A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should not be congruent numbers. Recently, Alex Smith has proved that at least 41.9% of $n=1,2,3$ satisfy (refined) BSD in rank $0$, and at least 55.9% of $n=5,6,7$ mod $8$ satisfy (weak) BSD in rank $1$. This implies in particular that at last 41.9% of $n=1,2,3$ mod $8$ are not congruent numbers, and 55.9% of $n=5, 6, 7$ mod $8$ are congruent numbers. I will explain the ingredients used in Smith's proof: including the classical work of Heath-Brown and Monsky on the distribution F_2 rank of Selmer group of E_n, the complex formula for central value and derivative of L-fucntions of Waldspurger and Gross-Zagier and their extension by Yuan-Zhang-Zhang, and their mod 2 version by Tian-Yuan-Zhang.[-]

The local Gan-Gross-Prasad conjectures concern certain branching or restriction problems between representations of real or p-adic Lie groups. In its simplest form it predicts certain multiplicity-one results for "extended" L-packets. In a recent series of papers, Waldspurger has settled the conjecture for special orthogonal groups over p-adic field. In this talk, I will present a proof of the conjecture for unitary groups which has the advantage of working equally well over archimedean and non-archimedean fields.[-]

Beyond endoscopy is the strategy put forward by Langlands for applying the trace formula to the general principle of functoriality. Subsequent papers by Langlands (one in collaboration with Frenkel and Ngo), together with more recent papers by Altug, have refined the strategy. They all emphasize the importance of understanding the elliptic terms on the geometric side of the trace formula. We shall discuss the general strategy, and how it pertains to these terms.[-]

Let G be a connected reductive p-adic group that splits over a tamely ramified extension. Let H be the fixed points of an involution of G. An irreducible smooth H-distinguished representation of G is H-relatively supercuspidal if its relative matrix coefficients are compactly supported modulo H Z(G). (Here, Z(G) is the centre of G.) We will describe some relatively supercuspidal representations whose cuspidal supports belong to the supercuspidals constructed by J.K. Yu.[-]