Speaker: Li Han, Yale University Title: Q-forms of PGL_2 and the Jacquet-Langlands correspondence Venue: B1044 Time: 10:00 – 11:30, 28 June Abstract: The Jacquet-Langlands correspondence, proved in 1970, is an important result in the theory of automorphic forms. Here we roughly state it in the representation theoretic language. Let $G$ be an inner $Q$-form (in the sense of algebraic group) of $PGL_2$. Notice that the automorphic representation of $G$ has discrete spectrum since $G(A)/G(Q)$ is compact, where $A$, $Q$ denotes the Adeles and the rationals respectively. The Jacquet-Langlands correspondence asserts that, for every infinite dimensional irreducible subrepresentation $rho$ of the automorphic representation of $G$, there exists an irreducible subrepresentation of the automorphic representation of $PGL_2$, which for all but finitely many local components agree with $rho$. It turns out that the regular representations of $SO(2,1)$ on all the closed orbits in $SL_3(R)/SL_3(Z)$ can be understood via this theory. The aim of this lecture is to explain these results, and discuss its applications in homogeneous dynamics. .