\titleofpresentation{\bf Thom polynomials and their positivity}

\presenter{Piotr Pragacz}

The global behavior of singularities is governed by their {\it Thom
polynomials} \cite{T}, \cite{AVGL}, \cite{FR}, \cite{Ka1}, \cite{Ka2}.
Knowing the Thom polynomial of a singularity, one can
compute the cohomology class represented by the points of a map,
having this singularity. For a more detailed account, we refer to \cite{O2}
in the booklet of Antalya Algebra Days 2006.

Computation of the Thom polynomial of a singularity is a difficult task, 
even for ``easiest'' singularities. We are especially interested 
in the computation of the Thom polynomials containing as a parameter 
the difference of dimensions between the target and source of a map 
\cite{Rim}. 

We widely use the {\it Schur function} techniques \cite {L}.

The talk has two goals:

\begin{enumerate}

\item
New, explicit computations of Thom polynomials for the singularities
$I_{2,2}$ and  $A_i$  via their Schur function expansions \cite{P0},
\cite{P1}, \cite{P2}, \cite{P3}, \cite{O1}. These computations use
the ``method of restriction equations'' of Rimanyi et al. for computing Thom
polynomials \cite{Rim}, \cite{FR}.

\item
Investigation of the structure of the Thom polynomials expanded in
the Schur function basis. Combining the approach to Thom polynomials
via classifying spaces of singularities \cite{Ka2} with
the Fulton-Lazarsfeld theory of cone classes and positive polynomials
for ample vector bundles \cite{FL}, we show -- with Andrzej Weber --
that the coefficients of the Schur function expansions of the Thom
polynomials of nontrivial stable singularities are nonnegative with
positive sum \cite{PW1}. Then we present a generalization of
this result to invariant cones in representations \cite{PW2}.
\end{enumerate}

\emph{Research supported by a KBN grant,
by T\"UB\.ITAK (during the stay at METU
in Ankara), and by the Humboldt Stiftung (during the stay at
the MPIM in Bonn).}

\begin{ourbiblio}{99}

\bibitem{AVGL} V. Arnold, V. Vasilev, V. Goryunov, O. Lyashko:
\emph{Singularities. Local and global theory,}
Enc. Math. Sci. vol. 6 (Dynamical Systems VI), Springer, 1993.

\bibitem{FR} L. Feher, R. Rimanyi,
\emph{Calculation of Thom polynomials and other cohomological obstructions
for group actions,} in: ``Real and complex singularities (San Carlos
2002)'' (T. Gaffney and M. Ruas eds.), Contemporary Math. {\bf 354},
(2004), 69--93.

\bibitem{FL} W. Fulton, R. Lazarsfeld,
\emph{Positive polynomials for ample vector bundles,}
Ann. of Math. {\bf 118} (1983), 35--60.

\bibitem{Ka1} M.E. Kazarian,
\emph{Characteristic classes of singularity theory,}
in: ``The Arnold-Gelfand mathematical seminars: Geometry and singularity
theory'' (1997), 325--340.

\bibitem{Ka2} M.E. Kazarian,
\emph{Classifying spaces of singularities and Thom polynomials,}
in: ``New developments in singularity theory'', NATO Sci. Ser. II Math.
Phys. Chem., {\bf 21}, Kluwer Acad. Publ., Dordrecht (2001), 117--134.

\bibitem{L} A. Lascoux,
\emph{Symmetric functions and combinatorial operators on polynomials},
CBMS/AMS Lectures Notes {\bf 99}, Providence (2003).

\bibitem{O1} O. Ozturk,
\emph{On Thom polynomials for $A_4(-)$ via Schur functions,}
Preprint, IMPAN Warszawa 2006 (670); to appear in Serdica Math. J. {\bf 1}
(2007).

\bibitem{O2} O. Ozturk,
\emph{Thom polynomials via Schur functions,}
in the booklet of ``Antalya Algebra Days 2006'', 39--40.

\bibitem{P0} P. Pragacz,
\emph{Thom polynomials and Schur functions I,}
math.AG/0509234.

\bibitem{P1} P. Pragacz,
\emph{Thom polynomials and Schur functions: the singularities
$I_{2,2}(-)$,}
Preprint MPIM Bonn 2006 (83), to appear in Ann. Inst. Fourier {\bf 57}
(2007).

\bibitem{P2} P. Pragacz,
\emph{Thom polynomials and Schur functions: towards the singularities
$A_i$,}
Preprint MPIM Bonn 2006 (139).

\bibitem{P3} P. Pragacz,
\emph{Thom polynomials and Schur functions: the singularities $A_3(-)$,}
in preparation.

\bibitem{PW1} P. Pragacz, A. Weber,
\emph{Positivity of Schur function expansions of Thom polynomials,}
Preprint, math.AG/0605308 and MPIM Bonn 2006 (60).

\bibitem{PW2} P. Pragacz, A. Weber,
\emph{Positivity of Thom polynomials revisited,}
in preparation.

\bibitem{Rim} R. Rimanyi,
\emph{Thom polynomials, symmetries and incidences of singularities,}
Inv. Math. {\bf 143} (2001), 499--521.

\bibitem{T} R. Thom,
\emph{Les singularit\'es des applications diff\'erentiables,}
Ann. Inst. Fourier {\bf 6} (1955--56), 43--87.

\end{ourbiblio}

\presenteraddress{Institute of Mathematics of Polish Academy of Sciences,
\'Sniadeckich 8, 00-956 Warszawa, Poland}

\email{P.Pragacz@impan.gov.pl}

