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\begin{center}\subsection*{ Contents 1983}\end{center}

\doc{1982}{1}  

December 31, 1982  Definition of a Connes cocycle.

January 1,2:  On Connes theory.  Discussion of the goal to find the character, expressed as an equivariant form, of the index of a family of operators parametrized by a space of connections.

January 3:  Characteristic classes for flat bundles.

January 6,7: Equivariant De Rham cohomology for the gauge group $\mathscr{G}$.

January 8:  Comment on regularizing Green's functions.

January 9,10:  Characteristic classes for the index of virtual bundles.  Characteristic classes on a Grassmannian and links with Connes theory.

January 11:  Representations of the symmetric group.

January 12:  Connes homology for $A=k\oplus\overline{A}$ where $\overline{A}^2=0$.

January 13:  Calculation of Connes homology and Hochschild homology for Lie algebras.

January 14:  Letter to Loday on maps between the Connes groups and Lie $K$-groups.

January 16:  Discussion of Witten's work with the operator $d+si(X)$ and connections between the Connes complex and the loop group.

January 18:   Invariant differential forms on a gauge group.

January 19:  Connes's basic examples of cocycles.  Lie theory versus the discrete group.

January 20, 21:  Preparation for lecture on the determinant line bundle.  Elliptic curve example.

January 23:  Determinant line bundle and singular set.  New idea for producing invariant forms.

January 24:  Brief discussion on using ideas of Atiyah and Bott to produce equivariant forms in the presence of a circle action.

January 25:  Computations with the free loop groups of $U_n$.  Discussion  of aim to define a character for the index of a family of Dirac operators.

\doc{1983}{2}

January 27:  Formulas relevant for calculating the equivariant form.

January 28:  Preparation for lecture on the determinant line bundle including the use of the formula $\mathrm{Tr}_{(\mathrm{reg})}(D^{-1}\delta D)$.

January 29:  Transgression in the Chern-Simons paper applied to $GL_n$-
bundles.

January 30:  Relating Lie $K$-theory to algebraic $K$-theory. Characteristic classes of representations and implications for de Rham cohomology, Connes cohomology and Deligne cohomology.

February 1:  On a paper of Benora, Cotta-Ramasino:  Remarks on BRS Anomolies and Gauge Transformation Groups.

February 2:  Constructing classes in $H^*(B\mathscr{G}_{\delta})$ where $\mathscr{G}_{\delta}$ is the discrete group underlying $\mathscr{G}$.

Febrary 4: Preparation for the third lecture on the determinant line bundle and its connection.

February 16, 17:  Continuation of work on defining the character of the index as a differential form, metric version.

February 18, 19:  Computing the differential of $\mathrm{Tr}_{(\mathrm{reg})}(D^{-1}\delta D)$ for the Riemann surface case.

February 20, 22:  Circle case:  $\mathscr{G}=\mathrm{Maps}(S^1, U_n)$ and $\mathscr{A}$, the unitary connections.

February 23:  Meaning of an analytic proof of an index theorem for families.

February 24:  Comment on problems encountered with current approach.

February 25:  Preparation for lecture 5:  $\zeta$-function determinants or analytic torsion.

February 26:  More on the problem of finding an analytic formula for the index of a family of operators.

February 27:  Formal structure of proof of the index theorem.

February 28:  Idea from Bott-Chern paper.

March 1:  Bott-Chern formulas in the holomorphic setting.

March 2:  Curvature calculations for the determinant line bundle belonging to a family of $\overline{\partial}$ operators over a Riemann surface.

March 3:  Generalizing $\mathrm{Tr}(e^{-tD^* D})=\mathrm{Tr}(e^{-tD D^*}).$  Review Schwinger calculation using Witten formulas.

March 6:  Preparation for 7th lecture:  Computation of the curvature for $\mathscr{L}$ using the analytic torsion metric.

March 5:  Bott-Chern theory applied to investigating:  $$  [ch(E)-ch(F)]-[ch(Ker(D))-ch(Coker(D)]=d(???).$$

March 6:  Eigenvalue calculations for variations of $D^* D$.

March 6, 9:  Discussion of the problem with finding an analytic proof of an index therem for a family of Dirac operators.  Review of spinors, Clifford algebras and their $K$-theory,  Dyer-Kan classification therem for diagrams of simplicial sets.

\doc{1983}{3}

March 10, 11:  Calculations for a family of Dirac operators on $\mathbb{R}^p/\mathbb{Z}^p\times\mathbb{R}^q/\mathbb{Z}^q$.

March 11:  Notes for 7th lecture.

March 12:  Clifford algebras and Dirac operators.

March 13:  Dirac operator on $(S^1)^{n-1}\times S^1$, $n=2m.$

March 14, 15:  Kasparov cup product.  Further work on familes of Dirac operators.


March 16:  Lagrangian viewpoint.

March 17: Path integral approach.  Heat kernel over a product of two tori.  Path integral formula for the Dirac operator over Euclidean space.

March 18, 20, 23:  Calculating diagonal values for the kernel $e^{-t\square}$.

March 24:  Family of Dirac operators and the Connes algebra assigned to a foliation.

March 27:  Calculation of the index of the standard harmonic oscillator $d+d^*$ on $\mathbb{R}^n$.  Comments on the Patodi approach and the Seeley approach to the asymptotic expansion of $e^{-t\square}$.

March 28, 29:  Physics approach to calculating terms in the heat kernel.

March 30: Further calculations for the heat kernel.

March 31:  Review $\zeta(-k)$, $k = 0,1,2\ldots$ and the Adams operations in $K$-theory. 

April 1:  Arekelov-Faltings intersection theory on arithmetic surfaces.  Related questions and ideas.

April 3:  Return to holomorphic vector bundles over Riemann surfaces.

April 4:  Calculating constant term in $\mathrm{Tr}(e^{tD^* D}D^{-1}B)$ as $t\rightarrow 0$.

\doc{1983}{4}

April 8, 9:  The fermion $C^*$ algebra.

April 10, 11:  A $C^*$ algebra  and its $K$-theory, particularly the Kronecker foliation algebra.

April 12:  Cross products and factors in $C^*$ algebras.  Atiyah's $L^2$ index theorem.

April 13:  Asymptotic behavious of $\mathrm{Tr}(e^{-tD^* D}D^{-1}\delta D)$ as $t\downarrow 0$ where $D=\overline{\partial}$ on a Riemann surface.

April 14:  The determinant line bundle over a space of constant coefficient $\overline{\partial}$-operators on a trivial line bundles over a 2-dimensional torus.

April 16, 17, 18:  Review the Ray-Singer calculation of torsion on elliptic curves.

April 19:  The GRR formula for a family of constant coefficient $\overline{\partial}$-operators over a torus. 

April 20, 22:   Connes $K$-theory  of a foliation, especially the Kronecker foliation.

April 23:  Identifying $\mathscr{S}(\mathbb{R})$ equipped with the operators translation by 1 and multiplication by $e^{2\pi i x}$ with sections of the trivial line bundle over $\mathbb{T}^2$.

April 24:  Calculating $\mathbf{Pic}$ of the orbit topos of the Kronecker foliation.  Review of $\theta$-functions.

April 25, 26:  A construction using von Neumann algebras of type II which  shows that a flat unitary bundle gives zero in $K(X)\otimes\mathrm{R}$.

April 27, 28: Principal bundles with discrete structure group.

April 29, 30:  Equivalence of holomorphic structures and metrics on a compact oriented surface of fixed volume.

May 1:  Conversation with George Wilson about the Yang-Baxter identity.

May 7:  Preparation for talk on Arakelov-Faltings theory and zeta determinants.

May 8:  Computing analytic torsion for line bundles.  Results from Falting's paper.

May 9:  Talk on Arakelov-Faltings theory.



\doc{1983}{5}

May 10, 11:  Discussion of the possibility of a $K$-theory of holomorphic vector bundles.

May 12: $K$-theory of non-unital rings.

May 18:  Preparation for $K$-theory conference in Luminy.

May 19:  Return to discussing families of holomorphic curves, seeking inspiration from the work of Connes work and Feigin- Tsygan.

May 20, 21, 22, 23:  Calculations inspired by the Connes results on the Kronecker foliation.

May 25:  Notes on cyclic homology.  Mixed Hodge structure for a non-singular variety which is not complete.  Removing points and discs from a Riemann surface.

May 26:  The KMS condition.

may 27:  Cyclic homology calculation- forward shift.

May 29: Why $\prod_{n=1}^{\infty}(1-q^n)$ is a modular function.

May 30:  Cyclic cohomology and cyclic graphs.

June 1:  On cyclic homology for rings without unit.

June 2:  Relative cyclic homology of $A$ modulo $k$.

June 3:  Relating the Quillen approach to cyclic homology from Hochschild homology to the Connes approach through the non-commutative de Rham complex.  Connes definition as a functor on cyclically ordered finite sets.

June 4:  The cyclic category.

June 5:  Hsiang and Staffeldt result that $HC_p(T(V),k)=0$ if $p\neq 1$.  Comparison with de Rham homology.

June 6:  Cyclic and de Rham homology.

June 9:  Preparation for a paper on cyclic homology giving an exposition of some aspects using the double complex.

\doc{1983}{6}

June 30- July 4:  Review of progress on the index theorem for families.

July 5:  Summary of analytic progress and chang of direction to a geometrical attack.  Introduction of form $\mathrm{Tr}(e^{-tL^2+\sqrt{t}dL+\Omega})$.

July 6:  Summary of progress on ideas on the cohomology of gauge groups.

July 7:  Invariant forms on $G$ give natural transformations from flat connections on $Y\times G/Y$ to $\Omega^*(Y)$.

July 8, 9: Review of Bott-Chern formulas and applications to flat bundles on the trivial principal $\mathscr{G}$-bundle over $Y$.

July 10:  Calculation of characteristic classes using Maurer-Cartan form.

July 14:  Using the transgression formula $\int^1_0 dt(e^{td''w+(t^2-t)w^2})$.

July 15:  Preparation for letter to Loday on the natural transformation from the Connes complex $\mathscr{C}(A)$ to the filtered de Rham compex based on using $\int^1_0 dt(e^{td''w+(t^2-t)w^2})$.

July 17, 18:  Connes periodicity operator.  Connes index:  $\mathrm{Index}(epe)=\mathrm{tr}((p^{-1}[p,e])^{2m+1})$.

July 19:  Loday's conjecture on the filtration of cyclic homology obtained from $\mathfrak{gl}_n$ for different $n$.

July 20:  Defining homology classes  using the Chern-Weil curvature process for $\tilde{\mathfrak{g}}=\mathfrak{gl}_n(A)$ with values in the filtered de Rham complex.

July 21, 22:  More work on Loday's conjecture.  Letter to Loday.

July 23, 24:  Fadeev-Popov Ansatz.  List of ideas and problems.  Formal category of a scheme and related de Rham complexes.

July 25:  Review of Feynmann diagrams, effective potentials and vertex functions.

July 26:  Review normalization (Lee model).

July 28, 29, 30, August 1:  Characteristic classes for $H*(\mathscr{G})$. How to realize $\mathrm{ch}(E_{\mathrm{invar}})$ on $H^*(B\mathscr{G}\times M)$ by equivariant forms on $\mathscr{G}\backslash\mathscr{A}$.

\doc{1983}{7}

August 7:  Determining $H^*(B\mathscr{G})$ and realising primitive generators of $S(\mathfrak{g}^*)$ as differential forms.  Contrast between compact group and gauge group cases.

August 8:  Comment on continuous cohomology. Is $H^*_c(\mathscr{G},\mathcal{M})=H^*((\mathcal{M}\times \Omega^*(\mathscr{A})^{\mathscr{G}})$?

August 9:  More work on $B(\mathscr{G})$ and $H^*(\mathscr{G})$.

August 10, 11, 12:  The Lie algebra of the gauge group and Gelfand-Fuks cohomology.  The map $H^*(\tilde{\mathscr{G}})\rightarrow H^*(\mathscr{G})$.  Rational cohomology of $B\mathscr{G}$ and $\mathscr{G}$.

August 13:  Continuation of the program to determine the continuous and Lie algebra cohomology of gauge groups.  Conjecture on primative generators of the cohomology of $\mathscr{G}$, $\tilde{\mathscr{G}}$, $B_c\mathscr{G}$, and $B\mathscr{G}$.

August 17:  On the Polyakov formula for $\mathrm{det}(\slashed{\partial}+\slashed{A})$ on $\mathbb{R}^2$.


August 19:  On normalization. 

August 20:  Feynmann's formula for $\frac{1}{ab}=\int_0^1 dt\frac{1}{[ta+(1-t)b]^2}$.  Field theory of a real-valued function $\phi(x)$ , $x\in\mathbb{R}^n$  given by action $S(\phi)=\int d^n x\{\frac{1}{2}\phi(-\Delta +m^2)\phi +\frac{\lambda}{4!}\phi^4\}$.

\doc{1983}{8}

August 21:  Motion of a particle on the line governed by the Hamiltonian (anharmonic oscillator) $H=\frac{p^2}{2}-\frac{w_0^2}{2}x^2+\frac{\lambda}{4!}x^4$.

August 29, 30:  Magnetism.

September 1:  Conversation with Jackiw on anomolies and $\sigma$-model approximation to low energy QCD.

September 9:  BRS and Dixon's work.

September 10:  Review problem left over from Loday letter.  Review determinant line bundle.

September 15: Discussion of whether there is a direct connection between cyclic cohomology and anomolies.

September 17, 18:  Discussion of Connes $\Lambda$-interpretation of cyclic cohomology and the discussion of compatibility of two maps from the Lie algebra homology to Deligne cohomology.

\doc{1983}{9}

September 20, 22:  Construction of character forms associated to an invariant connection on an equivariant bundle in equivariant cohomology.

September 24-29: Determining $H^*(B\mathscr{G})$.

October 2:  Lifting a $\mathfrak{gl}_n(A)$ cycle with values in the De Rham complex to one with values in the complex $\mathscr{B}(A)$.  Amitsur compex.

October 4:  Observations from a paper of Witten on baryons.

\doc{1983}{10}

October 9:  What is $\mathrm{Ext}^*_{\lambda}(k^{\natural}, A^{\natural})$?  Return to problem raised in  letter to Loday on constructing a cocycle for $\mathfrak{gl}_n(A)$ with values in the double compex $\mathscr{C}(A)$.

October 10:  Chain complexes $\Omega^*(Y, P\times^G V)$ and $C^*(\mathfrak{g},V)$, and connection with Lie algebra cohomology.  Formulas for the boundary operators $b$ and $B$ in $\mathscr{C}(A)$.

October 11:  Karoubi's non-commutative differential algebra of forms $\overline{\Omega}(A)=\Omega(A)/[\;\:,\;\:]$.  Application of Connes theorem to the Loday problem.

October 13, 14:  Chern characteristic classes of $\mathfrak{gl}_n(A)$ with values in non-commutative de Rham cohomology of $A$.

October 17, 18:  Curvature of the Grassmannian connection form $\mathrm{ch}_n=\frac{1}{n!}\mathrm{tr}(e(de)^{2n})$.  Index formula for $F$:  Index=$(-1)^n\mathrm{tr}(\epsilon e[F,e]^{2n})$.

\doc{1983}{11}

October 20:  Maps $K_0(A)\rightarrow HC_{2n}(A)$ and $K_1(A)\rightarrow HC_{2n-1}(A)$.


October 22, 23:  Proving $\mathscr{B}(A)_{\mathrm{red}}$ is quasi-isomorphic to $\overline{A}^{*(*+1)}/(1+t,b)$.  Lie algebra cohomology of the gauge group amd how it is related to the index and determinant questions.

October 26, 27:  Connes $S$-operator.

October 29, 30,:  Trace for 1-summable Fredholm modlues.


October 31, November 1:  Connes $S$-operator.

November 2:  Connes-Karoubi theorem:  $H^p(\overline{\Omega})=\mathrm{Im}\{S:\overline{HC}_{p+2}(A)\rightarrow\overline{HC}_{p}(A)\}$.

\doc{1983}{12}

November 10:  Computing transgression $H^*(B\mathscr{G})\rightarrow H^*(\mathscr{G})$.  $(BG)^{S^1}\equiv  PG\times^{G}(G_c)$ where $G_c$ denotes the $G$ space with $G$ acting as conjugation.

November 11:  Discussion of Singer's approach to calculating transgression.

November 12, 13:  Calculating transgression using the Chern-Simons form.

November 14, 15:  Transgression formula:  $\mathrm{tr}(e^{F_A})-\mathrm{tr}(e^{F_B})=d\int_0^1dt\;\mathrm{tr}(A-B)e^{(1-t)F_B+tF_A-t(1-t)(A-B)^2}$.

November 16:  Notes on an anomoly formula:  $c_1(\overline{\mathscr{L}})=\int_{M^{2n}}(\mathrm{ch}(E)\hat{A}(M))_{n+1}$.  Connections on the principal bundle.

November 17:  Calculations on the principal bundle.  Equivariant curvature in both $D$ and $A$ notations.

November 18:  Constructing characterisitic classes for $\mathscr{G}$-bundles out of character classes for $U$-bundles.

November 19, 20:  Formulas for gauge transformations on the principal bundle.  Action of gauge transformations on connections:  $g^{-1}\circ D\circ g$ on $\Omega(M,E) \leftrightarrow d+g^*A$ on $\Omega(P)\times V$.  Transgression map:  $W(\mathfrak{g})_{\mathrm{basic}}\rightarrow\Lambda(\mathfrak{g})^G$.

November 21:  Check transgression calculations.

\doc{1983}{13}

November 21:  Letter to Singer on transgression formulas.  Checking left and right actions of the gauge group.

November 26:  Review of local index formulas.  Some new ideas.

November 27:  Review construction of characterisitic classes for the Lie algebra of gauge trasnformations as in letter to Loday.

November 28:  Constructing invariant forms on $\mathscr{G}$.

November 29:  On Atiyah's suggestion that Quillen's formula associated with $e^{tL^2+\sqrt{t}[D,L]+F}=e^{tL^2+\sqrt{t}d^{\mu}[D_{\mu},L]+\frac{1}{2}dx^{\mu}dx^{\nu}F_{\mu\nu}}$ and Getzler's proof should be part of the same framework.

November 30, December 1,2:  Comment on Wess-Zumina Lagrangian as described by Witten.  Preparation for letter on transgression.

December 3:  Comment on Singer's intention on using his $\mathrm{vol}_B$ construction and how it works for flat connections.

December 4:  Summary of formulas and explanation of apparent paradox.

\doc{1983}{14}

December 6:  Talk in Jaffe's seminar.  Notes of a conversation with Luis and Ginzpang about anomoly formulas.

December 8:  Describing anomolies using cyclic cohomology.

December 11:  Note that Witten-Alvarez obtain the $\hat{A}$-genus by means of a constant EM-filed.  Fermion quantum mechanics.

December 13:  Lot's problem.

Decmber 14:  Fujikawa's approach to anomolies.

December 15:  Return to Lot's problem.  Witten's table and QCD.

December 16:  Facts about QCD.  Witten's observation about a physical interpretation of stable homotopy groups.

December 17:  On the Connes $S$-operator.

December 18:  The local index theorem  using path integrals on Euclidean space with arbitrary gauge potential.  Digression on spinor representations in 2n and almost complex manifolds.

December 19:  The index of the Rarita-Schwinger operator.

December 21:  Discussion of Clifford algebras.

December 22:  Weyl quantization and its fermion version.

December 23:  Developing a formal theory of path integral and fermion integration theory.

December 24:  Quantizing the Toeplitz process.  Fermionic analogues.

December 28:  Review transgression process for constructing differential forms on $\mathscr{G}$.

December 29:  Local index formula for a family of Dirac operators on $M$ parametrized by a family of connections quotiented by a gauge group.

\doc{1983}{15}

December 30:  Is the cohomology class of the form $\mathrm{tr}_E (\epsilon_E e^{L^2+[D,L]+D^2})$ independent of $L$?

December 31:  Super-trace.

\doc{1983}{Calculations relevant to cyclic theory}

\doc{1983}{Ideas}

\doc{1983}{Lecture Notes 0}

 Quillen: First lecture on local index theory

\doc{1983}{Lecture Notes 1}

 Donaldson: Generalization of a theorem of Narasimhan-Seshadri

\doc{1983}{Lecure Notes 2}

 Getzler:  New proof close to Kotake.

\doc{1983}{Lecture Notes 3}

 Kirwan: Theorems about convex bodies.

\doc{1983}{Lecture Notes 4}

 Atiyah: Integrals over fixpoint submanifolds.

\doc{1983}{Lecture Notes 5}

 Quillen: Review of local index theorem.

\doc{1983}{Lectyre Notes 6}

 Ginsparg: Obtaining the Chern-Simons form.

\doc{1983}{Lecture Notes 7}

 Witten: Equivariant index theorem.

\doc{1983}{Lecture Notes 8}

 Witten: Some QCD inequalities.

\doc{1983}{Lecture Notes 9}: 

 Witten: Tables.

\doc{1983}{Lecture Notes 10}

 Getzler: Heat kernel of $H=\frac{1}{2}[-\Delta+A^* A]$.

\doc{1983}{Lecture Notes 11}:

 Kazhdan and Thomas Parker:  
On Hecke algebra $C_c^{\infty}(G)$.  Super-symmetric $\sigma$-model.

\doc{1983}{Lecture Notes 12}

 Atiyah: Newton Polyhedra and Algebraic Geometry.

\doc{1983}{Lecture Notes 13}

  Cuntz and Connes: Quasi-homomorphisms.  Cyclic cohomology.

\doc{1983}{Lecture Notes 14}

 Schroer: Super-symmetric theory

\doc{1983}{Lecture Notes 15}

 Patterson: New results in ergodic theory.

\doc{1983}{Notes}

 On cyclic theory.

\doc{1983}{Review}

 Of Getzler material.





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