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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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