October 11, 2013
Abstract : Let $G=Z_{p}^k$ be the $p$-torus of rank $k$, $p$ a prime, or respectively $G=T^k = (S^1)^k$ be a $k$-dimensional torus and let V and W be orthogonal representations of G with $V^G= W^G=\{0\}$. The goal of this talk is to present a sufficient and necessary condition for the existence of G-equivariant map $f:S(V)-->S(W)$. Joint work with Denise de Mattos USP-Brazil and Waclaw Marzantowicz UAM-Poland.
Abstract : In the study of the moduli space of tropical curves of genus 1 with marked points, D. N. Kozlov shows that this moduli space has the homotopy type of a quotient space of the n-torus with respect to a conjugation Z2-action, where the latter is interesting in its own right. I will explain Kozlov's work on the analysis of the space and the computation of its mod 2 homology. Furthermore, I would like to talk about my joint work with Prof. T. Akita on the integral homology of this space.
Abstract : The transversality theorem has been formulated in various contexts after R. Thom. Let us consider equivariant manifolds with smooth actions of finite groups. Equivariant versions of the theorem were formulated by A. Wasserman, T. Petire, E. Bierstone, and so on. In this talk we introduce an equivariant version formulated analogously to a non-equivariant version by A. Hattori. As applications of the theorem, we obtain a degree sum theorem for semi-free actions with finite fixed points, and a necessary condition so that two real representation spaces could be Smith equivalent.
Abstract : Let W be a simply-connected 4-manifold, and let x_1, x_2, x_3, x_4 be 2-dimensional homology classes of W. If x_i\cdot x=0 for every x\in H_2(W,Z) and i=1,2,3,4, and all the Matsumoto triple < x_i,x_j,x_k > vanish for all distinct i,j,k's, we define a quadruple product < x_1, x_2; x_3, x_4 >. It is defined in Z/I(x_1,x_2;x_3,x_4), where I(x_1,x_2;x_3,x_4) is an ideal of Z generated by the sets, , , (x\in H_2(W,Z)). If x_1, x_2, x_3, x_4 can be represented by disjoint immersions, then this quadruple product vanishes. A relation with the second order non-repeating intersection invariant of Schneider-Teichner is discussed.
Abstract : We consider vector partition functions with integral weights and associated volume functions. We give explicit formulas for them, which generalize the results of Brion and Vergne. We apply these formulas to the study of the topology of certain symplectic quotients, such as multiple weight varieties.
Abstract : Goldin gave an explicit formula for the rational cohomology ring for weight varieties of type A, which are symplectic torus quotients of a coadjoint orbit of SU(n) by using a theorem of Tolman and Weitsman. In this talk, I will explain a generalization of this formula to multiple weight varieties of type A, which are symplectic torus quotients of direct products of coadjoint orbits of SU(n). I will also talk about its application.
Abstract : Springer varieties are subvarieties of a flag variety which are parametrized by nilpotent matrices. It is well-known that the cohomology ring of a Springer variety is the quotient of a polynomial ring by the ideal called Tanisaki's ideal. On the other hand, Springer varieties admit a natural S^1-action. In this talk, we give an explicit description of the S^1-equivariant cohomology ring of the special Springer varieties ((n-k,k) Springer varieties). Our description of the S^1-equivariant cohomology gives a description of the ordinary cohomology ring of (n-k,k) Springer variety by forgetting the S^1-action and the ideal obtained this way agrees with Tanisaki's ideal but our generators of the ideal are different from the generators given by Tanisaki.
Abstract : The GKM theory claims that the equivariant cohomology of some good manifold with a torus action is completely determined by the fixed point sets of subtori of codimension 1. First I will explain some results of Guillemin, Sabatini, and Zara with integral coefficients, which make us able to analyze some good fiber bundles by the GKM theory. Next I give a concrete description of the equivariant integral cohomology ring of an exceptional flag manifold $F_4/T$ and explain GKM theoretical aspects of fiber bundles which are useful for understanding of the cohomology. I would also like to talk about $E_6/T$ if it is possible.
Abstract : Let P be a simple n-polytope, and T^n be the n-dimensional torus (S^1)^n. A quasitoric manifold over P is a 2n-dimensional manifold M with a locally standard T^n-action, namely, an action locally modeled in the diagonal action of T^n on C^n, for which the orbit space M/T^n is homeomorphic to P as a manifold with corners. In this talk, we consider the topological classification of quasitoric manifolds from the viewpoint of the cohomological rigidity problem.
Abstract : Torus manifold is an even dimensional manifold which has a half dimensional torus action with fixed points. It is easy to check that the 2-dimensional torus manifold is equivariantly diffeomorphic to the 2-sphere with the standard circle action. In 1970, Orlik and Raymond prove that each 4-dimensional simply connected torus manifold can be obtained from an equivariant connected sum of 4-sphere, complex projective space or Hirzebruch surfaces. This is known as a construction theorem of 4-dimensional torus manifolds. In this talk, I introduce the construction theorem of 6-dimensional torus manifolds with vanishing odd degree cohomology (this is also called an equivariantly formal torus manifold). As a consequence of this result, we get the following theorem: Let M and M' be simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. Then, M and M' are equivariantly diffeomorphic up to automorphism of T if and only if their cohomology H^*(BT)-algebras are isomorphic up to automorphism of H^*(BT). If time permits, I also talk about some of the related problems of this topic.
Abstract : Chow stability is one of notions of Mumford's Geometric Invariant Theory to study the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is completely determined by its inherent 'secondary polytope', which is a polytope whose vertices are corresponding to regular triangulations of the associated (Delzant) polytope. In this talk, we would like to discuss combinatorial framework for the Chow form of a smooth polarized toric variety and its application.
Abstract : We shall construct complex contact similarity manifolds. Among them there exists a complex contact infranil-manifold which is a holomorphic torus fiber bundle over a quaternionic euclidean space form. We show that the connected sum of the infranil-manifold with the complex contact complex projective space admits a complex contact structure. Our examples are different from previously known complex contact manifolds such as the complex Boothby-Wang fibration or the twistor fibration.
Abstract : In this talk, we consider the rings of SL(2, C)-characters of free groups and free abelian groups. In our previous works, we studied a certain descending filtration of ideals of the ring of SL(2, C)-characters of a free group, on which the automorphism group of a free group naturally acts. By considering this action, and by using an argument similar to that in Johnson-Morita theory, we introduced a central filtration of the automorphism group of a free group. In order to investigate its graded quotients, we requiers detailed information about the graded quotients of the ideals in the ring of SL(2, C)-characters. However, they seem to be quite complicated, and not easy to handle in general. In this talk, we mainly consider the free abelian group analogue of the above situation. Let $H$ be a free abelian group, and $X(H)$ the ring of SL(2, C)-characters of $H$. For the ideal $J$ of $X(H)$ generated by $(\mathrm{tr} x)-2$ for any $x \in H$, we determine give a $\Q$-basis of each of $J^k/J^{k+1}$ for each $k \geq 1$. As an application, we obtain that $X(H)$ is an integral domain.
Abstract : We prove the basic theorems of P.A. Smith theory under minimal assumptions.