RESEARCH STATEMENT ALLAN YASHINSKI

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1 RESEARCH STATEMENT ALLAN YASHINSKI 1. Overview I study the role of deformation theory in Alain Connes program of noncommutative geometry [11]. Loosely speaking, a deformation of algebras is a family {A t } of algebras which depend continuously on a parameter t. One goal of my research is to identify when properties of algebras are preserved under deformations. This is a useful thing to understand because many complicated algebras can be viewed as deformations of simpler, better-understood algebras. In noncommutative geometry, deformations are also important because of their role in K-theory and index theory. Here are several examples which are relevant to my work. A prominent example from noncommutative geometry is the noncommutative torus [38], which is a deformation of the algebra C (T n ) of smooth functions on an n-torus. This is an instance of a strict deformation quantization in the sense of Rieffel [35], in which one is deforming an algebra from classical mechanics into an algebra from quantum mechanics [5]. Twisted group C -algebras are obtained as deformations of group C -algebras by applying a twist determined by a group 2-cocycle. For example, one obtains noncommutative tori by considering the case where the group is Z n. There are several other examples of noncommutative manifolds such as Rieffel s quantum Heisenberg manifolds [37], quantum homogeneous spaces [27], quantum lens spaces [23], and a variety of types of quantum spheres [13], each of which is a deformation of the algebra of functions on the manifold. The tangent groupoid construction of Alain Connes [11, Ch. 2.5] encodes the analytic index map Ind a : K 0 (T M) Z of the Atiyah-Singer index theorem [3] within a smooth deformation. The assembly map of the Baum-Connes conjecture [4] arises as a generalization of the tangent groupoid. The Baum-Connes conjecture is an important problem in noncommutative geometry with applications to many areas with mathematics including topology, geometry, operator algebras, and representation theory. George Mackey studied the relationship between the representation theory of a semisimple Lie group G and its Cartan motion group G 0 [29]. The group G can be viewed as a deformation of G 0 which is very similar to the tangent groupoid deformation [4], [24]. Further, the Connes-Kasparov conjecture is equivalent to the invariance of the K-theory groups for the corresponding reduced group C -algebras. My research thus far has focused on the rigidity of periodic cyclic homology and K-theory under deformations of algebras. For example, I would like to determine what conditions on a deformation {A t } of algebras imply that the periodic cyclic homology groups {HP (A t )} are pairwise isomorphic, and similarly for the K-theory groups {K (A t )}. These groups are noncommutative generalizations of de Rham cohomology and topological K-theory, which are homotopy invariants. Thus one expects a certain amount of rigidity under continuous perturbation. Additionally I am interested in studying the effects of deformation on the Chern-Connes pairing between cyclic cohomology and K-theory. Even in deformations where the K-theory and cyclic cohomology groups are not changing, this pairing can sometimes distinguish between the individual 1

2 2 ALLAN YASHINSKI algebras in the deformation, as in [34] for noncommutative 2-tori. Many interesting numerical invariants of K-theory classes arise from the Chern-Connes pairing, such as the analytic index of Atiyah and Singer mentioned above. Further, a complete understanding of the deformation of this pairing for the tangent groupoid deformation would lead to another proof of the Atiyah-Singer index theorem (see 4.1 below). In an application of noncommutative geometry to physics, Bellissard [6] showed that the integer quantum Hall effect is explained by the integrality of the Chern-Connes pairing with a cyclic 2-cocycle on the noncommutative torus, see also [11, Ch. 4.6]. This work was extended to the hyperbolic setting to study the fractional quantum Hall effect, which is explained by the Chern-Connes pairings in twisted group C -algebras associated to certain discrete groups of isometries of the hyperbolic plane [8], [30], [31]. My main tool in studying both the invariance of cyclic homology as well as the deformation of the Chern-Connes pairing is Getzler s Gauss-Manin connection [20]. This connection allows one the possibility to parallel translate (periodic) cyclic homology classes from one algebra in the deformation to another. The connection also has a compatibility with the Chern-Connes pairing which gives insight into how the pairings change with the deformation parameter. I have completed the following projects. I gave a complete analysis of the Gauss-Manin connection for the deformation of noncommutative tori. I proved the invariance of periodic cyclic homology by constructing parallel translation isomorphisms between the different algebras, and I also described the deformation of the Chern-Connes pairing. Using the Gauss-Manin connection, I proved an abstract rigidity result which guarantees that periodic cyclic cohomology is preserved for small enough deformations of Banach algebras which satisfy a certain finite cohomological dimension condition. As an example, this applies to the amenable Banach algebras first studied by Johnson [25]. Returning to the noncommutative torus example, I proved that this deformation extends to a deformation of differential graded algebras which is trivial, when viewed as a deformation of A -algebras. This proof generalizes easily to Rieffel s strict deformation quantizations for actions of R d [36]. As a corollary, one obtains the integrability of the Gauss-Manin connection, which proves invariance of periodic cyclic homology under these deformations. I am currently working on the following projects. Exploring the Gauss-Manin connection in the setting of the tangent groupoid, and using it to give a proof of the Atiyah-Singer index theorem. A proper understanding of this example could hopefully be generalized to prove other higher index theorems. Exploring the Gauss-Manin connection for the Mackey deformation of a semisimple Lie group into its Cartan motion group, and how it relates to both the Connes-Kasparov conjecture and the L 2 -index theorem used by Atiyah and Scmid to construct discrete series representations [2]. Extending my results with the Gauss-Manin connection to various other examples, including crossed product algebras of noncommutative tori by finite groups, twisted group algebras for surface groups, and quantum Heisenberg manifolds. Each of these examples presents a different additional difficulty relative to the noncommutative torus example. In the rest of this document, I provide more details regarding these past and current projects, as well as some background material. 2. Background 2.1. Notions from noncommutative geometry C -algebras and noncommutative geometry. The Gelfand-Naimark theorem gives a contravariant equivalence between the category of commutative C -algebras and the category of locally compact, Hausdorff topological spaces, see e.g. [32]. To such a space X, one associates the C -algebra

3 RESEARCH STATEMENT 3 C 0 (X) of continuous complex-valued functions on X that vanish at infinity. The philosophy of noncommutative geometry is to view noncommutative C -algebras as algebras of continuous functions on noncommutative spaces. Then one attempts to generalize ideas from topology to this larger class of spaces. Deformation theory provides a natural setting in which one can carry out this program, as many noncommutative C -algebras arise as deformations of commutative C -algebras K-theory. The topological K-theory of Atiyah and Hirzebruch is one such theory which has been generalized to arbitrary C -algebras, see e.g. [7]. K-theory for C -algebras enjoys many useful properties such as homotopy invariance, the existence of certain long exact sequences, and the Bott periodicity theorem; all of which make K-theory a computable invariant in many situations. K-theory is an indispensable tool in the study of C -algebras, and plays a central role in the classification of C -algebras (e.g. [15], [16]), the Baum-Connes conjecture, and index theory Cyclic homology. Cyclic homology was discovered independently by Connes [10] and Tsygan [41], see also [28]. It can be viewed as a noncommutative analogue of de Rham cohomology. As de Rham cohomology is defined in terms of a smooth structure, one does not typically consider an arbitrary C -algebra here, but rather a certain dense subalgebra which is thought of as the algebra of smooth functions on the noncommutative space. Periodic cyclic homology is a variant of cyclic homology that assigns to an algebra A two vector spaces HP 0 (A) and HP 1 (A). Connes showed that if one considers C (M), the algebra of smooth complex-valued functions on a smooth compact manifold M, then HP i (C (M)) = H i+2k dr (M), i = 0, 1, k 0 where H j dr (M) is the j-th de Rham cohomology group of M with complex coefficients. Here, one is considering C (M) with its natural Fréchet topology and using a version of periodic cyclic homology designed for topological algebras Chern-Connes character and pairing. The K-theory and the cohomology of a space are linked by the classical Chern character homomorphism. For any, possibly noncommutative, algebra A, this generalizes to the Chern-Connes character homomorphism ch : K (A) HP (A). Using the Chern-Connes character, one can define the Chern-Connes pairing HP (A) K (A) C, where HP (A) denotes the periodic cyclic cohomology of A, which is the cohomology theory dual to HP (A). This pairing can be used to assign numerical invariants to a K-theory class, as it is used in the classical setting Smooth deformations and the Gauss-Manin connection Deformations and rigidity. In a broad sense, a one-parameter deformation is a family of objects {X t } whose structure varies in a controlled fashion with a real parameter t, e.g. polynomially, smoothly, or continuously. Then one can view this collection of objects as a bundle over some parameter space J R, where X t is the fiber over t J. In my work, the fibers {X t } are topological vector spaces with extra structure, e.g. they are associative algebras or chain complexes, and the structure varies smoothly in the parameter t. The main tools I use to study such deformations are linear connections and their corresponding parallel translation maps, which are linear isomorphisms between any two fibers in the deformation. To prove that additional structure is preserved by the deformation, one must choose an appropriate connection so that the parallel translation maps respect this structure. Using this approach, there are two steps in proving a rigidity result for a deformation: (R1) Find a connection that is compatible with the additional structure of the fibers {X t }.

4 4 ALLAN YASHINSKI (R2) Prove the existence and uniqueness of solutions to the corresponding parallel translation differential equations. There is typically a cohomological obstruction to the existence of such a connection. Even if a connection of this type exists, it is a nontrivial problem to solve the corresponding parallel translation differential equations. If this is possible, we shall say the connection is integrable. The main issue is that once one considers topological vector spaces more general than Banach spaces, the existence and uniqueness of solutions to linear ordinary differential equations fails in general. Thus there are analytic obstructions as well as algebraic Deformations of associative algebras. Since parallel translation provides linear isomorphisms of the fibers, it is natural to discuss deformations of algebras in which the fibers have the same underlying topological vector space X. Given an open interval J R, a smooth one-parameter deformation of algebras is a collection {m t } t J of continuous bilinear associative multiplications on X which vary smoothly with respect to t in an appropriate sense. For each t J, we shall denote by A t the topological algebra which is X equipped with the product m t. One can view {A t } t J as a bundle of algebras over J, and then equip the space of smooth sections A J = C (J, X) with the fiberwise product (ab)(t) = m t (a(t), b(t)), a, b A J. We shall refer to A J as the algebra of sections of the deformation. Note that A J is an algebra over C (J), where the module action is given by pointwise scalar multiplication. By a connection on a C (J)-module M, we mean a complex-linear map : M M such that (f ξ) = f ξ + f (ξ), f C (J), ξ M. The appropriate notion of a compatible connection as in (R1) for a deformation {A t } t J of algebras is a connection on the algebra of sections A J that is a derivation, which means (ab) = (a)b + a (b), a, b A J. One can show that parallel translation maps induced by such a connection are algebra isomorphisms. Given any connection on A J which is not necessarily a derivation, consider the bilinear map E : A J A J A J defined by the equation (ab) = (a)b + a (b) + E(a, b). One can show that E defines a class in the Hochschild cohomology group H 2 (A J, A J ) (over the ground ring C (J)) which is independent of the chosen connection. Moreover, A J possesses a connection that is a derivation if and only if [E] = 0. Thus, [E] is a cohomological obstruction to (R1) in this case. This cocycle [E] is analogous to the infinitesimal direction of a formal deformation, appearing in the foundational work of Gerstenhaber on deformations [19], except [E] contains the infinitesimal direction of the deformation at each fiber t J Periodic cyclic homology and the Gauss-Manin connection. In many deformations of interest, e.g the noncommutative tori, the fibers are not pairwise isomorphic, and the obstruction [E] is nontrivial. To prove rigidity of periodic cyclic homology, one can attempt to construct a compatible connection on the periodic cyclic chain complex C per (A J ) of the algebra of sections A J, over the ground ring C (J). Here, C per (A J ) is isomorphic to the space of sections of the bundle of chain complexes {C per (A t )} t J. A connection is compatible with the chain complex structure if it commutes with the boundary map. The remarkable result of Getzler is that for any deformation {A t } t J, such a connection exists [20]. Theorem 2.1 (Getzler). There exists a connection GM on C per (A J ) that commutes with the boundary map. Consequently, GM descends to a connection on HP (A J ).

5 RESEARCH STATEMENT 5 Thus, there is no obstruction to (R1) for the deformation {C per (A t )} t J of chain complexes. If one can prove GM is integrable, then one obtains parallel translation isomorphisms HP (A t1 ) = HP (A t2 ) t 1, t 2 J. However, there are many deformations for which periodic cyclic homology is not preserved. Thus, one can expect to encounter analytic difficulties in general from the resulting differential equations The Gauss-Manin connection and the Chern-Connes pairing. For the algebra A J of sections of a deformation, the Chern-Connes pairing takes values in C (J):, : HP (A J ) K (A J ) C (J). For example, a smoothly varying family {φ t φ t is an even cyclic cocycle on A t } t J determines an element of HP 0 (A J ) and a smoothly varying family of idempotent matrices {e t M n (A t )} t J determine an element of K 0 (A J ). The Chern-Connes pairing done fiberwise produces a function in C (J). The dual Gauss-Manin connection GM on HP (A J ) satisfies d dt [φ t], [e t ] = GM [φ t ], [e t ]. Thus computations with GM allow us to understand how this pairing varies with t, and in some instances allow us to compute pairings for A t based entirely on pairings for one of algebras, say A Completed projects 3.1. The Gauss-Manin connection for noncommutative tori. For simplicity, I shall only discuss noncommutative 2-tori, though everything that follows can be generalized to noncommutative n-tori. Given a real parameter θ, the smooth noncommutative 2-torus A θ is a Schwartz completion of the universal algebra generated by two invertible elements u 1, u 2 subject to the commutation relation u 2 u 1 = e 2πiθ u 1 u 2, see [38]. In the case θ = 0, the algebra is commutative, and in fact A 0 is isomorphic to C (T 2 ), where T 2 is the 2-torus. Thus, the family of algebras {A θ } θ R is a smooth deformation of C (T 2 ). The cyclic cohomology of A θ was first computed by Connes in [10] by direct calculation. Nest generalized Connes computation for higher-dimensional noncommutative n-tori in [33]. In my work, I use GM to give a deformation-theoretic calculation of HP (A θ ). Let A J denote the algebra of sections of the noncommutative tori deformation. There is a canonical connection = d dθ on A J which satisfies the identity (ab) = (a)b + a (b) + 1 2πi δ 2(a)δ 1 (b), where δ 1, δ 2 are certain derivations on A J. At the commutative fiber θ = 0, the operators δ 1 and δ 2 are the usual partial differential operators on T 2. So itself is not a derivation, as the obstruction [E] is nontrivial, but it fails to be a derivation in a very precise way which is controlled by the abelian Lie algebra of derivations spanned by δ 1 and δ 2. This was the crucial fact used is my proof of the following theorem. Theorem 3.1 (Yashinski, [42]). The Gauss-Manin connection associated to the deformation {A θ } θ R is integrable. As a consequence, one obtains an alternative calculation of HP (A θ ). Using the Gauss-Manin connection, there is a parallel translation isomorphism HP (A θ ) = HP (C (T 2 )), and the latter group can be described in terms of the de Rham cohomology of T 2, as discussed above.

6 6 ALLAN YASHINSKI In my analysis of the Gauss-Manin connection for this example, I also provided a complete description of the operator GM on the generating cyclic cocycles of HP (A J ). This can be used to describe the Chern-Connes pairings. For example, it is now well-known [34] that the subgroup Z + θz is the image of the pairing of the canonical trace τ (a cyclic 0-cocycle) on A θ with K 0 (A θ ). This can be observed with the Gauss-Manin connection: if {e θ M n (A θ )} θ J is a smoothly varying family of idempotents, then τ(e θ ) = τ(e 0 ) + c 1 (e 0 )θ. Here, e 0 M n (C (T 2 )) corresponds to a smooth vector bundle on T 2 and the numbers τ(e 0 ) and c 1 (e 0 ) are the dimension and first Chern number of this bundle. Notice both numbers are integers. One can also define c 1 (e θ ) for each θ as a Chern-Connes pairing and show that c 1 (e θ ) doesn t depend on θ. Therefore c 1 (e θ ) Z, a fact related to the integer quantum Hall effect Deformation quantization and A -algebras. The algebra C (T 2 ) can be extended to form the de Rham algebra Ω (T 2 ) of differential forms on T 2. In an analogous way, the noncommutative torus A θ can be extended to a differential graded algebra (DGA) Ω (A θ ) with the property that Ω 0 (A θ ) = A θ. As in the commutative case θ = 0, the differential d can be described in terms of the derivations δ 1 and δ 2. In this way, one obtains a smooth deformation {Ω (A θ )} θ R of DGA s, in which the underlying cochain complexes are isomorphic, but the multiplication is changing. Let Ω (A J ) denote the corresponding algebra of sections of this deformation. Each derivation δ j extends to a Lie derivative L δj on Ω (A J ), and the connection = d dθ satisfies (ωη) = (ω)η + ω (η) + 1 2πi L δ 2 (ω)l δ1 (η) for all ω, η Ω (A J ). The advantage here is that the classical Cartan homotopy formula [d, ι X ] = L X implies that the last term is chain homotopic to zero in an appropriate sense. Since the defect of from being a derivation is chain homotopic to zero, we say that is a derivation up to homotopy. So one could expect parallel translation maps induced by such a connection to be algebra isomorphisms up to homotopy. What is meant here can be made precise in the language of Stasheff s A -algebras [39], see also [21]. An A -algebra is a generalization of a DGA in which associativity is only required to hold up to homotopy. Additionally, an A -algebra satisfies a sequence of relations involving higher homotopies. By considering DGA s as A -algebras, we obtain a more flexible notion of isomorphism. One can use and a correcting homotopy to construct a connection on the deformation {Ω (A θ )} θ R of A -algebras which is compatible as in (R1). This connection is integrable and so its parallel translation maps are A -isomorphisms. This leads to the following theorem. Theorem 3.2 (Yashinski, [43]). The differential graded algebras Ω (A θ1 ) and Ω (A θ2 ) are isomorphic as A -algebras, for all θ 1, θ 2 R. This is strictly weaker than being isomorphic as DGA s. This is remarkable because {A θ } θ R are in general pairwise non-isomorphic algebras, and consequently {Ω (A θ )} θ R are pairwise nonisomorphic as DGA s. Getzler and Jones extended the theory of cyclic homology to A -algebras in [21], and Getzler s original work on the Gauss-Manin connection defines it in the setting of deformations of A -algebras. One can show that the above A -isomorphisms induce ismorphisms HP (Ω (A θ1 )) = HP (Ω (A θ2 )) which coincides with the parallel translation of the Gauss-Manin connection. Further, it is a general fact that HP (Ω ) = HP (Ω 0 ) for any nonnegatively graded DGA Ω [22], and this isomorphism respects the Gauss-Manin connection.. Thus one obtains another proof of the integrability of GM, and in particular that HP (A θ ) is independent of θ.

7 RESEARCH STATEMENT 7 An added benefit to this approach with A -algebras is that the methods can be generalized to a large class of examples, namely Rieffel s deformation quantization for actions of R d [36]. Given a C -algebra A on which the group R d acts by -automorphisms and a skew-symmetric d d matrix Θ, Rieffel constructed a deformed product Θ on a smooth Fréchet subalgebra A A. In this way one obtains a smooth deformation of Fréchet algebras {A tθ } t R in which A 0 carries the original multiplication of A. This deformation is controlled by the Lie algebra g = Lie(R d ) in the same sense that the noncommutative torus deformation is controlled by Lie(R 2 ). As a result, the same machinery applies to produce A -isomorphisms between certain DGA extensions Ω (A tθ ), as above. All of the above remarks generalize to give the following theorem. Theorem 3.3 (Yashinski, [43]). Let A be a C -algebra on which the group R d acts and let Θ be a skew-symmetric d d matrix. (1) There is an A -isomorphism Ω (A Θ ) = Ω (A). (2) The Gauss-Manin connection is integrable for the deformation {A tθ } t R. In particular, HP (A Θ ) = HP (A) HP -rigidity for certain Banach algebras. The analytic difficulties in solving the parallel translation differential equations of the Gauss-Manin connection stem from the fact that the periodic cyclic chain complex is in some sense too big. It is an infinite product of the Hochschild chain groups, and so it will not be a Banach space even if the underlying algebra is a Banach algebra. Thus we cannot apply standard theorems on existence and uniqueness of solutions to linear ordinary differential equations in a Banach space. Following an idea of Khalkhali [26], we can replace the periodic cyclic cochain complex of a Banach algebra A with a smaller quasi-ismorphic subcomplex provided that A has a certain type of finite Hochschild cohomological dimension. This complex is obtained by truncating all Hochschild cochain groups past the cohomological dimension, and the resulting complex which computes HP (A) is actually a Banach space. Standard theorems from differential equations imply that the Gauss-Manin connection is automatically integrable in this setting. With these ideas, I proved the theorem below. The technical name for the cohomological dimension referenced above is the weak bidimension. Theorem 3.4 (Yashinski, [42]). Let A be a Banach algebra whose weak bidimension is finite. Given any smooth deformation {A t } t J of Banach algebras with A 0 = A, there is some ɛ > 0 such that HP (A t ) = HP (A 0 ) t < ɛ. The class of Banach algebras whose weak bidimension is zero is precisely the amenable Banach algebras first studied by Johnson [25]. In [26], Khalkhali showed that the entire cyclic cohomology HE (A), see [9], coincides with HP (A) if the weak bidimension of A is finite. As a corollary to the above theorem, we also obtain HE (A t ) = HE (A 0 ) t < ɛ. 4. Current projects I am currently working on the following projects, and I have made some progress on each one Applications of the Gauss-Manin connection to index theory. For a closed manifold M, Connes tangent groupoid construction [11, Ch. 2.5] produces a deformation of C -algebras {A t } t R such that A 0 = C0 (T M), where T M denotes the cotangent bundle, and A t = K(L 2 (M)), the algebra of compact operators on the Hilbert space L 2 (M), for t 0. An elliptic differential operator D on M determines a symbol class [σ D ] K 0 (A 0 ) = K 0 (T M), which one can canonically extend to a class [e t ] K 0 (A t ) = Z. As shown by Connes, the map which sends [e 0 ] := [σ D ] to this integer coincides with the analytic index map Ind a : K 0 (T M) Z, Ind a [σ D ] = dim(ker D) dim(coker D)

8 8 ALLAN YASHINSKI of Atiyah and Singer [3]. The index theorem asserts that the analytic index equals the topological index Ind t [σ D ] = ( 1) dim M ch(σ D ) Td(T M C), T M which is computed in terms of characteristic classes and the symbol [σ D ] K 0 (T M). The analytic index arises by a Chern-Connes pairing of [e t ] with the canonical trace tr on the compact operators Ind a [σ D ] = [tr], [e t ], (t 0). On the other hand, the topological index can be viewed as a cohomological pairing of a homology class with ch(σ D ). The idea then is to view the topological index as a Chern-Connes pairing Ind t [σ D ] = [ϕ], [e 0 ] at t = 0, for some cyclic cocycle ϕ (after passing to a suitable dense subalgebra of C 0 (T M)). One way to produce this ϕ is to use the Gauss-Manin connection to parallel translate the cyclic cohomology class [tr] at t = 1 to a class [ϕ] at t = 0. If we find a smooth family of cyclic cocycles {ϕ t } depending on the parameter t such that GM [ϕ t ] = 0 and [ϕ 1 ] = [tr], then properties of the Gauss-Manin connection imply [ϕ 0 ], [e 0 ] = [ϕ 1 ], [e 1 ] = Ind a [σ D ]. By the nature of the algebra at t = 0, the class [ϕ 0 ] necessarily corresponds to some de Rham homology class. To prove the index theorem, it remains to identify [ϕ 0 ] as the appropriate class giving the topological index formula. I have made partial progress on this project in that I have successfully produced the GM -parallel family {ϕ t } of cyclic cocycles. The form of ϕ 0 is somewhat complicated technically, and so far I have been unable to recognize it as the appropriate characteristic class. The exception is the case where the manifold has a flat metric, in which the deformation behaves similarly to a Rieffel deformation by an action of R d. For example, when M = R n, the result agrees with the work of Elliott, Natsume, Nest [17] The Mackey analogy and cyclic cohomology. The Mackey Analogy provides a correspondence between the representation theory of a semisimple Lie group G and its Cartan motion group G 0, which is a semidirect product of a certain vector space by a maximal compact subgroup of G. Higson studied this correspondence at the level of K-theory, and elaborated on its connection to the Connes-Kasparov conjecture (which is equivalent to the Baum-Connes conjecture for G) [24]. The motion group G 0 arises as a deformation of the group G, which is very similar to the tangent groupoid, discussed above. The same argument described above for the Atiyah-Singer index theorem can be applied to a canonical trace on Cc (G) to produce an index theorem, which should coincide with the index theorem used by Atiyah and Schmid to geometrically realize discrete series representations [2]. Through similar arguments, it should also be possible to prove higher index theorems by applying the same machinery to other cyclic cocycles on Cc (G). I have been studying this deformation for the low-dimensional case G = SL(2, R), and working towards proving the index theorem of Atiyah and Schmid in this case Behavior of the Gauss-Manin connection for other examples. The behavior and general properties of the Gauss-Manin connection are not well-understood. The sophisticated examples such as the tangent groupoid and Mackey deformation discussed above illustrate this point. Much can hopefully be learned by studying how it works for various other examples. The noncommutative tori and more general Rieffel deformations were my first step in this direction. I am currently working on understanding the following examples.

9 RESEARCH STATEMENT Twisted group algebras. Noncommutative tori arise as twisted group algebras of free abelian groups. It is natural to consider the behavior of the Gauss-Manin connection for other, more complicated groups. One such group to consider is a surface group, that is the fundamental group Γ g of an orientable surface of genus g. This group possesses a family of group 2-cocycles which can be used to deform the group algebra. In the g = 1 case, we obtain noncommutative 2-tori. For general g > 1, the abelianization of Γ g is the free abelian group Z 2g, and the 2-cocycles on Γ g arise as pull-backs of 2-cocycles on Z 2g under the quotient homomorphism Γ g Z 2g. It is for this reason that this example lends itself to a similar analysis to that of a noncommutative 2g-torus. These methods would apply to other examples where the group cocycles are pull-backs from a free abelian group Crossed products of noncommutative tori by finite groups. The group SL(2, Z) acts on the noncommutative 2-torus A θ by automorphisms, and therefore so does any finite subgroup F of SL(2, Z). The only such finite subgroups satisfy F = Z/kZ for k = 2, 3, 4, 6. With such an action, one can form the crossed product algebra A θ Z/kZ and hence a smooth deformation {A θ Z/kZ} θ R. These algebras have been studied recently in [14] and [40]. They can also be viewed as twisted group algebras of the semidirect product group Z 2 Z/kZ. Despite the close relationship to the noncommutative tori deformation, it is nontrivial to extend previous results to this deformation. I have made progress the k = 2 case, but would also like to understand the k = 3, 4, 6 cases as well as higher-dimensional examples Quantum Heisenberg manifolds. The quantum Heisenberg manifolds of Rieffel [37] give another interesting example of deformation quantization. In contrast to the noncommutative tori deformation, which is controlled by an action of an abelian Lie algebra by derivations, the quantum Heisenberg manifold deformation is controlled by an action of the nonabelian Heisenberg Lie algebra. This may add a new interesting element to the computations with the Gauss-Manin connection. There is a trace whose pairings with the K 0 group were computated by Abadie [1]. Further, Gabriel computed the periodic cyclic cohomology groups and calculated the Chern-Connes pairings for the higher cyclic cocycles [18]. Gabriel s results suggest what the parallel translation of the Gauss-Manin connection is, and it appears to be possible to directly prove it is. Further, these direct computations with the Gauss-Manin connection would give the pairings with the higher cyclic cocycles for free, assuming Abadie s trace computations. 5. Future projects The following are projects I would like to work on in the future The Gauss-Manin connection, the Baum-Connes conjecture, and the Novikov conjecture. Let Γ denote the fundamental group of a closed aspherical manifold M. A generalization of the tangent groupoid construction yields a deformation of groupoids, and consequently a deformation of convolution C -algebras {A t } t R, which has the property that A 0 = C0 (T M) and A t is Morita-equivalent to the reduced group C -algebra C r (Γ) for each t 0. The analog of the analytic index map here is the Baum-Connes assembly map µ : K (T M) K (C r (Γ)), which can be viewed as a more refined analytic index. The Baum-Connes conjecture for Γ asserts that µ is an isomorphism. Essentially it asserts that K-theory is preserved for this deformation. If this holds, then one expects periodic cyclic cohomology to be preserved as well. I would like to prove this directly using the Gauss-Manin connection. Using Chern-Connes pairings, the integrability of the Gauss-Manin connection would imply the rational injectivity of µ. The latter implies the (Strong) Novikov conjecture for Γ, an important outstanding problem in differential topology. It would be interesting to draw a connection between these methods and those used in [12] for hyperbolic groups.

10 10 ALLAN YASHINSKI 5.2. Towards a general rigidity theorem. I am very interested in extending Theorem 3.4 to apply to deformations of Fréchet algebras. Such a theorem would apply to many of the previously mentioned examples of interest. I see two potential strategies for proving such a theorem. The first is to consider Fréchet algebras which are projective limits of Banach algebras. Here, one can hope to reduce the problem to Theorem 3.4. The difficulty is that smooth deformations of such Fréchet algebras do not necessarily induce smooth deformations on the terms in the limit of Banach algebras. The second approach would be prove a result along the lines of Theorem 3.2. Every algebra can be embedded as the degree 0 part of the DGA C (A, A) of Hochschild cochains. In this way we can extend a deformation {A t } t J of algebras to a deformation {C (A t, A t )} t J of DGA s. From the finite Hochschild dimension assumption, one could potentially truncate these DGA s to reduce them something more manageable. The idea is then to try to construct a connection that is a derivation up to homotopy, and integrate it to provide A -isomorphisms Rigidity of K-theory. I would like to better understand the implications of the A -isomorphisms of Theorem 3.3. For instance, I would like to use this result to conclude that K (A Θ ) = K (A). What is needed is a variant of topological K-theory designed for DGA s (or possibly A -algebras) that is invariant under A -isomorphism and satisfies K dg (Ω ) = K (Ω 0 ) for any DGA Ω. If such a theory can be shown to exist, then the rigidity result K (A Θ ) = K (A) is immediate. References [1] B. Abadie, The range of traces on quantum Heisenberg manifolds, Trans. Amer. Math. Soc. 352 (2000), no. 12, (electronic), DOI /S MR (2001k:46113) [2] M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), MR (57 #3310) [3] M. F. Atiyah and I. M. Singer, The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), MR (38 #5243) [4] P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and K-theory of group C -algebras, C -algebras: (San Antonio, TX, 1993), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp , DOI /conm/167/ MR (96c:46070) [5] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), no. 1, MR (58 #14737a) [6] J. Bellissard, A. van Elst, and H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35 (1994), no. 10, , DOI / Topology and physics. MR (95h:81114) [7] B. Blackadar, K-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, MR (99g:46104) [8] A. L. Carey, K. C. Hannabuss, V. Mathai, and P. McCann, Quantum Hall effect on the hyperbolic plane, Comm. Math. Phys. 190 (1998), no. 3, , DOI /s MR (99f:58195) [9] A. Connes, Entire cyclic cohomology of Banach algebras and characters of θ-summable Fredholm modules, K- Theory 1 (1988), no. 6, , DOI /BF MR (90c:46094) [10] A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), MR (87i:58162) [11], Noncommutative geometry, Academic Press, Inc., San Diego, CA, MR (95j:46063) [12] A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), no. 3, , DOI / (90) MR (92a:58137) [13] L. Da browski, The garden of quantum spheres, Noncommutative geometry and quantum groups (Warsaw, 2001), Banach Center Publ., vol. 61, Polish Acad. Sci., Warsaw, 2003, pp , DOI /bc MR (2004k:58009) [14] S. Echterhoff, W. Lück, N. C. Phillips, and S. Walters, The structure of crossed products of irrational rotation algebras by finite subgroups of SL 2 (Z), J. Reine Angew. Math. 639 (2010), , DOI /CRELLE MR (2011c:46127) [15] G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, MR (53 #1279)

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