDF
Multivariate linear recurrences and power series division (with C. Koutschan), Discrete Math. 312 (2012), 3553-3560. PDF
Arcs, Cords and Felts: Six Instances of the Linearization Principle (with C. Bruschek), Amer. J. Math. 132 (2010), 941-986. PDF
Perfect bases for differential equations (with S. Gann). J. Symb. Comp. 40 (2005), 979-997. PDF
Three power series techniques. Proc. London Math. Soc. 88 (2004), 1-24. PDF
Continuous division for differential operators (with L. Narváez). Ann. Inst. Fourier 51 (2001), 769-778. PDF
Comparing modules of differential operators by their evaluation on polynomials. Compos. Math. 69 (1989), 295-307. PDF
A rank theorem for analytic maps between power series spaces (with G. Müller). Publ. Math. I.H.E.S. 80 (1994), 95-115. PDF
Semi-universal unfoldings and orbits of the contact group (with. G. Müller). Abh. Math. Sem. Hamburg 66 (1996), 1-9. PDF
Analytic curves in power series rings (with G. Müller). Compos. Math. 76 (1990), 197-201. PDF
Automorphism groups in local analytic geometry, infinite dimensional rank theorem and Lie groups (with G. Müller). Comptes Rendus Acad. Sci. 313, série I (1991), 751-756. PDF
Sur la construction de la déformation semi-universelle d'un germe d'espace analytique. Ann. Éc. Norm. Sup. Paris 18 (1985), 1-56. PDF
An algorithm of construction of the semi-universal deformation of an isolated singularity. Proc. Symp. Pure Appl. Math. Amer. Math. Soc. 40 (1983), 567-573.
Sur la déformation semi-universelle d'une singularitée isolée. Thèse de Troisième Cycle, Université Paris-Sud, Orsay, 1980.
Vector Fields in Algebraic Geometry
The reduced Bautin index of planar vector fields (with J.-J. Risler, B. Teissier). Duke J. Math. 100 (2000), 425-446. PDF
Dérivations et idéaux réels invariants (with J.-J. Risler). Publ. RIMS, Kyoto Univ. 35 (1999), 585 - 598. PDF
Lie algebras of polynomial vector fields and the Jacobian conjecture (with G. Müller). Monatshefte 126 (1998), 211-213. PDF
On the Lie algebra Θ(X) of vector fields on a singularity (with G. Müller). J. Math. Sci. Univ. Tokyo 1 (1994), 239-250. PDF
Affine varieties and Lie algebras of vector fields (with G. Müller). Manuscr. Math. 80 (1993), 309-337. PDF
Analytic varieties versus integral varieties of Lie algebras of vector fields (with G. Müller). Bull. Amer. Math. Soc. 26 (1992), 276-279. PDF
Frobenius type theorems and the Gröbner correspondence (with G. Müller). In: Congrès Singularités, Lille 1991 (ed. J.-P. Brasselet), IRMA, Université de Lille.
The Lie algebras I D X, Θ(X), and D(X) of a variety (with G. Müller). In: Workshop on Lie algebras of vector fields, Mainz 1993. PDF
Singularities in Complex Analytic Geometry
The cancellation property for direct products of analytic space germs (with G. Müller). Special volume: 60th anniversary of Hans Grauert, Math. Ann. 286 (1990), 209-223. PDF
The trivial locus of an analytic map germ (with G. Müller). Ann. Inst. Fourier 39 (1989), 831-844. PDF
Automorphisms of direct products of algebroid spaces (with G. Müller). Singularity Theory and its Applications. Warwick 1989, Part I (Eds.: D. Mond, J. Montaldi). Springer Lecture Notes in Mathematics 1462 (1991).
Algebraic singularities have maximal reductive automorphism groups (with G. Müller). Nagoya Math. J. 113 (1989), 181-186.
Characterizing complex analytic functions. In: Géometrie algébrique et applications (ed.: Aroca et al.), II. Singularités et géométrie complexe, 133-139. Travaux en cours, Hermann, Paris 1987.
Harmonic and dissonant singularities (with G. Müller). In: Proc. Conf. Algebraic Geometry (ed.: Kurke et al.), Berlin 1985, 123-134. Teubner-Texte Bd. 92, Leipzig 1986.
Characterizing singularities of varieties and of mappings (with T. Gaffney). Invent. math. 79 (1985), 427-447. PDF
On the singular subspace of an analytic variety. Adv. Studies Pure Math. 8 (1986), 125-134.
Simplification et factorisation unique des germes d'espaces analytiques complexes (with G. Müller). Comptes Rendus Acad. Sci. 309, série I (1989), 617-619.
Books
The Resolution of Singular Algebraic Varieties, Obergurgl 2012 (eds. D. Ellwood, H. Hauser, S. Mori, J. Schicho), Clay Mathematics Proceedings vol. 20, Amer. Math. Soc. 2012.
Resolution of Singularities, Obergurgl 1997 (eds. H. Hauser, J. Lipman, F. Oort, A. Quiros), Progress in Mathematics vol. 181, Birkhäuser 2000.
Bilderbuch der Singularitäten, 2010, 80 pp.
Grants & Projects
FWF-Project P-34765 "Algebraic Aspects of Fuchsian Differential Equations", 2021-2025
FWF-Project P-31338 "Artin Approximation, Arc Spaces, and Resolution of Singularities", 2018-2021
Chaire Jean-Morlet, Aix-Marseille University und CIRM, Spring 2015: Special semester on Artin Approximation
FWF-Project P-25652 "Resolution of Singularities", 2013-2018
Clay Summer School 2012 "Resolution of Singularities", Obergurgl June 2012
FWF-ANR-Project "Approximation Theorems in algebra and Cauchy-Riemann geometry: When Michael Artin meets Henri Poincaré" 2010-2014; with B. Lamel, N. Mir
FWF-Project P-21461 "Solving Algebraic Equations II" 2009-2013; with J. Schicho
FWF-Project P-15551 "Explicit Resolution and Related Methods in Algebraic Geometry and Number Theory" 2002-2006; with J. Schicho
FWF-Project P-18992 "Solving Algebraic Equations" 2006-2009; with J. Schicho
Visualizations
Curriculum Vitae
1956 Born in Innsbruck, Austria. Austrian citizen. Married, three children
1974 - 80 Studies at the Universities of Innsbruck and Paris
1980 - PhD in Paris with Teissier and Douady on the semi-universal deformation of singularities
1980 - 88 Assistent professor at the University of Innsbruck
1982 - 84 Visiting Brandeis and Northeastern University
1988 - Habilitation
1988 - 2007 Associate Professor at the University of Innsbruck
1992 - 94 Visiting Professor at the Universidad Autónoma de Madrid
2015 Chaire Jean Morlet, Aix-Marseille University und CIRM, Spring 2015
2007 - 2021 Professor at the University of Vienna
2021 - present Mathematician & Researcher
PhD Theses:
Sergey Yurkevich, PhD 2023, on integer sequences, algebraic series and differential operators, PDF
Hana Melánová, PhD 2020, on the curvature of singular algebraic surfaces PDF
Christopher Chiu, PhD 2020, on the geometry of arc spaces PDF
Stefan Perlega, PhD 2018, on resolution of singularities in positive characteristic PDF
Eleonore Faber, PhD 2011, on normal crossings in singularity theory PDF
Clemens Bruschek, PhD 2009, on infinite dimensional algebraic geometry and arc spaces PDF
Dominique Wagner, PhD 2009, on resolution of surfaces in positive characteristic PDF
Sebastian Gann, PhD 2005, on polynomial solutions of differential equations PDF
Dominik Zeillinger, PhD 2005, on resolution of singularities and the polyhedral game PDF
Master Theses:
Nicholas Merkl Master 2022, on ordinary linear differential equations with irregular singularities, PDF
Florian Fürnsinn Master 2022, on normal forms of ordinary linear differential equations, PDF
Johannes Droschl Master 2020, on blowups in monomial ideals and stacks, PDF
Sergey Yurkevich, Master 2020, on algebraic power series in zero characteristic, PDF
David Stinner, Master 2020, on Hilbert functions and polynomials
Hana Melánová, Master 2016, on resolution of curves via higher curvatures, PDF
Valerie Roitner, Master 2015, on lattice walks and generating functions, PDF
Christopher Chiu, Master 2015, on Zariski's Main Theorem, PDF
Markus Müller, Master 2014, on the Rogers-Ramanujan identity, PDF
Alexandra Fritz, Diploma 2010, on platonic stars and invariant theory, PDF
Matthias Schiechtl, Diploma 2008, on zero dimensional algebraic varieties, PDF
Martin Kreidl, Diploma 2007, on methods of commutative algebra in positive characteristic, PDF
Eleonore Faber, Diploma 2007, on tame ideals and blowups, PDF
Dominique Wagner, Diploma 2005, on algebraic power series and division theorems, PDF
Clemens Bruschek, Diploma 2005, on mappings between formal power series spaces,
Andreas Hochenegger, Diploma 2004, on Bernstein's theorem, PDF
Online Lectures: Fuchsian Differential Equations, Vienna, Fall 2022
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