Department of Algebra and Algebraic Geometry

Head:

• Piotr Pragacz (Professor)
  email
  pok. 423, tel.: 22 5228 236

Staff:

• Ada Boralevi (Assistant Professor)
  email
• Weronika Buczyńska (Assistant Professor)
  email
  pok. 421, tel.: 22 5228 151
• Jarosław Buczyński (Assistant Professor)
  email
  pok. 421, tel.: 22 5228 151
• Sławomir Cynk (Professor)
  email
  Division in Kraków
• Grzegorz Kapustka (Assistant Professor)
  email
• Mateusz Michałek (Assistant Professor)
  email
• Karol Palka (Assistant Professor)
  email
• Elisa Postinghel (Assistant Professor)
  email
  421, tel.: 22 5228 151

Phd Students:

• Łukasz Kubat
  email

About the Department

The Department of Algebra and Algebraic Geometry was founded in 2000. It is, in some sense, a continuation of the IM PAN Department of Algebra in Toruń, directed at first by Jerzy Łoś and then by Stanisław Balcerzyk. For the history of the latter, see the summary by S. Balcerzyk published in the booklet edited on the occasion of the 50th anniversary of IM PAN, or here.

The following mathematicians have worked in the Department of Algebra and Algebraic Geometry in the period 2000–2008: Janusz Adamus, Marcin Chałupnik, Sławomir Cynk, Piotr Hajac, Joanna Jaszuńska, Zbigniew Jelonek, Paweł Kasprzak, Jarosław Kędra, Oskar Kędzierski, Mariusz Koras, Urlich Kraehmer, Adrian Langer, Tomasz Maszczyk, Jerzy Płonka, Piotr Pragacz (head), Agata Smoktunowicz, Tomasz Szemberg, Marek Szyjewski, Halszka Tutaj-Gasińska, Bronisław Wajnryb, Andrzej Weber, and Bartosz Zieliński.

The spectrum of the interest of mathematicians in the department is marked by the following topics (displayed in alphabetical order):

1. Affine algebraic geometry (Jelonek)
2. Algebraic topology (Chałupnik, Weber)
3. Classical algebra and combinatorics (Pragacz)
4. Complex projective algebraic geometry (Cynk, Kedzierski, Langer, Pragacz, Tutaj-Gasińska, Szemberg)
5. Enumerative theory of singularities (Pragacz, Weber)
6. Geometry and topology of surfaces (Wajnryb)
7. K-theory (Szyjewski)
8. Moduli spaces (Langer)
9. Noncommutative geometry and quantum groups (Hajac, Kasprzak, Kraehmer, Maszczyk, Zieliński)
10. Noncommutative rings (Smoktunowicz)
11. Symplectic geometry and topology (Kędra)
12. Universal algebra (Płonka)

The "center" of mathematical life of the department is the seminar IMPANGA. Its leading subject is complex algebraic geometry. The seminar meets each two weeks (for two sessions with a break for discussions), and gather algebraic geometers from all around of Poland. The seminar IMPANGA hosted as speakers: F. Hirzebruch, H. Esnault, G. van der Geer, J. Kollar and many others.

IMPANGA also organizes at the Banach Center mini-schools, directed especially towards young researchers from all around of Europe. Let us mention here, e.g., the following mini-schools of IMPANGA : "Characteristic classes" (2002), "Schubert varieties" (2003), and "Moduli spaces" (2005). In 2003, IMPANGA organized jointly with Institutes of Mathematics of Bulgarian, Hungarian and Romanian Academy of Sciences, the Conference and Summer School "Algebraic Geometry, Algebra, and Applications" in Borovetz (Bulgaria). Two sessions, prepared by IMPANGA at the Banach Center for a wider audience (from history and philosophy of sciences), were devoted to Grothendieck (2004) and Hoene-Wroński (2007). Consult the above web-page for the list of IMPANGA events and details of these activities.

The outgrowth of seminars and schools of IMPANGA was published in two volumes:
Topics in cohomological studies of algebraic varieties (2005),
and
Algebraic cycles, sheaves, shtukas, and moduli (2007), edited by Birkhauser-Verlag,
and in:
Hoene-Wroński: Życie, Matematyka i Filozofia (2008), edited by IM PAN.

The seminar IMPANGA also stimulated research of mathematicians not working directly in the department. Let us mention here, e.g., the papers: "A cascade of determinantal Calabi-Yau threefolds" by M. and G. Kapustka from the Jagiellonian University (math.AG/08023669), and "On Thom polynomials for A4(-) via Schur functions" by O. Ozturk from METU in Ankara (Serdica Math. J. 33 (2007), 301-320).

Another very active seminar in the department, organized by P. Hajac and T. Maszczyk, was
"Noncommutative geometry and quantum groups".
For the content of this seminar (among the speakers were the Fields medalists: A. Connes and M. Kontsevich), and for many related activities on noncommutative geometry at IM PAN, including the Conference in honor of Paul Baum's 70th Birthday (2007), see http://ncg.mimuw.edu.pl.

SELECTED PUBLICATIONS
of mathematicians from the department

• J. Adamus, E. Bierstone and P.D. Milman, Uniform linear bound in Chevalley's lemma, to appear in Canad. J. Math. (2008),
• T. Bauer, T. Szemberg, Local positivity of principally polarized abelian threefolds, J. Reine Angew. Math. 531 (2001) 191-200,
• P.F. Baum, P.M. Hajac, R. Matthes and W. Szymański, The K-theory of Heegaard-type quantum 3-spheres, K-Theory 35 (2005), 159-186.
• P.F. Baum, P.M. Hajac, R. Matthes and W. Szymański, a chapter in the forthcoming volume "Quantum Symmetry in Noncommutative Geometry", European Mathematical Society Publishing House.
• B. Bojarski, A. Weber, Generalized Riemann-Hilbert transmission and boundary value problems, Fredholm pairs and bordisms, Bull. Polish Acad. Sci. Math. 50 (2002), 479-496.
• P. Cassou-Nogues, M. Koras and P. Russell, Smooth embeddings of C^* into C², part I., to appear in J. Algebra (2008)
• S. Cynk, Defect of a nodal hypersurface, Manuscripta Math. 104 (2001), 325-331.
• T. Hadfield, U. Kraehmer, Twisted Homology of Quantum SL(2), K-Theory 34 (2005), 327-360.
• P.M. Hajac, R. Matthes and W. Szymański, A locally trivial quantum Hopf fibration, Algebr. Represent. Theory 9 (2006), 121-146.
• S. Janeczko, Z. Jelonek, Linear automorphisms that are symplectomorphisms, J. of the London Math. Soc. 69 (2004), 503-517.
• Z. Jelonek, The Łojasiewicz exponent and effective Nullstellensatz, preprint (2003).
• Z. Jelonek, K. Kurdyka, On asymptotical critical values of a complex polynomial, J. Reine Angew. Math. 565 (2001), 1-11.
• J. Kędra, D. McDuff, Homotopy properties od Hamiltonian group actions, Geometry and Topology, vol.9 (2005), Paper no.3, pp. 121-162.
• A. Langer, Semistable principal G-bundles in positive characteristic, Duke Math. J. 128 (2005), 511-540.
• A. Langer, Moduli spaces and Castelnuovo-Mumford regularity of sheaves on surfaces, Amer. J. of Math. 128 (2006), 373-417.
• A. Langer, T. Gomez, A. Schmitt, I. Sols, Moduli spaces for principal bundles in arbitrary characteristic, preprint (2006).
• A. Lascoux, P. Pragacz, Double Sylvester sums for subresultants and multi-Schur functions, J. Symb. Comp. 35 (2003), 689-710.
• T.H. Lenagan, A. Smoktunowicz, An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension, Journal of the A.M.S. 20 (2007), 989-1001.
• T. Maszczyk, A pairing between super Lie-Rinehart and periodic cyclic homology, Comm. in Math. Physics, 263 (2006), 737-747.
• T. Maszczyk, One-dimensional infinitesimal-birational duality through differential operators, Fund. Math. 191 (2006), 23-43.
• A. Parusiński, P. Pragacz, Characteristic classes of hypersurfaces and characteristic cycles, J. Alg. Geom. 10 (2001), 63-79.
• J. Płonka, Subvarieties of the clone extension of the variety of distributive lattices, Algebra Universalis 55 (2006), 175-186.
• P. Pragacz, Multiplying Schubert classes, in "Topics in cohomological studies of algebraic varieties", Trends in Math., Birkhauser (2005), 163-174.
• P. Pragacz, Thom polynomials and Schur functions: the singularities I22(-), Annales de l'Institut Fourier 57 (2007), 1487-1508.
• P. Pragacz, V. Srinivas and V. Pati, Diagonal subschemes and vector bundles, math.AG/0609381 - to appear in the special volume of Quarterly J. of Pure and Appl. Math., dedicated to J-P. Serre on his 80th Birthday (S.T. Yau et al. eds.).
• P. Pragacz, A. Weber, Positivity of Schur function expansions of Thom polynomials, Fund. Math. 195 (2007), 85-95.
• A. Smoktunowicz, There are no graded domains with GK dimension strictly between 2 and 3, Inv. Math. 164 (2006), 635-640.
• A. Smoktunowicz, Some results in noncommutative ring theory, Proceedings of the International Congress of Mathematicians, Madrid, Spain, August 22-30, 2006, Vol 2: Invited talks, 259-269.
• A. Weber, Pure homology of algebraic varieties, Topology 43 (2004), 635-644.

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