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{\bf Math 622: Algebraic Geometry \hfill Spring 1994}
\centerline{\bf Syllabus}
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\halign{\strut\hfil #\ &#\quad\hfil&#\hfil\cr
\week
Jan. 12 &{\bf 1. Lie Groups} {\it a}) Definitions,
		examples, semidirect product, Aut~$X$\cr
     14 &1.{\it b}) Lie algebras, exp\cr
\week
     17 &1.{\it b}) Lie algebras\cr
     19 &1.{\it c}) Levi-Malcev decomposition, $G = R\times S$\cr
     21 &1.{\it d}) Root decomposition of semisimple\cr
\week
     24 &1.{\it d}) Classification of semisimple\cr
     26 &1.{\it d}) Standard bases\cr
     28 &1.{\it d}) Computing roots from Cartan matrix\cr
\week
     31 &{\bf 2. Homogeneous Spaces} {\it a}) Definitions, groups actions,
	 cosets, manifold structure, fibrations\cr
Feb.  2	&2.{\it a}) Examples ${\bf P}^n$, ${\rm Gr}(k,n)$, $Q^n$, Iwasawa,
		${\bf C}^n$, ${\bf C}^n\setminus 0$, ${\bf B}^n$, $\Delta^n$\cr
      4 &2.{\it b}) Parallelizable manifolds, Wang's Theorem\cr
\week
      7 &2.{\it c}) Equivariantly imbedded, clsd, homogeneous, in ${\bf P}^n$\cr
      9 &2.{\it c}) $B$-fixed point, classification of parabolic\cr
     11 &2.{\it d}) Standard fibrations: Normalizer, Albanese, radical\cr
\week
     14 &2.{\it d}) Borel-Remmert\cr
     16 &2.{\it d}) Reductions: holomorphic, meromorphic, hyperplane,$\ldots$\cr
     18 &2.{\it e}) Non-compact homogeneous, affine/Stein, Matsushima\cr
\week
     21 &2.{\it e}) Generalizations, extensions of representations\cr
     23 &2.{\it e}) Winkelmann's example\cr
     25 &2.{\it e}) Quasi-affine, observable subgroups\cr
\week
     28 &{\bf 3. Homogeneous Bundles, I} {\it a}) Definitions, examples, coset
		maps, vector bundles, $T_X$, $\bigwedge^p T_X^*$\cr
Mar.  2 &3.{\it b}) Line bundles, homogeneous (proof), $K_X$, characters\cr
      4 &{\sl Extra}\cr
\week
  5--13 &{\bf Midsemester Break}\cr
\week
     14 &{\bf 4. Elements of Representation Theory} {\it a}) weights,
		 standard bases, roots\cr
     16 &4.{\it b}) Weyl group, chambers, length, singular\cr
     18 &4.{\it b}) How to compute orbits, examples\cr
\week
     21 &4.{\it c}) How to calculate weights of a representation\cr
     23 &4.{\it d}) Formulas: Weyl character and dimension, examples\cr
     25 &4.{\it d}) Freudenthal's formula, examples\cr
\week
     28 &4.{\it e}) Computer implementations\cr
     30 &{\sl Extra}\cr
Apr.  1 &{\sl Easter Break}\cr
\week
      4 &{\sl Easter Break} {\sl Quiz 9}\cr
      6 &{\bf 5. Homogeneous Bundles, II} {\it a}) Borel-Weil (proof)\cr
      8 &5.{\it b}) Bott's Theorem (proof)\cr
\week
     11 &5.{\it b}) Applications, ${\bf P}^n$, rigidity\cr
     13 &5.{\it c}) Dolbeault Cohomology, hermitian symmetric spaces\cr
     15 &5.{\it c}) General, filtrations\cr
\week
     18 &5.{\it c}) Computer algorithms, examples\cr
     20 &5.{\it d}) Nef value, how to compute $K_X^*$\cr
     22 &5.{\it d}) Related vanishing theorems\cr
\week
     25 &5.{\it d}) Classification of positive defect\cr
     27 &{\sl Extra}\cr
     29 &{\sl Study Day}\cr
\week
}	


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