I will discuss a proof of the existence of infinitely many solutions for the singular Yamabe problem in spheres using bifurcation theory and the spectral theory of hyperbolic surfaces.
The Langlands and Fontaine–Mazur conjectures in number theory describe when an automorphic representation f arises geometrically, meaning that there is a smooth projective variety X, or more generally a Chow motive M in the cohomology of X, such that there is an equality of L-functions L(M,s) = L(f,s). We explicitly describe how to produce such a variety X and Chow motive M in the case of powers of certain automorphic representations, called algebraic Hecke characters. This is joint work with J. Lang.
My talk will aim to be a friendly introduction for condensed matter friends, mathematicians, and QFT theorists alike --- I shall quickly review and warm up the use of higher symmetries and anomalies of gauge theories and condensed matter systems. Then I will present the results of recent work [arXiv:1904.00994].
Deligne's "Weil II" paper includes a far-reaching conjecture to the
effect that for a smooth variety on a finite field of characteristic p,
for any prime l distinct from p, l-adic representations of the etale
fundamental group do not occur in isolation: they always exist in
compatible families that vary across l, including a somewhat more
mysterious counterpart for l=p (the "petit camarade cristallin"). We
explain in more detail what this all means, indicate some key