\documentclass[12]{article} \usepackage{latexsym, amssymb, amsmath, mathrsfs, mathtools, slashed, hyperref, a4wide, parskip, slashed} \hypersetup{colorlinks=true,allcolors=blue, extension=pdf} %\newcommand{\doc}[1]{\hspace{-0.5cm}{\bf\href{#1}{#1}}} \newcommand{\doc}[2]{ \hspace{-0.5cm} {\bf\href{http://www2.maths.ox.ac.uk/cmi/library/Quillen/Working_papers/quillen\%20#1/#1-#2.pdf}{#1-#2}}} \begin{document} \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. \end{document}