Curriculum vitae et studiorum


Personal information
Name: Stefano Pasotti
Address: Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica - DICATAM
Sezione di Matematica
Università degli Studi di Brescia
Via Valotti 43,
25123 Brescia, Italy
e-mail:
Telephone:
Fax:
+39 030 3715931
+39 030 3715745
Home page http://www.ing.unibs.it/~stefano.pasotti
Current position Research associate (ricercatore) in Geometry
Languages
  • Italian
  • English
Education / Fellowships From 04/2008 to 12/2008 Postdoc fellow (assegnista di ricerca) in geometry, working on "Parallelisms in geometric spaces embedded into projective spaces" (Parallelismi in spazi geometrici immersi in spazi proiettivi) at Università Cattolica del Sacro Cuore of Brescia, under the supervision of Prof.ssa Silvia Pianta

From 08/2005 to 10/2005 and from 01/2006 to 02/2006 Marie Curie Fellow at the University of Liverpool (UK) under the supervision of Prof. Peter Newstead.

30/11/2006 - PhD in Mathematics at the University of Trento (Italy) with the thesis "Vector bundles on algebraic curves: holomorphic triples in low genus", under the supervision of Prof. Edoardo Ballico.

12/04/2002 - Degree in Mathematics at Università Cattolica del Sacro Cuore of Brescia with a thesis on "Partizioni di Gruppi e strutture geometriche" under the supervision of Prof.ssa Silvia Pianta; final score 110/110 cum laude.
Schools and courses
April 2005 Course on "Geometric structures and their Applications" held by Prof. S. Segura Gomis of the University of Alicante (Spain) at Dipartimento di Matematica of the University of Brescia, Italy
July 2004 Summer school on "Topologia delle varietà di basse dimensioni e Geometria Algebrica Reale" at Dipartimento di Matematica of University of Trento, Italy
September 2003 Summer school on Convexity and Discrete Geometry at Departamento de Análisis Matemático of University of Alicante, Spain
June 2003 Summer school on "Algebraic surfaces" at Dipartimento di Matematica of University of Trento, Italy
June 2003 Mini-course on "Some applications of Number Theory to Digital Signal Processing" held by Prof. Vassil Dimitrov of Calgary University at Dipartimento di Matematica of University of Trento, Italy
July 2002 Summer school "Giuseppe Tallini" on Combinatorial Geometry at Dipartimento di Matematica of Università Cattolica del Sacro Cuore of Brescia, Italy
September 2001 Summer school "Giuseppe Tallini" on "Finite Geometries and Linear Codes" at Ascea Marina (SA), Italy
July 2000 Summer school "Giuseppe Tallini" on Combinatorial Geometry at S. Felice del Benaco (BS), Italy
June 2000 Workshop on "Mathematical Control Theory and Robotics" at SISSA/ISAS of Trieste, Italy
Conferences and talks
9-14 September 2012 Combinatorics 2012, international conference on pure and applied combinatorics and its connections with Geometry, Graph Theory and Algebra, Perugia. Talk "Loops, regular permutation sets and graph colorations"
13-14 February 2012 Giornate di Geometria, Vicenza. Talk "The limit rotation loop of a hyperbolic plane".
15-19 May 2011 "36th Conference on Geometry and Algebra", Berlin.
4 March 2011 "Giornata di Geometria", Università di Padova. Talk "Slid product di loops".
June 22-July 3 2010 "Combinatorics 2010", international conference on pure and applied combinatorics and its connections with Geometry, Graph Theory and Algebra, Verbania, Italy. Talk "Clifford-like parallelisms".
26 May 2010 Talk "Quaternioni reali e generalizzati: un parallelismo", Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore di Brescia.
20 January 2010 Giornata in onore di Mario Marchi, Università Cattolica del Sacro Cuore, Brescia. Talk "Strutture di traslazione e prodotto semidiretto di gruppi".
5-10 September 2009 IX International Conference on Geometry and Applications, Varna, Bulgaria. Talk "Translation structures and semidirect product of groups".
24 April 2009 Talk "Parallelismi di Clifford generalizzati", Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università di Padova, IT.
14 November 2008 Talk "Al limite, ruoriamo", Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore di Brescia.
22-28 June 2008 "Combinatorics 2008", international conference on pure and applied combinatorics and its connections with Geometry, Graph Theory and Algebra, Costermano(VR), Italy. Talk "Regular parallelisms in linear spaces".
29 February - 3 March 2008 35th Conference on Geometry and Algebra, Berlin, Germany. Talk "Translation structures with a principal line".
24-29 September 2007 XVIII Congresso dell'Unione Matematica Italiana, Bari, Italy. Talk "Parallelismi di Clifford generalizzati".
3-8 September 2007 8th International Conference on Geometry and Application, Varna, Bulgaria. Talk "Generalized Clifford parallelisms".
2-6 July 2007 "Design Theory of Alex Rosa", Bratislava, Slovakia. Talk "Infinite translation transversal divisible designs".
9 June 2006 Talk "Holomorphic triples (e sistemi coerenti) in genere basso", Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Brescia.
18-22 April 2006 "Algebraic Geometry, Singularities and Knot Theory", Department of Mathematical Sciences of the University of Liverpool, UK. Talk "Holomorphic triples and coherent systems in low genera".
14 February 2006 Talk "Holomorphic triples and coherent systems (and an historical riddle)", Department of Mathematical Sciences of the University of Liverpool, UK.
August 2005 Workshop on "Vector bundles over algebraic curves" at the Department of Mathematical Sciences of the University of Liverpool, UK
November 2004 Talk "Modelli canonici per curve algebriche", Università Cattolica del Sacro Cuore, Brescia.
September 2004 "Recent Advances in Complex and Real Geometry", Levico Terme (Trento), Italy.
September 2004 Combinatorics 2004, 12th Italian conference in Combinatorics, Acireale (CA), Italy
June 2004 International conference "Trends in Geometry - in memory of Beniamino Segre", Roma, Italy
April 2004 Workshop on "Algebraic curves, monodromy and related topics" at Dipartimento di Matematica of Politecnico of Milano, Italy
November 2003 "Giornate di Geometria" at University of Brescia, with the talk "Traslazioni normali e algebre speciali"
June 2002 Combinatorics 2002, Maratea (PO), Italy
Teaching experiences
  • 2010/2011 - Course of "Algebra e Geometria" for students in Engineering of Università degli Studi of Brescia.
    Course of "Istituzioni di Geometria Superiore I" for students in Mathematics of Università Cattolica del Sacro Cuore of Brescia.
    Esercitazioni for the course of Algebra e geometria for students in Engineering of Universtià degli Studi of Brescia.
  • 2009/2010 - Course of "Algebra e Geometria" for students in Engineering of Università degli Studi of Brescia.
    Esercitazioni for the course of Algebra ed elementi di geometria for students in Engineering of Universtià degli Studi of Brescia.
    Esercitazioni for the course of Matematica - Geometria for students in Engineering of Universtià degli Studi of Brescia.
  • 2008/2009 - Esercitazioni for the course of Algebra ed elementi di geometria for students in Engineering of Universtià degli Studi of Brescia.
    Esercitazioni for the course of Matematica - Geometria for students in Engineering of Universtià degli Studi of Brescia.
    Esercitazioni for the courses of Geometria 1 and Geometria 2 for students in Mathematics and Physics of Università Cattolica del Sacro Cuore of Brescia.
  • 2007/2008 - Esercitazioni for the course of Matematiche Complementari (1st and 2nd part) for students in Mathematics of Università Cattolica del Sacro Cuore of Brescia.
    Esercitazioni for the course of Matematica - Geometria for students in Engineering of Universtià degli Studi of Brescia.
    Esercitazioni for the courses of Geometria 1 and Geometria 2 for students in Mathematics and Physics of Università Cattolica del Sacro Cuore of Brescia.
  • 2006/2007 - Esercitazioni for the course of Matematiche Complementari (1st and 2nd part) for students in Mathematics of Università Cattolica del Sacro Cuore of Brescia
  • 2005/2006 - Esercitazioni for the course of Matematiche Complementari (1st and 2nd part) for students in Mathematics of Università Cattolica del Sacro Cuore of Brescia
  • 2004/2005 - Esercitazioni for the course of Matematiche Complementari (1st and 2nd part) for students in Mathematics of Università Cattolica del Sacro Cuore of Brescia
  • 2003/2004 - Esercitazioni for the course of Matematiche Complementari (1st and 2nd part) for students in Mathematics of Università Cattolica del Sacro Cuore of Brescia
  • 2002/2003 - Esercitazioni for the course of Matematiche Complementari (1st and 2nd part) for students in Mathematics of Università Cattolica del Sacro Cuore of Brescia
    Esercitazioni for the course of Elementi di Analisi Matematica e Geometria for students in Engineering of Politecnico of Milano
Awards and grants
  • Grant by "Giuseppe Tovini" Foundation for the academic years 2006/2007 and 2007/2008.
  • Marie Curie fellowship at the Department of Mathematical Sciences of the University of Liverpool (UK) in 2006.
  • Degree award by "Don Bartolomeo Grazioli" Foundation and Istituto Lombardo Accademia di Scienze e Lettere for a thesis in Mathematics discussed in the year 2002.
  • XLIII Premio "Agostino Gemelli" of Ludovico Necchi Association as best graduated student in the Faculty for the year 2002.
Research interests

Linear spaces with parallelism

  • Affine parallel structures and affine parallel translation structure (André structures)
  • Finite and infinite groups with a partition in subgroups
  • Kinematic spaces (double prismatic spaces) and their geometry
  • Apt-structures and kinematic spaces with a principal line (a line which meets all the lines not in its parallel class)
  • Parallelisms in projective spaces and in incidence spaces
  • Loops derived from the hyperbolic plane
Recently I am working, in collaboration with S. Pianta from the Università Cattolica of Brescia and with A. Blunck from the University of Hamburg, on problems concerning linear spaces endowed with one or more parallelism relations. In particular we extended the classical definition of Clifford parallelisms in the real projective space and some equivalent formulations to 3-dimensional projective spaces over a generic field of characteristic different from 2. In our constructions we used generalized quaternion algebras, the theory of quadratic forms (see [Lam05]) and the properties of Baer subspaces and indicator sets of a parallelism (see [BU93, Hav94]). We proved a nice geometric characterization of generalized Clifford parallelisms, similar to the classical one, but dealing with the fact that in general, if the ground field is different from the field of real numbers, many different quadratic extensions may appear. This led the way to a situation that has no counterpart in the classical case end allowed us to introduce a technique for building a new "Clifford like" class of parallelisms, which will be the target of more investigations. We worked also on kinematic spaces (see [Kar73]). In this context, strictly related to fibered groups, we considered in particular those geometric structures with a principal line, that is a line which acts mostly like an "affine" line. In the finite case these structures are linked with transversal group divisible designs (see [Sch85]), and in fact many geometric properties have a corresponding formulation in terms of designs, but I considered also the infinite case from a geometric point of view. In particular I proved an algebraic characterization of these structures as particular semidirect products, and I investigated the properties of their automorphisms group. Recently, in an attempt to contribute to the study of the translation group of a kinematic space and of its normality in the group of automorphisms, i proposed an extension of the notion of regular parallelism, classically introduced in 3-dimensional projective spaces, to the case of kinematic spaces not necessarily embedded into projective ones.
As already mentioned the linear spaces with parallelisms I am dealing with are strictly related to fibered groups. Actually I am working, in collaboration with H. Karzel, M. Marchi and S. Pianta, on those algebraic structures known as loop, which are in some sense a weakening of the group structure. We try to construct a new class of loop starting from the motions of the hyperbolic plane, and in particular the group of limit rotations and from the horocycles of the hyperbolic plane.

References

[BU93]
A. Beutelspacher and J. Ueberberg. Bruck's vision of regular spreads or What is the use of a Baer superspace?. Abh. Math. Sem. Univ. Hamburg 63 (1993), 37-54.
[Hav94]
H. Havlicek. Spreads of right quadratic skew field extensions. Geom. Dedicata 49 (1994), no. 2, 239-251.
[Kar73]
H. Karzel. Kinematic spaces. Symposia Mathematica, Vol. XI (Con- vegno di Geometria, INDAM, Rome, 1972), Academic Press, London, 1973, pp. 413-439.
[Lam05]
T. Y. Lam. Introduction to quadratic forms over fields. Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005.
[Sch85]
R.H. Schulz. On the classification of translation group-divisible designs. European J. Combin. 4 (1985), 369-374.

Augmented vector bundles on algebraic curves

  • Vector bundles on smooth algebraic curves
  • Augmented bundles on algebraic curves (coherent systems and holomorphic triples)
  • Stable and semi-stable triples on curves of genus 0 and 1
  • Weierstrass points and n-tuples on algebraic curves
I worked on coherent systems (see [BGPMN03]) and holomorphic triples (see [BGP96, BGPG04]) on algebraic curves, which are particular cases of the so called augmented bundles (see [BDGPW95]), vector bundles on a curve with additional data (global sections, maps,...). In particular a coherent system is a pair (E,V) where E is a vector bundle on a curve X and V is a vector subspace of the vector space of its global sections, while a holomorphic triple is a triple (E1, E2, φ) where E1 and E2 are vector bundles on X and φ: E2 ---> E1 is a map. I dealt with curves of genus 0 and 1 and with problems concerning the moduli spaces of α-stable coherent systems and holomorphic triples (irreducibility, smoothness, dimension). For coherent systems these problems were already solved for elliptic curves in [LN05], and deeply inspected in [LN04] and ahead papers by the same authors for the projective line. In this last case, in collaboration with E. Ballico and F. Prantil from the University of Trento (IT), we succeeded in proving some existence results for some values of the parameter α. For holomorphic triples, our main investigation subject, the investigation on their moduli space was already considered in the literature, but almost entirely for curves of genus g>1. For elliptic curves we proved necessary and sufficient conditions for the moduli spaces to be non-empty, and, whenever these conditions are fulfilled, we proved that the moduli spaces are smooth and irreducible. Dealing with the projective line, as it happens for coherent systems, the situation is more difficult, even if Grothendieck provided a complete classification of the vector bundles. In this case we proved some general properties of holomorphic triples and some necessary conditions for α-stable triples to exist, but we succeeded in proving sufficient conditions only for some particular values of the parameters involved. We dealt also with bielliptic curves, double covering of elliptic ones. Our aim was to use our results on elliptic curves to deduce some results on bielliptic ones. The natural tools to achieve this task are pullback and elementary transformations, so we investigated their properties related to α-stability, and succeed in proving some existence results from the corresponding results on elliptic curves.

References

[BDGPW95]
S. Bradlow, G. D. Daskalopoulos, O. García-Prada, and R. Wentworth. Stable augmented bundles over Riemann surfaces, Vector bundles in algebraic geometry (Durham, 1993). (Cambridge University Press, ed.), London Math. Soc. Lecture Notes Ser., vol. 208, 1995, pp. 15-67.
[BGP96]
S. B. Bradlow and O. García-Prada. Stable triples, equivariant bundles and dimensional reduction. Math. Ann. 304 (1996), 225-252.
[BGPG04]
S. B. Bradlow, O. García-Prada, and P. B. Gothen. Moduli spaces of holomorphic triples over compact Riemann surfaces. Math. Ann. 328 (2004), no. 1-2, 299-351.
[BGPMN03]
S. Bradlow, O. García-Prada, V. Munoz, and P. E. Newstead. Coherent systems and Brill-Noether theory. International Journal of Mathematics 14 (2003), no. 7, 683-733.
[LN04]
H. Lange and P. E. Newstead. Coherent systems of genus 0. International Journal of Mathematics 15 (2004), no. 4, 409-424.
[LN05]
H. Lange and P. E. Newstead. Coherent systems on elliptic curves. Internat. J. Math. 16 (2005), no. 7, 787-805. MR MR2158960 (2006d:14031)
Papers Please follow this link to see a list of my papers.