RESEARCH STATEMENT Timothy E. Goldberg 1. INTRODUCTION My research is primarily in the field of equivariant symplectic geometry, in which one studies symmetries of symplectic manifolds represented by group actions. This is a rich area of research which has its roots in classical physics and whose modern incarnation has considerable intersection with many other fields, including complex geometry, algebraic geometry, representation theory, combinatorics, and several branches of physics. In my earlier doctoral work I proved some results about holomorphic, Hamiltonian actions on integral Kähler manifolds with compatible anti-holomorphic, anti-symplectic involutions, as described in [Gol09a]. More recently I have been studying analogues of equivariant symplectic constructions in the relatively new field of generalized complex (GC) geometry. This field has its origins in the physics of mirror symmetry and supersymmetry, and encompasses both symplectic and complex geometry as special cases, in the sense that every symplectic or complex structure naturally induces a GC one. The perspective of GC geometry as a sort of generalization of symplectic geometry has been particularly fruitful, as many notions and results from the latter have proved extendable to the former. My current work and plans for future work are devoted to expanding upon this program. I am presently working on an analogue of singular reduction of symplectic manifolds and stratified symplectic spaces in the GC context, ([Gol09b]). In the following sections, I describe the background of my research area, my plans for future research, and connections to undergraduate study and research. 2. BACKGROUND 2.1. Symplectic geometry. Definition 1. A symplectic manifold is a differentiable manifold M equipped with a symplectic form ω, that is, a closed and non-degenerate differentiable 2-form. Symplectic manifolds are the natural mathematical setting for classical mechanics. Although the non-degeneracy assumption forces the dimension of M to be even, examples abound. For instance, the cotangent bundle (phase space) of any manifold can be given a canonical symplectic structure. Also, any oriented smooth surface in R 3 has a symplectic structure given by its oriented area form. Further examples include all coadjoint orbits of Lie groups, and all smooth complex projective varieties. Because ω is non-degenerate, it induces an isomorphism between 1-forms and vector fields on M. Therefore each smooth real-valued function f on M gives rise via its differential to a vector field X f, called its Hamiltonian vector field. Because ω is skew-symmetric, each f is constant in the direction of X f. Symmetries of (M, ω) are captured by smooth Lie group actions G M preserving ω, in the sense that the diffeomorphism corresponding to each g G pulls ω back to itself, called symplectic actions. A symplectic action is called Hamiltonian if its features can be encoded in a single vectorvalued map, called a moment map.
Timothy E. Goldberg RESEARCH STATEMENT Page 2 of 7 Definition 2. A moment map for a symplectic action G (M, ω) is a smooth map Φ: M g from M to the dual of the Lie algebra of G which is G-equivariant and satisfies the equation dφ ξ = ω (ξ M, ) (1) for each ξ g. Here Φ ξ is the map M R, p Φ(p), ξ, and ξ M is the vector field on M induced by ξ. Equation (1) states exactly that each ξ M is the Hamiltonian vector field for the function Φ ξ. Features of Hamiltonian actions and the moment map have been much studied. A central result in the area is the 1982 convexity theorem of Atiyah/Guillemin Sternberg, ([Ati82], [GS82]). They proved that if T is a torus and Φ: M t is a moment map for a compact and connected Hamiltonian T-manifold (M, ω), then the moment image Φ(M) is a convex polytope, given as the convex hull of the moment images of the T-fixed points: ( ) Φ(M) = conv Φ(M T ). Several years later this was expanded by Kirwan in [Kir84] to Hamiltonian actions of arbitrary compact and connected Lie groups. Here the moment image itself is not usually a convex polytope, but its intersection with any Weyl chamber is. These results have been generalized to many different contexts, and it has been found that much information about the manifold and the action upon it can be extracted from the moment image. This is especially true for Hamiltonian torus actions. 2.2. Generalized complex geometry. Generalized complex geometry can be viewed as growing out of Dirac geometry, a subject which was developed by Theodore Courant and Alan Weinstein as a way of generalizing Poisson structures, ([CW88], [Cou87], [Cou90]). Nigel Hitchen defined generalized complex structures in [Hit03], which can be viewed as complex Dirac structures satisfying a certain non-degeneracy condition. Marco Gualtieri developed these ideas further in [Gua03], laying a thorough groundwork for the field. Let M be an n-dimensional manifold. The Pontryagin bundle, or generalized tangent bundle, of M is TM := M M. For each x M, the fiber T x M carries a natural, non-degenerate, symmetric bilinear form of signature (n, n), defined by v 1 + λ 1, v 2 + λ 2 := 1 2 (λ 1(v 2 ) + λ 2 (v 1 )) for all v i + λ i T x M. This construction induces a smoothly varying metric on the bundle, and by complex linear extension also one on the bundle s complexification T C M := ( M M) R C, called the standard metrics on TM and T C M. The other key ingredient for defining Dirac and GC structures is the Courant bracket on the space Γ(TM) of smooth sections of TM M. This is a skew-symmetric bilinear bracket extending the standard Lie bracket on vector fields, defined by [ X + α, Y + β ] := [X, Y] + LX β L Y α 1 2 d( β(x) α(y) ) for all X + α, Y + β Γ(TM). This extends complex linearly to a bracket on Γ(T C M). For a closed differential three-form H on M, there is also the H-twisted Courant bracket, defined by [ X + α, Y + β ] H := [ X + α, Y + β ] + H(X, Y, ) for X + α, Y + β Γ(TM). Definition 3. An almost Dirac structure on M is a smooth subbundle E TM such that for each x M the subspace E x T x M is a maximal isotropic subspace with respect to the standard metric, i.e. the restriction of the standard metric to E x is identically zero and dim R (E x ) = n. An almost Dirac structure E on M is a Dirac structure if its space Γ(E) of smooth sections is closed under
Timothy E. Goldberg RESEARCH STATEMENT Page 3 of 7 the Courant bracket, and it is an H-twisted Dirac structure if Γ(E) is closed under the H-twisted Courant bracket. Almost Dirac and Dirac structures on M in the complex situation, where one instead considers smooth subbundles of T C M, are defined entirely analogously. Trivial examples of Dirac structures on M are its tangent and cotangent bundles, M and M. The original motivating example is given by a Poisson structure. Let Π Γ ( 2 ( M) ) be a smooth bivector on M. This defines a bundle map Π : M M defined by λ Π(λ, ) for λ M, under the identification ( M) = M. Define E Π TM by E Π = graph(π) As proved in [Cou90], this is an almost Dirac structure on M, and it is a Dirac structure if and only if Π is a Poisson bivector, i.e. if and only if the Schouten Nijenhuis bracket of Π with itself vanishes. Definition 4. An almost GC structure on M is a complex almost Dirac structure E T C M such that T C M = E E. An almost GC structure E on M is a GC structure, respectively an H-twisted GC structure, if Γ(E) is Courant-closed, respectively H-twisted Courant-closed. Let ρ: T C M M C be the natural projection. For each x M, the type of an almost GC structure E T C M at x is the codimension of ρ(e x ) in x M C. An almost GC structure E on M defines an orthogonal bundle automorphism J: TM TM such that J 2 = id by setting E and E to be its (+i) and ( i) eigenbundles, respectively. This is a complex structure on the generalized tangent bundle TM, hence the name generalized complex geometry. Conversely, the (+i)-eigenbundle of such a map J: TM TM defines an almost GC structure, and so one can use J as the defining structure instead of E. Two important examples of GC structures come from symplectic and complex structures, as described in detail in [Gua03, Chapter 4]. A symplectic structure ω gives rise to a GC structure J ω : TM TM of constant type 0, which exchanges tangent vectors and covectors via the isomorphism defined by ω. A complex structure J: M M on M induces a GC structure J J : TM TM of constant type n, which maps vectors to vectors and covectors to covectors via J and its dual. Furthermore, in these cases the GC integrability condition, (that sections of the (+i)- eigenbundle be Courant-closed), corresponds exactly to the symplectic and complex integrability conditions, (that dω = 0 and the Nijenhuis tensor for J vanish, respectively). In [Gua03, Section 4.7], the author proves an important local structure theorem for GC manifolds. For a generic point where the GC structure has type k, the GC manifold is locally equivalent to the product of the GC manifolds R 2n 2k and C k, endowed with their standard symplectic and complex structures, respectively. The equivalence is up to diffeomorphism and a certain canonical class of transformations of GC structures, called B-transforms. Thus, the type of a GC structure can be interpreted as a measure of where it falls in the continuum between symplectic and complex structures. Additionally, every (untwisted) GC structure J: TM TM induces a Poisson structure Π on M, given by Π := ρ (J M) : M M, as described in [Gua07, Section3.4]. If J = J ω for a symplectic form ω on M, then Π is the Poisson structure on M induced by the symplectic structure, up to sign. Let M be an H-twisted GC manifold whose structure is given by J: M M and E T C M, and let G be a Lie group acting smoothly on M. This induces an action of G on TM by simultaneous pushforward and inverse pullback: g ( g, (g 1 ) ) for g G. We say that G acts by symmetries of the GC structure if (1) H is G-invariant and (2) the action G TM commutes with J, or equivalently if the complexified action G T C M preserves E. If the GC structure is given by a symplectic, respectively complex, structure, then it is easy to check that G acts by symmetries of J if and only if G acts by symplectic, respectively holomorphic, transformations. The natural class of
Timothy E. Goldberg RESEARCH STATEMENT Page 4 of 7 symmetries of a GC structure is actually larger than that induced by smooth group actions on the underlying manifold; it also includes all B-transforms. However, for this discussion we will focus on group actions. In [LT06], the authors developed a generalization of Hamiltonian actions in symplectic geometry to the GC setting. Definition 5. An action of a Lie group G by symmetries of an H-twisted GC manifold M is twisted generalized Hamiltonian if there exists a G-equivariant map µ: M g and a g -valued one-form α Ω 1 (M, g ) such that (1) H + α is an equivariantly closed three-form in the Cartan Model for G-equivariant cohomology, and (2) for all ξ g J(ξ M + α ξ ) = dµ ξ, or equivalently ξ M i dµ ξ + α ξ Γ(E). Here ξ M denotes the vector field on M induced by ξ via the action G M, and µ ξ : M R and α ξ are the R-valued function and one-form, respectively, defined by pairing the outputs of µ and α with ξ. The map µ and one-form α are called a generalized moment map and moment one-form, respectively. If H = 0 and α = 0, then the G-action is called (untwisted) generalized Hamiltonian. This is a generalization of the original definition from symplectic geometry in the sense that a symplectic moment map is also a generalized moment map with respect to the induced (untwisted) GC structure, and hence symplectic Hamiltonian actions are also generalized Hamiltonian. However, if the GC structure comes from a complex one, then the only generalized Hamiltonian actions are trivial actions. Lastly, as shown in [LT06], a generalized moment map is also a Poisson moment map with respect to the Poisson structure induced by the GC one. These twisted generalized Hamiltonian actions have been studied by multiple parties, who have proved a series of results analogous to previously known results for symplectic Hamiltonian actions. Reduction: In [LT06], the main use to which the authors put their definition of generalized Hamiltonian actions is to develop a reduction procedure, akin to the symplectic reduction initially developed by [MW74]. They prove that if µ: M g is a generalized moment map for a generalized Hamiltonian action of a compact group, if a g is a regular value of µ, and if the stabilizer G a of a acts freely on the fiber µ 1 (a), then the quotient M a := µ 1 (a)/g a inherits a natural GC structure. There is also a reduction procedure for twisted GC manifolds, but the structure on the quotient is then natural only up to B-transform. Furthermore, their reduction procedure works also for twisted generalized Kähler structures, as defined in [Gua03], allowing them to produce new and interesting examples of these structures. Convexity: In [Nit09], the author proves that if T M µ g is an H-twisted generalized Hamiltonian torus action and M is compact and connected, then µ(m) is a convex polytope which can be described as the moment image of the T-fixed points, µ(m T ). This is in perfect analogy to the symplectic convexity result for Hamiltonian torus actions of Atiyah/Guillemin Sternberg. Topology: The topology of twisted generalized Hamiltonian G-manifolds has been extensively studied in [Lin07], [Lin08], [BL08], and [BL09]. In addition to studying the usual equivariant cohomology of these spaces, Lin developed a twisted equivariant cohomology theory. In particular, Lin and later Lin and Baird showed that the properties of equivariant formality and Kirwan injectivity and surjectivity hold for both cohomology theories in many contexts, as they do for usual equivariant cohomology in symplectic geometry.
Timothy E. Goldberg RESEARCH STATEMENT Page 5 of 7 3. DIRECTIONS FOR FUTURE RESEARCH From my experience with symplectic geometry and my studies of the work mentioned above on Hamiltonian actions in GC geometry, I see many interesting directions for future study. Generally speaking, I wish to contribute to the process of exploring and expanding the results I mentioned above. Yi Lin and I are currently planning a collaboration, in which we hope to study further the twisted Kirwan map for generalized Hamiltonian manifolds, as well as develop a systematic method of constructing examples. Some directions for future individual work are described below. Consider an H-twisted generalized Hamiltonian action of a compact Lie group G on a manifold M with generalized moment map µ: M g and moment one-form α Ω 1 (M, g ). Let M 0 = µ 1 (0)/G. If 0 is a regular value of µ and G acts freely on the fiber µ 1 (0) then M 0 is a twisted GC manifold, as proved in [LT06]. However, if the action on the fiber is not free, or if 0 is a singular value of µ, then the quotient M 0 is not a manifold, and Lin and Tolman s results do not directly apply. 3.1. Stratified generalized complex spaces. Suppose 0 is a singular value of µ. Although the quotient M 0 = µ 1 (0)/G is generally only a topological space and not a manifold, it can be written as the disjoint union of a collection of strata, each of which is a manifold. This is the orbit-type stratification. First M is stratified by classifying each point p M according to the conjugacy class of its stabilizer G p in G. Because points in the same G-orbit have conjugate stabilizers, this stratification descends to one on the orbit space M/G, each strata of which can be proved to be a manifold. (See, for instance, [OR04].) Since µ is G-equivariant and 0 g is G-stable, we can similarly stratify both µ 1 (0) and M 0 = µ 1 (0)/G. In the case of symplectic Hamiltonian actions and moment maps, in [SL91] Lerman and Sjamaar prove that each strata of M 0 inherits a natural symplectic structure. Furthermore, they construct a Poisson structure on a certain subspace of the continuous functions M 0 R, and show that the symplectic strata of M 0 can be viewed as the symplectic leaves given by this Poisson structure. This motivates their definition of a symplectic stratified space. In my recent doctoral work, I have proved some similar results regarding the inherited structures on the orbit-type strata in the GC and generalized Kähler cases. I am currently working to determine whether there is some global structure on M 0 or M that ties together the structures on the individual strata. This may allow me to develop a notion of a stratified GC space, analogous to Lerman and Sjamaar s definition. 3.2. Generalized complex orbifolds. Suppose M is an untwisted GC manifold. Then as is the case for symplectic moment maps, if 0 is a regular value of µ then the action of G on the fiber µ 1 (0) is automatically locally free, so the quotient M 0 is an orbifold. It seems reasonable to expand the definition of GC structures on manifolds to orbifolds. Nitta considers this to a certain extent in [Nit09], but from the point of view of Satake s V-manifold definition of orbifolds, [Sat56]. I intend to make a more thorough study of this topic, hopefully integrating the modern notion of orbifolds in terms of Lie groupoids and differentiable stacks, for which an introduction and bibliography can be found in [Ler08]. 3.3. Comparing different reduction methods. The generalized reduction developed by Lin and Tolman is not the only proposed method of reducing GC structures by symmetry groups. Such reduction has also been considered in [SX08], [Hu05], and [Vai07], as well as [BCG07] in the more general context of reduction of Courant algebroids. Additionally, reduction of Dirac structures by symmetry groups has been studied, for example in [BC05] and [JRS09]. In particular, in the latter
Timothy E. Goldberg RESEARCH STATEMENT Page 6 of 7 the authors develop a notion of singular reduction of Dirac structures. Some comparison and contrast of these sometimes differing notions of reduction has been made, but I believe a careful and complete sorting would be very beneficial to myself as well as other researchers in this area. In particular, I am eager to compare the work I am doing on the singular reduction of GC structures, based on the Lin Tolman reduction, to that done in the Dirac context in [JRS09]. 4. CONNECTIONS TO UNDERGRADUATE STUDY AND RESEARCH The study of group actions in symplectic geometry is a rich and beautiful area for mathematical study and research, with fertile connections to complex and algebraic geometry, representation theory, combinatorics, and physics. Although an understanding and appreciation of its full scope requires background knowledge beyond the undergraduate level, there are many aspects that can be shared with and explored by undergraduate students. This is possible largely because of the material s many connections to other fields of study, increasing its accessibility to people with various interests and backgrounds. One useful portal to symplectic geometry is classical mechanics. The definition of symplectic structures originated from the Hamiltonian formulation of classical mechanics, and indeed the problem of solving Hamilton s equations for a given dynamical system amounts exactly to finding a function whose Hamiltonian vector field describes this system. The abstract ideas behind the definition of symplectic moment maps arise naturally from discussions about properties of linear and angular momentum. Another entirely accessible introduction is through questions involving isospectral sets of Hermitian matrices, i.e. those with the same eigenvalues, about whom there are many classical results. The Schur Horn Theorem describes these sets in terms of inequalities involving their diagonal entries and eigenvalues. The solution to Horn s problem describes the possible eigenvalues of the sum of two Hermitian matrices in terms of inequalities involving the eigenvalues of the two summands. The Gelfand Cetlin inequalities describe the possible eigenvalues of a principal minor of a Hermitian matrix in terms of inequalities involving the eigenvalues of the matrix and its minor. These isospectral sets can be viewed as symplectic manifolds, and these results can be seen as instances of the symplectic convexity theorem, as described in [GS05, Introduction and Chapter 3]. Symplectic geometry provides an interesting and unifying framework for these classical results, while the results provide an understandable introduction to the more advanced ideas. Another natural area in which to explore many of the ideas that are fundamental to my research field is the study of matrix groups. The use of groups to study symmetry and geometry is ubiquitous in mathematics. Matrix groups are the natural way to explore many such features of Euclidean spaces, and are also fascinating geometric objects themselves. I had the opportunity to grade for the course Matrix Groups at Cornell University, and was impressed by the wide variety of topics it is possible to include in the context of such a course. It has few prerequisites beyond vector calculus and linear algebra, but can give students a taste of and serve as a bridge to many different advanced topics in mathematics, such as representation theory and Lie groups. REFERENCES [Ati82] Michael Francis Atiyah. Convexity and commuting Hamiltonians. Bull. London Math. Soc., 14(1):1 15, 1982. [BC05] Henrique Bursztyn and Marius Crainic. Dirac structures, momentum maps, and quasi-poisson manifolds. In The breadth of symplectic and Poisson geometry, volume 232 of Progr. Math., pages 1 40. Birkhäuser Boston, Boston, MA, 2005. [BCG07] Henrique Bursztyn, Gil R. Cavalcanti, and Marco Gualtieri. Reduction of Courant algebroids and generalized complex structures. Adv. Math., 211(2):726 765, 2007.
Timothy E. Goldberg RESEARCH STATEMENT Page 7 of 7 [BL08] Thomas Baird and Yi Lin. Topology of generalized complex quotients, 2008. 33 pages, Preprint, arxiv:math.dg/0802.1341. [BL09] Thomas Baird and Yi Lin. Generalized complex hamiltonian torus actions: Examples and constraints, 2009. 19 pages, Preprint, arxiv:math.dg/0904.1178. [Cou87] Theodore James Courant. Dirac manifolds. PhD thesis, University of California, Berkeley, 1987. [Cou90] Theodore James Courant. Dirac manifolds. Trans. Amer. Math. Soc., 319(2):631 661, 1990. [CW88] Ted Courant and Alan Weinstein. Beyond Poisson structures. In Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986), volume 27 of Travaux en Cours, pages 39 49. Hermann, Paris, 1988. [Gol09a] Timothy E. Goldberg. A convexity theorem for the real part of a Borel invariant subvariety. Proc. Amer. Math. Soc., 137(4):1447 1458, 2009. [Gol09b] Timothy E. Goldberg. Singular reduction of generalized complex manifolds, 2009. in preparation. [GS82] [GS05] Victor Guillemin and Shlomo Sternberg. Convexity properties of the moment mapping. Invent. Math., 67(3):491 513, 1982. Victor Guillemin and Reyer Sjamaar. Convexity properties of Hamiltonian group actions, volume 26 of CRM Monograph Series. American Mathematical Society, Providence, RI, 2005. [Gua03] Marco Gualtieri. Generalized complex structures. PhD thesis, University of Oxford, 2003. [Gua07] Marco Gualtieri. Generalized complex geometry, 2007. 49 pages, Preprint, arxiv:math.dg/0703298. [Hit03] Nigel Hitchin. Generalized Calabi-Yau manifolds. Q. J. Math., 54(3):281 308, 2003. [Hu05] Shengda Hu. Hamiltonian symmetries and reduction in generalized geometry, 2005. 18 pages, Preprint, arxiv:math.dg/0509060. [JRS09] Madeleine Jotz, Tudor S. Ratiu, and Jedrzez Śniatycki. Singular reduction of dirac structures, 2009. [Kir84] Frances Kirwan. Convexity properties of the moment mapping. III. Invent. Math., 77(3):547 552, 1984. [Ler08] Eugene Lerman. Orbifolds as stacks?, 2008. arxiv:math.dg/0806.4160. [Lin07] Yi Lin. The generalized geometry, equivariant -lemma, and torus actions. J. Geom. Phys., 57(9):1842 1860, 2007. [Lin08] Yi Lin. The equivariant cohomology theory of twisted generalized complex manifolds. Comm. Math. Phys., 281(2):469 497, 2008. [LT06] Yi Lin and Susan Tolman. Symmetries in generalized Kähler geometry. Comm. Math. Phys., 268(1):199 222, 2006. [MW74] Jerrold Marsden and Alan Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Mathematical Phys., 5(1):121 130, 1974. [Nit09] [OR04] Yasufumi Nitta. Convexity properties of generalized moment maps, 2009. 25 pages, to appear in J. Math. Soc. Japan., arxiv:math.dg/0901.0361. Juan-Pablo Ortega and Tudor S. Ratiu. Momentum maps and Hamiltonian reduction, volume 222 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 2004. [Sat56] Ichirô Satake. On a generalization of the notion of manifold. Proc. Nat. Acad. Sci. U.S.A., 42:359 363, 1956. [SL91] Reyer Sjamaar and Eugene Lerman. Stratified symplectic spaces and reduction. Ann. of Math. (2), 134(2):375 422, 1991. [SX08] Mathieu Stiénon and Ping Xu. Reduction of generalized complex structures. J. Geom. Phys., 58(1):105 121, 2008. [Vai07] Izu Vaisman. Reduction and submanifolds of generalized complex manifolds. Differential Geom. Appl., 25(2):147 166, 2007. DEPARTMENT OF MATHEMATICS, CORNELL UNIVERSITY, ITHACA, NY 14850-4201 E-mail address: goldberg@math.cornell.edu URL: http://www.math.cornell.edu/ goldberg