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\begin{center}\subsection*{ Contents 1986}\end{center}

\doc{1986}{1}

January 4:  $\mathbb{C}$-valued Gaussian processes.

January 5:  Gaussian integrals with complex exponents.

January 6:  Gaussian integrals in infinite dimensions with non-real exponents.

January 8, 9:  Gaussian measure on finite dimensional space $V$ is a state in the Weyl algebra $W(V\oplus V^*)$.

January 10:  Equivalences associated to a symplective vector space.

January 13, 14:  Summary of work so far this month.

January 15:  KMS condition and Gaussian states on a Weyl algebra.  Fermion case.

January 16:  Further discussion of Gaussian measures on infinite dimensional spaces.

January 17:  A distribution theory viewpoint towards Gaussian measures on infinite dimensionals.

January 18:  Families of Gaussian probabilities $\mu_h=e^{-\frac{1}{2h^2}|x|^2}dx/\mathrm{norm}$ and Fock spaces.  Solutions to the heat equation $\partial_t f=(\partial_z^2+z\partial_z)f$.

January 20:  Rescaling.

\doc{1986}{2}

Quillen's own index January 30, 1986 - March 11, 1986.

January 31, February 1,3:  Quantum mechanics of the forced harmonic oscillator.

February 5: Gaussian integrals with complex exponents.

January 7;  Discrete serie of $\widetilda{SL(2\mathbb{R})}$.

Direct construction of unitary representation of $\widetilda{SU(1,1)}$.

February 10,11:  Geberal discussion of Gaussian integrals iwth complex exponents.  

February 13:  Gaussian integrals such as $\int e^-\int\overline{\psi}(\partial_t =W)dt}(?)\mathscr{D}\overline{\psi}\mathscr{D}\psi$.

February 14:   Lagrangian subspaces of a symplectic vector space.  Cyclic cohomology of a Weyl algebra.

February 15:  Cyclic cohomology of  a Weyl algebra.

March 1:  Connes construction of cyclic cocycles attached to a Dirac operator using a  transgression operator.

March 2,3:  Transgression of the closed 1-form $\mathrm{tr}\left(\frac{1}{1-L^2}dL\right)$ where $L=\partial_x+\alpha$ over $S^1$.

March 4:  Smoothing a singularity by a Connes trick.  Construction of transgression using a smoothed Heaviside function.

March 10: Calculation of $\phi(a)=\mathrm{tr}(T^{-1}[a,T})$.

March 8: Discussion of transgression problem.

March 11: Construction of cyclic cocycles by $\mathrm{tr}(F[F,a][F,b])=2\mathrm{tr}(a[F,b])$ where $F$ is an involution.


\doc{1986}{3}
 
 Quillen's own index March 11-April 20

March 12:  Atiyah-Singer proof of Bott periodicity using spaces of fredholm operators.

March 14:  The map $U_{\mathrm{res}(H)\rightarrow U(B^+/K^+)$ is a homotopy equivalence.

March 15:  Spectral theor3em for subspaces $W\subset V^+\oplus V^-$.  The Cayley yransform from self-adjoints commutng with $\epsilon$ to unitaries carried into theor inverses by $\epsilon$.

March 16:  The restricted unitary group is homotopy equivalent to the space of Fredholm operators.

March 17,18, 19:  Gaussian or quasi-free stat3es on the Weyl and Clifford algebras,  A simple proof that the restricted unitary group is homotopyequivalent to the unitary group of the Calkin algebara.  Dilating $A$ where $-1\leq A\leq 1$ to an involution.

March 20:  Forming a graded involution from an ungraded self-adjoint operator; forming an ungraded involution from a graded self-adjoint operator.  Massive "Dirac" operator.  Square root of the Cayley transform.

March 21:  Maps $\mathscr{F}_0\stackrel{\mathrm{Toelitz}{\leftarrow}U_{\mathrm{res}\rightarrow \mathscrl{G}_{\mathrm{res}$.

March 22:  Atiyah-Singer periodicity proof using Cayley maps.

March 23:  Identifying involutions with a set of unitary maps.  Higher $K$-groups identified as lower $K$-groups of certain categories rather than as homotopy groups.

March 24:  Canonical involution Mod $\mathscr{K}$ in $L^2(M,S\otimes E)$.  Graded versions of $U_{\mathrm{res}$.

March 25:  Formula for the puul back of $\mathrm{tr}(F(dF)^k)$ on $\mathcal{I}(H)$ to $U(V)$.

March 26:  Index in $K$-theory for a family of Dirac operators parametrized by $\mathscr{A}/\mathscr{G}$ consistent with the canonical map $\mathscr{G}\rightarrow U_{\mathrm{res}}$.



\doc{1986}{4}

March 27: Index of a family of Dirac operators over a circle and Kuiper's Theorem. 

March 28:  Milnor classifying space.  Kuiper trivialization and associated constructions.

March 29 - April 2:   Index of family of Dirac operators on $S^1$ parametrized by $U(n)$ as an element in $K^1(u(n))$.

April4:  Connes connections $\nabla:\mathscr{E}\rightarrow\mathscr{E}\otimes_{\mathscr{A}}\hat{\omega}^1$.

April 6-13:  Linking even forms on $U_{\mathrm{res}$ and odd forms on $\mathscr{F}_1$.

April 16-19:  Promising approach to transgression problem.  Restricted Grassmannians.

April 20:  Graeme Segal's ideas for constructing an extension of $U_{\mathrm{res},(0)}$ by $U(\mathscr{K}$.  Natural frame bundle over the Grassmannian $\mathscr{I}_{\mathrm{res},(0)}$ for $U(\mathscr{K}^+)$.

\doc{1986}{5}

Quillen's own index April21-June 24, 1986.

April 21,22:  Letter to Mathai on excision in $K$-theory.  Index over $G\mathscr{G'}$ classifying pairs $(E_0,E_1)$ of Hilbert bundles with an index zero isomorphism modulo complact operators.  Explicit models for $BU_{\mathrm{res}}$ and index maps.

April 23:  Index map for pairs of projections congruent modulo compact operators.

April 24:  Transgression.

April 25:  More work on $U_{\mathrm{res}}$ and associated classifying spaces.

April 26,27:  Const5ucting a 1-form on a model for $G\mathscr{G}$ consisting of a projection on $H^+\oplus H^-$ commuting with the grading $\epsilon$.

April 29:  On a map $B\mathscr{G}' \rightarrow \mathcal{F}_0$.

April 30:  Alternative to the Bott map $\mathrm{Grass(V)\rightarrow\Omega(U(V), 1,-1)$ and formulas for characteristic forms.

May 2:  Laurent polynomila loops.  Ideas for an alternative proof of periodicity.

May 5:  Real periodicity.

May 7:  Brief discussion of a model for $B\mathscr{G}'$ and an old problem of linking super-connections with Grassmannian graph methods.

May 8:  The 1-form  $B\mathscr{G}$.

May 9:  Berry (Bristol) talk on quantum chaos of the Riemann zeta function.  Anothr model of $\Omega(\mathrm{Grass})$.

May 11:  Transgressing cyclic cocycles on the restricted unitary group to differential forms on a model of its classifying space.

May 14:  Maps from connections on $C^{\infty}(S^1,\mathbb{C}^n)$ to $\mathscr{I}_{\mathrm{res}}$.

may 16:  Connections on the canonical bundle over $U_n\times S^1$.  Linking odd character forms for $U_n$ with even character forms on $\mathscr{G}=\mathscr{L}U(n)$.

\doc{1986}{6}

May 18:  Lifting a character form $\gamma_{2k+1}$ on $U_n$ and write it as $\gamma_{2k+1}=d\mu_{2k}$ where $\mu_{2k}$ is $\mathscr{G}$ invariant.

May 19:  Loops on the Grassmannian and the loop group $\mathscr{G}^{\sigma}$.

May 22,23,26:  Using infinite repetition to construct $B\mathscr{G}\rightarrow \mathscr{I}(\mathscr{Q})$, where $\mathscr{G}-U_{\mathrm{res}}$.

May 29:  More on the problem given a Fredholm module $(\mathscr{A}, H F)$ to construct differential forms on the space of projectors over $\mathscr{A}$.

June 1:  Left invariant forms on $\mathscr{G}=\mathrm{Aut}(E)$ and embedding $E
\hookrightarrow E'$.

June 3,7:  Understanding the Bott map and the $S$-operator.

June 8:  Linking the $S$-operator with periodicity.

June 11: Existence of a fibration $\mathbb{Z}\times BU\rightarrow *\rightarrow U$.  Quasi-fibration $\mathscr{I}_{\mathrm{res}}\rightarrow \mathcal{F}_{1,\eta}\rightarrow U(\mathcal{K}$.

June 13:  Review of odd forms on $\mathscr{G}$.  Families of Dirac operators on $S^1$.

June 14:  Review of what is known about producing chractr forms on $\mathscr{G}$, including Bismut forms.

June 17, 18:  Proof of periodicity based on quasi-fibration ideas.

June19:  Review of unsolved problem of constructing forms on a Grassmannian.

June 22:  A third way of constructing forms, closely related to Bismut's construction.

June 23:  Exactness of forms $\pi_*(\mathrm{tr}e^{\Delta_X+\nabla})$.

June 24:  Odd forms on $B\mathscr{G}$, $\mathscr{G}=\mathscr{L}(U_n)$ defined by holonomy are spacial cases of those formed by integrating over $S^1$ the character forms of a connection on $B\mathscr{G}\times S^1$.

\doc{1986}{7}

Quillen's own index of July 11-Sept 12, 1986.

July 11,18:  Linking super-connections with Grassmannian gra[h ideas.  Curvature of graph subbundles of $\mathrm{GR}(E^0\oplus E^1)$.

July 22:  More on the Grassmannian graph.

July 23:  Review of Bott-Chern forms.

July 24:  Connes-Moscavici transgression.

July 25,26: On the limit graph:  for $V_0$, $V_1\cong\mathbb{C}^n$, $\phi_t=\lect(\begin{array}{cc} 1&0\\0&t\end{array}\right)$ on $V_0\oplusV_1$, $Z_t=\mathrm{graph}(\phi_t)$, $\lim Z_t=Z_{\infty}\subset\mathrm{Gr}(V_0)\times\mathrm{Gr}(V_1)$.

July 29,30:  Computing cocycles in $\mathrm{Gr}$.  

July 31,  August 2:  On  correspondance $zmasthrm{Gr}_n\leftarrowV\rightarrow\mathbb{P}V^0\times\widetilde{\mathbb{P}}V^1$.

August 3: Calculations related to a corresponance $\begin{array}{ccc}\{(K,I,\Gamma\W}\}&\rightarrow&\mathrm{GL}_n(V)\\\downarrow& &\\\mathrm{GL}_n(V)\end{array}$.

August 4:  Further calculations on $\mathrm{Gr}_n(V^0\times V^1)$. Decomposition of cohomology of the Grassmannian and correspondances effecting it.

August 5:  Jacobi triple identity and decomposition.  Conversation with Graeme Segal on proofs of $\frac{1}{\prod_{i>0}(1-q^i}}=\sum_{s>0}\frac{q^s^2}{(\prod_{i=1}^s(1-q^i))^2}$.

August 6:  Composition of correspondances.  Where does class $c_k$ come from?

\doc{1986}{8}

August 16:  Natural corresponances between $H^*(\mathbb{Z}\times BU)$ and the fermion Fock space of $L^6(S^1)$, and the relationship between the Jacobi formula and a previous decomposition of $H^*(BU)$.

August 17:  Integration over the fibre for $D_{1\cdota d}(V)\rightarrow \mathrm{Gr}_d}(V)$.

August 18:  Canonical isomorphism $\\Lambda^k(H^*(\mathbb{P}V)\simeq H^*(\mathrm{Gr}_k(V))$.

August 19:  Review of Berry's talk on the Grand Unitary Ensemble.

August 20,21,22:  Grassmannian graph ungraded case.  Morse decomposition.  Twisted case occurs as a desingularisation in the untwisted case.  Character lies in the $s-1$ in the twisted case.

August 25:  Chern-Simons difference form.  Investigating $\lim \phi^*(\gamma_k)$ where $\gamma_k$ is the $k$th character form.

September 2:  Super-connection forms associated to $V_0\oplus V_1$.

September 3:  Super-connection forms are globbally defined on the Grassmannain bundle.]

September 5:  $\lim_{\lambda\rightarrow 0}\mathrm{tr}\left(\frac{\sqrt{\lambda}}{\lambda-X^2}dX\right)^{2k}$ near skew-hermitian $X_0$ of $corank 1$

September 7,8:  Limit of odd character forms for $U(N)$>

August 8:  Calculations and proofs related to the limit of odd character forms.

September 10,11:  Restriction property for super-connection forms.

September 12:  Review of material on $\Gamma$-function lnks between asymptotic expansion of $\mathrm{tr}(e^{-tA})$ as $t\right\0$, residues of $\zeta_A(s)=\mathrm{tr}\left(A^{-s}\right)$ and asymptotics of $\mathrm{tr}\left(\frac{1}{\lambda-A}\right)$ as $\lambda\right\infty$.  Euler-Maclaurin formula.

\doc{1986}{9}

September 15:  Connes-Moscavici form: $w_k=\mathrm{tr}_S\left(\frac{1}{1-X^2}dX\right)^{2k}$, $\left(\begin{array}{cc}0&-T^*\\T&0\end{array}\right)$.

September 17:  Review of April's work and understanding a letter from Connes. 

Sptember 18-23:  Transgression problem and super-connections.

September 24:  Family of operators parametrized by $\Sigma(\mathscr{G})$, $\mathscr{G}=\mathrm{Aut}(E_0)$.  Graeeme Segal's ideas on using spectral flow.

September 26,27:  Using adiabatic approximation and loop groupm$(\mathcal{L}U_{2n})^{\sigma}$.

September 28:  Canonical Hilbert bundle over $\Sigma(\mathscr{G}$ and $KK$-theory.

September 29:  RElating family over $\SIgma(\mathscr{G})$ to the map $\mathscr{G}\rightarrow \mathscr{I}_\mathrm{res}$ via periodicity in $K$-theory.

September 30:  Adiabatic approximation.
\doc{1986}{10}

October 4:  $KK$-theory and the Hilbert bundle over $\Sigma\mathscr{G}$.

October 5:  $KK$ description of an element of $K^1((0,1))$ and the operator $\partial_t+A_t$ on $L^2((0,1), H_0)$.

October 7:  Ihara's talk on cyclotomic units.

October 11,12:  Is the model of the loop group for Grassmannians useful for the transgression problem?

October 16:  Review of the Index theorem over $S^1$ from Shanahan's notes.

October 17, 19:  Return to family of Dirac operators over $\mathrm{Grass}(E)$.

October 20:  Adiabatic expansion.  Return to Dirac operators over $S^1$.

october 21: The index of a family of Dirac operators over $M$ give a class in $K(B\mathscr{G})$, where $\mathscr{G}$ is the group of gauge transformations, transgress to give  $K$-class on $\mathscr{G}$.  Relationship to super-connections on $B\mathscr{G}$.

October 22:  Symmetric representation $\mathrm{Gr}(V^0)\times\mathrm{Gr}(V^1)\rightarrow\mathrm{Gr}(V^)\oplus V^1)$.

October 23,24:  Non-adiabatic limit.  Calculating $\frac{1}{i}\partial_t+(1-\rho(t))\slashed{D}+\rho(t)g^{-1}\slashed{D} g$ where $\rho$ is a smooth approximation to the Heaviside function.

October 26:  Return to loop space model of the Grassmannian.

October 27,28,29:  Bound states for the operator $\left(\begin{array}{cc}  0&-\partial_t+F_t\\  \partial_t+F_t&0\end{array}\right)$ where the $F_t$ are certain involutions.

October 31: Evaluating the loop group model for $\Omega(\mathrm{Gr}\mathbb{C}^{2n};\epsilon,-\epsilon)$.

\doc{1986}{11}

Quillen's own index for November 1, 1986-January 28, 1987.

November 1:  Remark on another approach to Connes $S$-operator.

November 5: For unitary $g\in U_2$, produce a canonical deformation $u_t$ of 
$\left(\begin{array}{cc}g&0\\ o&g^{-1}\end{array}\right)$ such that $u_t$ has no eigenvalue equal to -1 for $t\neq 0$, and possible constructions of the periodicity map.

November 6:  For graded $(E,D)$ associate forms to each $\Gamma\subset E$.

November 7:  Remark on unitary group flow.

November 9:  On $\log(\lambda -A)$ where $A$ has its spectrum in $\mathbb{R}_{\leq 0}$.

November 10:  Isomorphism of superalgebras $A\hat{\otimes}B\simeq A\otimes B$.

November 11,12:  Super-connection $\epsilon D+X$     

November 13:  Is $\log(\lambda-X^2-dX\sigma)-\log(\lambda-X^2)$ defined on the unitary group?

November 18:  Account of conversation with Friedan on conformal field theory.

\doc{1986}{12}

November 20:  On the form 
$cw_{2k+1}^t=(4t)^{2k+1}\mathrm{tr}\left(\frac{1}{t(g+1)+g-1}dg\frac{1}{t(g+1)-(g-1)\right)^{2k+1}$
where $g\in-U_p(H)=\left{g\in U(H):g\equiv -1\mathrm{mod}L^p(H)\}$.  Narashiman-Ramanan theorem.

November 21,25:  More on the Narashiman-Ramanan theorem.  Proof for a non-compact space.  Character forms for $-U_p(H)$.

November 26:  On the differential form $R_{\lambda}=\frac{1}{\lambda-X^2-dX\sigma}$.

November 28:  Normalizing character forms.

December 1:  Proof that smooth $\phi:S^1\rightarrow\mathbb{C}$ gives  smooth $g\mapsto\phi(g)$ from $U_p$ to $\phi(1)+L^p$.

December 3:  Discussion of Connes-Moscavici form from Quillen's viewpoint.

December 5,9:  $U^p(H)$ and $\mathrm{Gr}^p(H,\epsilon)$ are Banach manifolds.

December 13:  Polynomial growth in $L^{p^'}$

January 4, 1987:  On a simple proof of the Index theorm by imbedding methods.  Resolvant $\frac{1}{\lambda-L^2}$ for $L=h\gamma^{\mu}\partial_{\mu}+\sigma X$.

January 6, 7:  To extend the Dirac operator $L=\slashed{\partial}+X\sigma$ to the Cayley transform of $X$.  Von Neumann's construction.

January 20:  Transmission lines and strings.

January 21, 23:  Summary of some synplectic geometry and the manifold of Lagrangian subspaces.

January 24:  Discussion of $L=\left(\begin{array}{cc} 0&\partial_x-f(x)\\ \partial_x+f(x)&0\end{array}\right)$ where $f:S^1\rightarrow\mathbb{R}$ is smooth.

January 25: Transmission lines and Krein strings.

January 28:  Discussion of $\gamma^{\mu}\partial_{\mu}+\sigma X$ where $X$ is allowed to become infinite.





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