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\section{Plan of the Talks}  \subsection{Introduction to K-theory}  \paragraph{Talk 1: Higher Algebriac K-theory}  In this talk the speaker should recall the definition of Higher K-Theory for rings. Explain the $BGL^{+}$ definition for rings. The speaker should recall the basic properties of the K-theory with out proofs.   \begin{itemize}  \item Porduct in the K-theory and teh structure of $H$-space  \item Localization theorem  \item Projective Bundle Formula  \item Fundamental exact sequence   \item Genstern resolution for smooth-semilocal ring over $k$.   \item The speaker should explain why it is important compute the K-theory of algebraically closed fields.  \item Also it would be nice to give explicity computations of $K_0$, $K_1$, $K_2$ for fields.   \end{itemize}  \subsection{Borel's theorem}  \paragraph{Talk 6: Rank of $K_n$ over number fields}  \begin{itemize}  \item Show how to reduce the problem to the computation of the real cohomology of $SL(F)$. Borel determined the ring $H^*(SL_m(R);\mathbb{Q})$ and its dual coalgebra. Relate the primitive elementes of to the indecomposables elementes and finally to $H^*(SL(O_F),\mathbb{R})$.  A more precise survey is in Soul\'e's paper Lecture one two papers of Borel.  (Survey of Borel's paper)  (could be related later with $K_3$, Bloch groups and regulators)  \begin{theo}  Let $A$ be a finite-dimensional semisimple $\mathbb{Q}-algebra$. Then for every order $R$ in $A$ we have $K_n(R)\otimes\mathbb{Q}\cong K_n(A)\otimes\mathbb{Q}$.   \end{theo}  \begin{theo}\label{Borel-Ranks}  Let $F$ a number field and and let $A$ be a central simple $F$-algebra then rank $K_n(A)\otimes\mathbb{Q}$ is periodic with period four and equal to $0$, $r_1+r_2$, $r_2$   \end{theo}  The pourpuse of this talk is assuming the computation of the real cohomology of $SL(F)$ prove Theorem \ref{Borel-Ranks}.  \end{itemize}  \paragraph{Talk 2: K-theory for finite fields}  Quillen's work in Adam's conjecture  Follow Mitchell, \emph{Notes on the K-theory of finite fields}.  See Quillen, \emph{On the cohomology and K-theory of GL over a finite field} for more details.  \paragraph{Talk 5: }  \paragraph{Talk 11: K-theory of algebraically closed fields}  The purpuse of this section is prove one of the Quillen-Lichtembaum conjectures  \begin{theo}  If $F$ is an algebraically closed field, then for $n\geq1$, $K_n(F)$ is divisible and the torsion subgroup in $K_n(F)=0$ if $n$ is evan and isomorphic to $\coprod_{l\neq char F}\mathbb{Q}_l/Z_{l}(n)$ if $n$ is odd.   \end{theo}  (First Talk of this part) In this talk we are going to discus the Rigidity Theorem  and prove Theorem 1.1 y Thm 1.2 in Weibel's book.  (Second Talk of this part) From the preliminar results describe the structure of $K_*(F)$   Corollary 1.3.1 and Proposition 1.4   The main references for this part are: Weibel's book \cite{Kbook} VI.1 and Suslin's paper  \subsection{$K_3$ of fields and the relation with the Bloch group}  \paragraph{Talk 12: $K_3$ of fields and the relation with the Bloch group}  \subsection{Introduction to K-theory}  %\paragraph{Talk 1: ``Classical'' K-theory} 
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The pourpuse of this talk is assuming the computation of the real cohomology of $SL(F)$ prove Theorem \ref{Borel-Ranks}.  \end{itemize}  \paragraph{Talk 7: Cohomology of $GL_n$}  \paragraph{Talk 8: The Borel regulator.}  (Burgos Gil Book)  \subsection{Rigidity theorem}  The aim of this section   \paragraph{Talk 9: The rigidity theorem}  \paragraph{Talk 11: K-theory of algebraically closed fields}  The purpuse of this section is prove one of the Quillen-Lichtembaum conjectures