### Schedule for Algebra/Geometry Seminar

 Feb 7   3:00 pmP-131 Pavel Etingof, M.I.T. and Columbia University. Algebras and varieties related to finite subgroups of $Sp(2n)$ Abstract: The talk is about the following joint work of the speaker with Victor Ginzburg. Let $V$ be a finite dimensional symplectic vector space over $C$, and $\Gamma$ be a finite subgroup of $Sp(V)$, generated by symplectic reflections. Then one can consider the (singular) algebraic variety $V/\Gamma$. It is expected that in many cases it has a symplectic desingularization (for instance, if $V=C^{2n}$, $\Gamma=S_n$, then this desingularization is the Hilbert scheme of $\C^2$). It is also expected that if such a desingularization exists, then it admits a nontrivial multiparameter deformation (as a symplectic variety), with the number of parameters equal to the number of conjugacy classes of symplectic reflections in $\Gamma$. We give a construction of a family of varieties which is generically expected to be the same as the above deformation. This is actually true in the case of $S_n$. We also provide a quantization of these varieties. The construction is based on representation theory, and is related to the Calogero-Moser integrable systems and the double affine Hecke algebras, introduced by Cherednik to prove the famous Macdonald conjectures. Feb 14   3:00 pmP-131 Mark de Cataldo, SUNY at Stony Brook. The Topology of semismall maps I First of a series of talks on the subject. Proof of the Lefschetz Theorems for lef bundles. Feb 21   3:00 pmP-131 Mark de Cataldo, SUNY at Stony Brook. The Topology of semismall maps, II An introduction to derived categories. Feb 28   3:00 pmP-131 Mark de Cataldo, SUNY at Stony Brook. The Topology of semismall maps, III An introduction to intersection cohomology Mar 7   3:00 pmP-131 Mark de Cataldo, SUNY at Stony Brook. The Topology of semismall maps, IV The Decomposition Theorem for semismall maps. Mar 14   11:30 am5-127 Andrei Caldararu, University of Massachusetts, Amherst. Counterexamples to Torelli via Fourier-Mukai transforms The Torelli theorem for K3 surfaces states that the isomorphism type of such a surface is completely determined by the intersection form and Hodge decomposition on the second cohomology group. In my talk I shall show that the corresponding statement does not hold for Calabi-Yau threefolds, by exhibiting an explicit counterexample. The proof that our spaces are indeed a counterexample to Torelli will take us through various areas of algebraic geometry: derived categories and Fourier-Mukai transforms, elliptic fibrations and Ogg-Shafarevich theory, Mori theory. This is a survey of work in progress. Mar 21   : pm SPRING BREAK, SUNY at Stony Brook. Mar 28   3:00 pmP-131 Tony Knapp, SUNY at Stony Brook. Branching theorems for compact symmetric spaces A compact symmetric space, for purposes of this talk, is a quotient G/K, where G is a compact connected Lie group and K is the identity component of the subgroup of fixed points of an involution. A branching theorem describes how an irreducible representation of a group decomposes under restriction to a subgroup. The talk deals with some old and one new branching result of this type---with an eye toward providing new data that might help in identifying the unitary dual of a noncompact semsimple group. The new result involves a curious "duality" between three kinds of classical compact symmetric spaces related to indefinite Hermitian forms and three kinds of classical compact symmetric spaces related to complexifications. Hints of this kind of duality date back to a 1974 paper by S. Gelbart. Apr 11   4:30 pmP-131 Riccardo Longoni, SUNY at Stony Brook. Topological quantum field theories and (co)homology of knot spaces in any dimension The real cohomology of the space of imbeddings of $S^1$ into $R^n$, $n>3$, is studied. Cohomology classes are obtained generalizing Bott and Taubes's construction of Vassiliev knot invariants. A diagrammatic representation of these classes can be given in term of Kontsevich graph homology. In fact one can build a complex generated by certain decorated graphs and an integration map'' that defines a morphism from this complex to the de~Rham complex of forms on the space of imbeddings. If only trivalent diagrams are considered these maps are proved to be injective, by explicitely constructing the dual cycles in homology. NOTE: DIFFERENT TIME! Apr 18   3:00 pmP-131 Agnes Szilard, Columbia University. Resolution graphs of some complex surface singularities In this talk I would like to present a geometric construction related to an isolated complete intersection singularity (icis) determined by analytic functions $f,g:{\bf C}\to{\bf C}$ such that $f$ has a one dimensional singular locus. The endproduct is a weighted graph $\Gamma$, which stores an enormous amount of information about the icis. As an application a purely combinatorial algorithm is obtained for finding each of the (minimal) resolution graphs of the hypersurface singularities $(\{ f+g^k=0\},0)$ ($k>>0$), that uses the graph $\Gamma$ and the number $k$ as input. This result is a highly non-trivial generalization of the cyclic covering case of $({\bf C}^2,0)$. Apr 25   3:00 pmP-131 John Wermer, Brown University. The Argument Principle and Boundaries of Analytic Varieties May 2   3:00 pmP-131 Bin Zhang , Penn State University. Equivariant Todd Classes for Toric Varieties For a complete toric variety, we give an explicit formula for the localized equivariant Todd class in terms of the combinatorial data -- the fan. This is based on the description of the equivariant cohomology and equivariant homology of toric varieties. May 4   2:00 pm5-127 Bill Watson , St. Johns University. Superminimal fibres in an almost hermitian submersion Please note different time, day and place!