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.[-]

In 1981, Drinfeld enumerated the number of irreducible $l$-adic local systems of rank two on a projective smooth curve fixed by the Frobenius endomorphism. Interestingly, this number looks like the number of points on a variety over a finite field. Deligne proposed conjectures to extend and comprehend Drinfeld's result. By the Langlands correspondence, it is equivalent to count certain cuspidal automorphic representations over a function field. In this talk, I will present some counting results where we connect counting to the number of stable Higgs bundles using Arthur's non-invariant trace formula.[-]

In this talk we shall discuss our recent results on the Adams' conjecture on theta correspondence. In more words, given a representation of a classical group (in our case, symplectic or even orthogonal) belonging to a local Arthur packet, Adams predicts that, under certain assumptions, its theta lift (i.e. a corresponding irreducible representation of the other group in a dual reductive pair), provided it is non-zero, is also in A-packet which can be easily described in terms of the original one. Mœglin gave some partial results, specifically, in case when the original representation is square-integrable. We are able to extend her results to the case of so called Arthur packets with the discrete diagonal restriction. Moreover, it seems that Arthur packet encapsulates lot of additional information even in relation to theta correspondence, e.g. we can easily read of from it the first occurrence index for the given representation in it. Adams conjecture takes an unexpectedly elegant form for the representations in discrete diagonal restriction packets. Also, we are able to pinpoint exactly how low in theta towers we can go with this description of the theta lifts which belong to Arthur packets, we can also address some other related conjectures due to Mœglin. This is joint work with Petar Baki.[-]

Motives represent hidden building blocks for both number theory and geometry. Automorphic representations are spectral objects with the analytic power to resolve some of the deepest questions in modern harmonic analysis. It has long been thought that there were fundamental relations between these very different sides of mathematics. We shall describe conjectures on the explicit nature of some of these relations, as expressed in terms of the automorphic and motivic Galois groups. If time permits, we shall comment on how these universal groups might extend to the broader theories of mixed motives and exponential motives.[-]