We give a modification of Yu’s construction of supercuspidal representations of a connected reductive group G over a non-Archimedean local field F. This modification restores the validity of certain key intertwining property claims made by Yu, which were recently proved to be false for the original construction. This modification is also an essential ingredient in the construction of supercuspidal L-packets in a preprint by the second author. As further applications, we prove the stability and many instances of endoscopic character identities of these supercuspidal L-packets, subject to some conditions on the base field F. In particular, for regular supercuspidal parameters, we prove all instances of standard endoscopy. In addition, we prove that these supercuspidal L-packets satisfy a recent conjecture by the second author, which, together with standard endoscopy, uniquely characterizes the local Langlands correspondence for supercuspidal L-packets (again subject to the conditions on F). These results are based on a statement of the Harish-Chandra character formula for the supercuspidal representations arising from the twisted Yu construction.

We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence ${\mathrm{\mu }}^2\equiv Dmodm$, with $D>0$ square-free and $D\not\equiv 1mod4$. This is achieved by translating the problem to convergence of certain geodesic random line processes in the hyperbolic plane. This geometric interpretation allows us in particular to derive an explicit expression for the pair correlation density of the roots.

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains a ${\mathbb{Z}}^{\mathrm{\infty }}$-subgroup and a $\mathbb{Z}$-summand. Our proof proceeds by introducing an algebraic variant of the involutive Heegaard Floer package of Hendricks, Manolescu, and Zemke. This is inspired by an analogous argument in the setting of knot concordance due to the second author.

We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality depending on the analytic rank. Moreover, we show that a plausible improvement of our field cardinality requirement would imply that the ranks are equal up to $1+o(1)$ in the exponent over every finite field. At the core of the proof is a technique for lifting decompositions of multilinear polynomials in an open subset of an algebraic variety, and a technique for finding a large subvariety that retains all rational points such that at least one of these points satisfies a finite-field analogue of genericity with respect to it. Proving the equivalence between these two ranks, ideally over fixed finite fields, is a central question in additive combinatorics, and was reiterated by multiple authors. As a corollary, we prove—allowing the field to depend on the value of the norm—the polynomial Gowers inverse conjecture in the d versus $d-1$ case.