We identify all Anosov representations of compact hyperbolic triangle reflection groups into the higher rank Lie group $\mathrm{SL}(3, \mathbb{R})$. Specifically, we prove that such a representation is Anosov if and only if either it lies in the Hitchin component of the representation space, or it lies in the "Barbot component" and the product of the three generators of the triangle group has distinct real eigenvalues. Unlike representations in the Hitchin component, Anosov representations in the Barbot component have non-convex boundary maps.

Suppose that a countably $n$ -rectifiable set $\Gamma_0$ is the support of a multiplicity-one stationary varifold in $\mathbb{R}^{n+1}$ with a point admitting a flat tangent plane $T$ of density $Q \geq 2$ . We prove that, under a suitable assumption on the decay rate of the blow-ups of $\Gamma_0$ towards $T$ , there exists a non-constant, genuinely time-dependent Brakke flow starting with $\Gamma_0$ . The result, which applies, in particular, to a large class of (possibly stable) minimal immersions with branch singularities, shows non-uniqueness of Brakke flow under these conditions. Furthermore, it suggests that stationary varifolds which are dynamically stable, i.e. stable with respect to mean curvature flow, may be free from flat singularities.

An influential result of McDuff and Schlenk asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase. This work has led to considerable interest in understanding when the ellipsoid embedding function for other symplectic fourmanifolds is partly described by an infinite staircase. We provide a general framework for analyzing this question for a large family of targets, called finite type convex toric domains, which we prove generalizes the class of closed toric symplectic fourmanifolds. When the target is of finite type, we prove that any infinite staircase must have a unique accumulation point $a_0$ , given as the solution to an explicit quadratic equation. Moreover, we prove that the embedding function at $a_0$ must be equal to the classical volume lower bound. In particular, our result gives an obstruction to the existence of infinite staircases that experimentally seems strong. Our methods apply equally well to any closed manifold which is a blowup of the complex projective plane. In the special case of rational convex toric domains, we can say more. We conjecture a complete answer to the question of existence of infinite staircases, in terms of six families that are distinguished by the fact that their moment polygon is reflexive. We then provide a uniform proof of the existence of infinite staircases for our six families, using two tools. For the first, we use recursive families of almost toric fibrations to find symplectic embeddings into closed symplectic manifolds. In order to establish the embeddings for convex toric domains, we prove a result of potentially independent interest: a four-dimensional ellipsoid embeds into a closed toric symplectic four-manifold if and only if it can be embedded into a corresponding convex toric domain. For the second tool, we find recursive families of convex lattice paths that provide obstructions to embeddings. We conclude by connecting our conjecture that these are the only infinite staircases among rational convex toric domains to a question in number theory related to a classic work of Hardy and Littlewood.

We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class. We employ the notion of eigenvalues of measures and show that if a normalized eigenvalue is close to its maximal value, the corresponding measure must be close in the Sobolev space $W^{-1,2}$ to the set of maximizing measures. In particular, this implies a qualitative stability result: metrics almost maximizing the normalized eigenvalue must be $W^{-1,2}$ -close to a maximal metric. Following this approach, we prove sharp quantitative stability of the celebrated Hersch's inequality for the first eigenvalue on the sphere, as well as of its counterpart for the second eigenvalue. Similar results are also obtained for the precise isoperimetric eigenvalue inequalities on the projective plane, torus, and Klein bottle. The square of the $W^{-1,2}$ distance to a maximizing measure in these stability estimates is controlled by the difference between the normalized eigenvalue and its maximal value, indicating that the maxima are in a sense nondegenerate. We construct examples showing that the power of the distance can not be improved, and that the choice of the Sobolev space $W^{-1,2}$ is optimal.

We exhibit the first set of examples of non-bumpy metrics on the ( $n+1$ )-sphere $(2 \leq n \leq 6)$ in which the varifold associated with the two-parameter min-max construction must be a multiplicity-two minimal $n$ -sphere. This is proved by a new area-and-separation estimate for certain minimal hypersurfaces with Morse index two inspired by an early work of Colding-Minicozzi. We also construct non-bumpy projective spaces in which the first min-max hypersurfaces are one-sided, and non-bumpy balls in which the free boundary min-max hypersurfaces are improper.

This paper gives the first description of derived monodromy on the stringy Kähler moduli space (SKMS) for a general irreducible flopping curve C in a 3-fold $X$ with mild singularities. We do this by constructing two new infinite helices: the first consists of sheaves supported on C , and the second comprises vector bundles in a tubular neighbourhood. We prove that these helices determine the simples and projectives in iterated tilts of the category of perverse sheaves, and that all objects in the first helix induce a twist autoequivalence for $X$ . We show that these new derived symmetries, along with established ones, induce the full monodromy on the SKMS. The helices have many further applications. We (1) prove representability of noncommutative deformations of the sheaves $\mathcal{O}_{\mathrm{C}}, \ldots, \mathcal{O}_{\ell \mathrm{C}}$ associated to a length $\ell$ flopping curve, via tilting theory, (2) control the representing objects, characterise when they are not commutative, and their central quotients, and (3) give new and sharp theoretical lower bounds on Gopakumar-Vafa invariants for a curve of length $\ell$ . When $X$ is smooth and resolves an affine base, we furthermore (4) prove that the second helix classifies all tilting reflexive sheaves on $X$ , and thus that (5) all noncommutative crepant resolutions arise from tilting bundles on $X$ .

Let (X, p) be a Ricci limit space. We show that for any ǫ > 0 and x ∈ X, there exists r < ǫ, depending on ǫ and x, so that any loop in Br(x) is contractible in Bǫ(x). In particular, X is semi-locally simply connected. Then we show that the generalized Margulis lemma holds for Ricci limit spaces of n-manifolds.

In this article, we show that there exist discrete isometry groups of the 2-and 3-dimensional complex hyperbolic spaces with critical exponents arbitrarily close to but strictly smaller than the maximum possible value. This result shows no gap in the values of critical exponents for complex hyperbolic Kleinian groups.

We establish a positive characteristic analogue of intersection cohomology theory for variations of Hodge structure. It includes: a) the de Rham-Higgs comparison theorem for the intersection de Rham complex; b) the E1-degeneration theorem for the intersection de Rham complex of a periodic de Rham bundle; c) the Kodaira-Saito vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle. These results generalize the decomposition theorem of Deligne-Illusie [DI] and the de Rham-Higgs theorem of Ogus-Vologodsky [OV], the E1-degneration theorem of Deligne-Illusie [DI], Illusie [I90], Faltings [Fa] and the Kodaira-Saito vanishing theorem of Arapura [Ar]. As an application, we give an algebraic proof of the E1-degeneration theorem due to Cattani-Kaplan-Schmid [CKS] and Kashiwara-Kawai [KK], and the vanishing theorem of Saito [Sa] for VHSs of geometric origin. Contents 1. Introduction 2. Intersection λ-complexes 2.1. The construction 2.2. Base change and cohomology 3. De Rham-Higgs comparison theorem 3.1. Homotopy on complexes 3.2. An infinity homotopy for nilpotent Higgs modules 3.3. Cartier isomorphism 4. E 1 -degeneration theorem 4.1. Periodic de Rham bundles 4.2. Intersection adaptedness theorem 4.3. E 1 -degeneration and vanishing theorem 5. Applications 6. Appendix 6.1. Higgs-de Rham ring 6.2. An initial value problem and its solution 6.3. The ∞-homotopy References Bibliography