Schedule-at-a-Glance

42 PRIMA 2013 AbstractsJoint work with M. Kazarian. Generic surfaces in projective3-space have isolated characteristic points (invariantunder projective trasformations and stable under smallperturbations of the surface) called elliptic nodes, hyperbolicnodes and godrons (also known as cusps of Gauss).These characteristic points have interesting propertiesand deserve special attention. We give a geometric definitionof the characteristic points (there are several equivalentdefinitions) and we define natural local invariants forthem (a sign ±).Our formulas for global counting of these invariants onthe surface (or on domains in it) involve the Euler characteristicof the surface (or of the respective domains)providing restrictions on the coexistence of characteristicpoints.One way to prove our counting formulas (we know threeways) is considering the fundamental cubic form: in thesame way as the second fundamental quadratic form measuresthe quadratic deviation of a surface from its tangentplane, the cubic fundamental form measures the cubic deviationof the surface from its quadratic part. The fundamentalcubic form is a local intrinsic invariant definedfor every point of the surface, which vanish at the characteristicpoints.Our exposition will be rather geometrical and for a wideaudience.Topological classification of multiaxial U(n)-actionsMin Yan (joint with S. Cappell and S. Weinberger)Hong Kong University of Science and Technology, HongKong, Chinamamyan@ust.hkA manifold under the action of the unitary group U(n)is multiaxial if all the isotropy groups are unitary subgroups.Such manifolds are often modeled on kC n ⊕ jR,where C n has the canonical U(n)-action and R has trivialU(n)-action. One of the major achievements aboutmultiaxial manifolds was M. Davis and W.C. Hsiang’sconcordance classification in late 1970s for the smoothcategory and under the assumption k ≤ n.We study the structure set S U(n) (M) of a multiaxialU(n)-manifold M, which is the homeomorphism classesof the topological U(n)-manifolds equivariantly homotopyequivalent to M. We show that S U(n) (M) can be decomposedinto simpler structure sets. We discuss the implicationof the decomposition and compute explicitly for thecase M is the canonical representation sphere. All theresults do not assume k ≤ n.Milnor fibers of real line arrangementsMasahiko YoshinagaHokkaido University, Japanyoshinaga@math.sci.hokudai.ac.jpThe Milnor fiber of a line arrangement is a certain cycliccovering space of the complexified complement equippedwith monodromy action. We will discuss the monodromyaction on the first homology group. We will present anew algorithm computing multiplicities of monodromyeigenvalues, which uses real and combinatorial structures(chambers). I will also give an upper bounds and severalconjectures.Special Session 21Symplectic Geometry and Hamiltonian DynamicsContacting the moonUrs FrauenfelderSeoul National University, Koreafrauenf@snu.ac.krThe restricted three body problem has an intriguing dynamics.In joint work with Peter Albers, Gabriel Paternainand Otto van Koert we showed that below andslightly the first critical level energy hypersurfaces ofthe restricted three body problem are of contact type.This result allows us to apply global methods comingfrom Gromov-Witten theory, Floer theory, and SymplecticField theory to this old problem. I will explain howwe can use these global methods to detect global surfacesof section were the perturbative methods used so far fail.Mean Euler characteristic:caseViktor L. GinzburgUniversity of California, Santa Cruz, USAginzburg@ucsc.eduthe degenerateThe mean Euler characteristic (MEC) is an invariant ofa contact manifold, which is on the one hand sensitiveenough to distinguish interesting contact structures and,on the other, relatively easily computable in terms of thedynamics of the Reeb flow. This invariant was introducedby van Koert. As has been shown by Kerman and thespeaker, the MEC can be expressed in terms of local,purely topological, invariants of closed Reeb orbits wheneverthe Reeb flow is non-degenerate and has finitely ofsuch orbits. This expression generalizes the resonancerelation for the mean indices of closed characteristics establishedby Viterbo for convex hypersurfaces, and it hasbeen further refined and generalized by Espina.In this talk, based on a joint work with Yusuf Goren, wewill discuss a version of such a local formula for the MECwhen the closed orbits are degenerate. This work buildson the previous results of Long et al. and Hryniewicz andMacarini, and its main new feature is that, as in the nondegeneratecase, the contributions of individual orbits arepurely topological. We will also touch upon applicationsof this resonance relation to dynamics.The Hill-type formula and Krein-type traceformulaXijun HuShandong University, Chinaxjhu@sdu.edu.cnIn the study of the motion of lunar perigee, Hill got a formulawhich relates the characteristic polynomial of themonodromy matrix for a periodic orbit and a kind of infinitedeterminant. Hill’s formula is useful in understandingthe property of the monodromy matrix. Motivatedby the previous works, we build up the Hill-type formulafor S-periodic solutions for first-order ordinary differentialequations and Lagrangian systems. The S-periodic solutionis a solution such that x(t) = Sx(t + T ) for some orthogonalmatrix S, which comes naturally from the studyof symmetry periodic orbits in n-body problem. Basedon it, we get the Trace formula for linear ODE, includethe case of standard Sturm-Liouville systems, which canbe consider as a generalization of Krein’s work in 1950’s.Especially, for the eigenvalue problem Lu + λR(x)u = 0,where L is the Sturm-Liouville operator, ∑ 1/λ k j can becomputed by the trace formula. Some applications forthe stability problem is given.The extremal Kahler metrics on toric manifoldsAn-Min LiSichuan University, Chinamath_li@yahoo.com.cnWe study the prescribed scaler curvature problem on toricmanifolds, following Donaldson’s program. Let ∆ be an