Using the conjugation symmetry on Heegaard Floer complexes, we define a $3$-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to ${\mathbb{Z}}_4$-equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, $\underline{d}$ and $\overline{d}$, and two invariants of smooth knot concordance, ${\underline{V}}_0$ and ${\overline{V}}_0$. We also develop a formula for the involutive Heegaard Floer homology of large integral surgeries on knots. We give explicit calculations in the case of L-space knots and thin knots. In particular, we show that ${\underline{V}}_0$ detects the nonsliceness of the figure-eight knot. Other applications include constraints on which large surgeries on alternating knots can be homology-cobordant to other large surgeries on alternating knots.

The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint $\gamma$-factor of its $L$-parameter. In this article, we prove the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case, which, combined with Arthur’s work on the local Langlands correspondence, implies the conjecture in the nongeneric case.

In this article we explore the connection between asymptotic base loci and Newton–Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplices contained in Newton–Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.

This article studies the geometry of proper open convex domains in the projective space $\mathbb{R}{\mathbf{P}}^n$. These domains carry several projective invariant distances, among which are the Hilbert distance $d^H$ and the Blaschke distance $d^B$. We prove a thin inequality between those distances: for any two points $x$ and $y$ in such a domain,

$$d^B(x,y)<d^H(x,y)+1.$$

We then give two interesting consequences. The first one answers a conjecture of Colbois and Verovic on the volume entropy of Hilbert geometries: for any proper open convex domain in $\mathbb{R}{\mathbf{P}}^n$, the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$. The second consequence is the following fact: for any Hitchin representation $\rho$ of a surface group $\Gamma$ into $PSL(3,\mathbb{R})$, there exists a Fuchsian representation $j:\Gamma \to PSL(2,\mathbb{R})$ such that the length spectrum of $j$ is uniformly smaller than that of $\rho$. This answers positively a conjecture of Lee and Zhang in the $3$-dimensional case.