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<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B9V1i<T(B</span><span
lang=EN-US>:] Professor Mark Behrens, %Department of Mathematics, <o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>MIT<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B9V1iBjL>(B</span><span
lang=EN-US>:] Topological automorphic forms<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>%\ \linebreak<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>%\hbox{</span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B9b<!85%2!<%8M}O@$H@5B'(B</span><span lang=EN-US>Casson</span><span
style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$BITJQNL(B</span><span lang=EN-US>}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>%\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>%}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>%Lecture series: <o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B%"%V%9%H%i%/%H(B</span><span
lang=EN-US>:] <o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will describe methods relating stable
homotopy groups of<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>spheres to the arithmetic of automorphic
forms. There is a filtration<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>on the stable stems called the chromatic
filtration. The first layer<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>may be detected by K-theory, and the second
layer may be detected by<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>the cohomology theory of Topological
Modular Forms (TMF). After<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>reviewing how this works, I will describe
how the nth layer may be<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>studied using cohomology theories of
Topological Automorphic Forms<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>associated to Shimura varieties.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\vspace{3mm}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{center}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US> {\Large\bf </span><span
style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B%W%m%0%i%`(B</span><span lang=EN-US>}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{center}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>%\small<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>%\vspace{1mm}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[{\bf </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B#17n#2#1F|!J7n(B</span><span lang=EN-US>) </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B8a8e(B</span><span lang=EN-US>: </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#5#2!#39fEo#1#1#09V5A<<(B</span><span
lang=EN-US>}]<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>%\vspace{1mm}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#3(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#5(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US>, %</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#1#09V5A<<!'(B</span><span
lang=EN-US>%,\qquad 15:40 $\sim$ 17:00 </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B9V1i#0#2(B</span>
<span lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Number theoretic background}]\
\newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will survey Hilbert's 12th problem, which
asks for an explicit form<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>of class field theory, and explain how it
is solved in different<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>contexts using the muliplicative group,
moduli of elliptic curves, and<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>Shimura varieties.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#6(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#8(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US>, %</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#1#09V5A<<!'(B</span><span
lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Homotopy theoretic background}]\
\newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will review some aspects of the chromatic
filtration, and explain<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>how it is studied using the multiplicative
group and K-theory (first<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>layer), and moduli of elliptic curves
(second layer).<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[{\bf </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B#17n#2#2F|!J2P(B</span><span lang=EN-US>) </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B8aA0(B</span><span lang=EN-US>: </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#5#2!#39fEo#1#0#39V5A<<(B</span><span
lang=EN-US>}]<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#0(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#2(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US>, %</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#0#39V5A<<!'(B</span><span
lang=EN-US>%,\qquad 15:40 $\sim$ 17:00 </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B9V1i#0#2(B</span>
<span lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Classification of abelian
varieties in characteristic $p$}]\ \newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will outline the Honda-Tate
classification of abelian varieties over<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>finite fields, using the concept of
$p$-divisible group.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[{\bf </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B#17n#2#2F|!J2P(B</span><span lang=EN-US>) </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B8a8e(B</span><span lang=EN-US>: </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#5#2!#39fEo#1#0#39V5A<<(B</span><span
lang=EN-US>}]<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#3(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#5(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US>, %</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#0#39V5A<<!'(B</span><span
lang=EN-US>%,\qquad 15:40 $\sim$ 17:00 </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B9V1i#0#2(B</span>
<span lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Tate modules and level
structures}]\ \newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will describe some theorems regarding
Tate modules and how to<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>represent isogenies with the data of a
level structure.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#6(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#8(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US>, %</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#0#39V5A<<!'(B</span><span
lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Polarizations}]\ \newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will review the theory of polarizations,
and classify polarizations<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>of abelian varieties over finite fields.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[{\bf </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B#17n#2#3F|!J?e(B</span><span lang=EN-US>) </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B8aA0(B</span><span lang=EN-US>: </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#5#2!#39fEo#1#1#0!#0#79V5A<<(B</span><span
lang=EN-US>}]<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\ <o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#9(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#2#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#0(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#2#0(B</span><span
lang=EN-US>, </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#1#09V5A<<!'(B</span><span
lang=EN-US>%,\qquad 15:40 $\sim$ 17:00 </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B9V1i#0#2(B</span>
<span lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Shimura varieties}]\ \newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>After briefly reviewing the notion of a
stack, I will define the<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>Shimura varieties we will be studying.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#0(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#1(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#5#0(B</span><span
lang=EN-US>, </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#0#79V5A<<!'(B</span><span
lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Deformation theory}]\ \newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will discuss the deformation theory of
formal groups, p-divisible<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>groups, and mod p points of our Shimura
varieties.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[{\bf </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B#17n#2#3F|!J?e(B</span><span lang=EN-US>) </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$BCk(B</span><span lang=EN-US>: </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#5#2!#39fEo#1#0#39V5A<<(B</span><span
lang=EN-US>}]<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#2(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#3(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US>, (</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B0lHL3X@88~$1(B</span><span
lang=EN-US>)</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$BFCJL9V1i!'(B</span><span
lang=EN-US>%,\qquad 15:40 $\sim$ 17:00 </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B9V1i#0#2(B</span>
<span lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Wrapping spheres around
spheres}]\ \newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will discuss the topological problem of
classifying ways to wrap<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>$n$-dimensional spheres around
$k$-dimensional spheres. I will describe<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>the patterns that emerge in this
classification, and relate them to<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>some elementary number theory.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[{\bf </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B#17n#2#4F|!JLZ(B</span><span lang=EN-US>) </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B8a8e(B</span><span lang=EN-US>: </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#5#2!#39fEo#1#1#09V5A<<(B</span><span
lang=EN-US>}]<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#3(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#5(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US>, </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#1#09V5A<<!'(B</span><span
lang=EN-US>%,\qquad 15:40 $\sim$ 17:00 </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B9V1i#0#2(B</span>
<span lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Height $n$ locus}]\ \newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will describe a zero dimensional
subvariety of particular importance<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>to the nth chromatic layer in homotopy
theory.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#6(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#8(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US>, </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#1#09V5A<<!'(B</span><span
lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Topological automorphic forms}]\
\newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will describe a generalization of the
Hopkins-Miller theorem (due to<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>Jacob Lurie) and explain how the
deformation theory allows us to<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>construct the cohomology theory of
Topological Automorphic forms<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>(TAF).<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[{\bf </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B#17n#2#5F|!J6b(B</span><span lang=EN-US>) </span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B8a8e(B</span><span lang=EN-US>: </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#5#2!#39fEo#1#1#09V5A<<(B</span><span
lang=EN-US>}]<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#3(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#5(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0(B</span><span
lang=EN-US>, </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#1#09V5A<<!'(B</span><span
lang=EN-US>%,\qquad 15:40 $\sim$ 17:00 </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B9V1i#0#2(B</span>
<span lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{$Q$-spectrum}]\ \newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will show that the adelic points of a
unitary acts on TAF, the<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>homotopy fixed points of this action is a
spectrum $Q_U.$ This spectrum<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>$Q_U$ is the main object of interest.
I will prove<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{enumerate}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item the spectrum admits a resolution
given by the action of U on its building<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item the $K(n)$-localization of $Q_U$ is
an approximation to the $n$th<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>chromatic layer in homtopy theory.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{enumerate}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#6(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US> $\sim$ </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#8(B</span><span
lang=EN-US>:</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#0#0(B</span><span
lang=EN-US>, </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#1#1#09V5A<<!'(B</span><span
lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\underline{Examples}]\ \newline<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>I will describe some examples<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\begin{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[$n = 1:$] Class field theory and
K-theory<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[$n = 2:$] Basically reduces to
elliptic curve theory and TMF<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\item[$n = p-1:$] I will show that maximal
finite subgroups of the Morava<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>stabilizer group may be realized.<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\end{description}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\vspace{5mm}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>Behrens</span><span style='font-family:
"$B#M#S(B $B%4%7%C%/(B"'>$B$5$s$N$42wBz$rF@$F!$A49V1i$O(B</span><span lang=EN-US>Hard Disk</span><span
style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$BIU$-%S%G%*%+%a%i$K$F<}O?$5$l$^$9!%(B</span><span lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B$3$l$i$N%S%G%*%U%!%$%k$r$44uK>$NJ}$O!$==FsJ,$JMFNL$r;}$C$?(B</span><span
lang=EN-US>(</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#3#0#G(B</span><span
lang=EN-US>B</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B$"$k$3$H$,(B</span><span
lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$BK>$^$7$$(B</span><span
lang=EN-US>)</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B#P#C$b$7$/$O%]!<%?%V%k$N(B</span><span
lang=EN-US>Hard Disk</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B$r$4;};22<$5$$!%(B</span><span
lang=EN-US><o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US><o:p> </o:p></span></p>
<p class=MsoNormal><span lang=EN-US>\vspace{8mm} \hspace{40mm}<o:p></o:p></span></p>
<p class=MsoNormal><span lang=EN-US>{\bf </span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$BLd$$9g$o$;@h(B</span><span
lang=EN-US>}</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$B!'(B</span> <span
style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$BFn(B</span> <span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$BHOI'(B</span><span
lang=EN-US> (</span><span style='font-family:"$B#M#S(B $B%4%7%C%/(B"'>$BL>8E209)6HBg3X!&$*$b$RNN0h!K(B</span><span
lang=EN-US> nori@nitech.ac.jp<o:p></o:p></span></p>
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