Geometric analysis seminar

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Abstract: In their seminal work in 1994, Bershadsky-Cecotti-Ooguri-Vafa introduced a particular Ray-Singer analytic torsion of Calabi-Yau manifolds which coincides with the genus one topological string partition function. They also proved a holomorphic anomaly formula for this torsion which is related to the variation of Hodge structure and the Weil-Petersson geometry of deformation spaces. In this joint work with Shu Shen and Jianqing Yu, we consider the similar object in Landau-Ginzburg models. We prove an index theorem for the associated Dirac operator and rigorously define the BCOV torsion. We also obtain a partial result towards proving a holomorphic anomaly formula.

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Abstract: Recent progress in understanding spatially closed, vacuum spacetimes with Killing fields has been characterized by instabilities. This is in contrast to the open case where, for example, Minkowski space is known to be stable. With Profs. Berger and Isenberg, we prove several results related to the stability of such solutions, expanding on previous work of LeFloch and Smulevici. We also present recent numerical work exploring the space of \( T^2 \)-symmetric spacetimes which are not covered by our theorems. We will give a picture of these instabilities, as well as potential attractors far from most previously identified classes.

Abstract:TBA This is work joint with D. Wraith. We use harmonic maps to study the map of moduli spaces of Ricci-non-negative metric on a Riemannian manifold \( M \) to the one on the product \( M\times S^1 \). In particular, we show that such moduli space on \( M\times T^p \) has infinite number of path components for some particular manifolds \( M \).

Abstract: Given a singular Hermitian-Yang-Mills connection A and assuming certain conditions, we will characterize the analytic tangent cones of A at an isolated singularity p by using the local algebraic geometric data near p given by the holomorphic bundle E determined by A. The uniqueness is a consequence. (Joint with Song Sun.)

Abstract: In this talk, by adding a technical assumption, we give an affirmative answer to Donaldsons tamed to compatible question.

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Abstract: I will report on joint work with I. Adeboye and Guofang Wei on computating explicit uniform lower bounds for the volumes of orbifold quotients of irreducible symmetric spaces (without compact factors) via differential geometric methods. This extends previous work by Adeboye and Wei for real and complex hyperbolic spaces, and work of others for quaternionic hyperbolic spaces.

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Abstract: A gravitational instanton is a noncompact complete Calabi-Yau surface with faster than quadratic curvature decay. In this talk, I will discuss the classification of gravitational instantons. This is a joint work with Xiuxiong Chen.

Abstract: In joint work with Catherine Searle we classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism. I will describe the classification result and give an outline of the proof.

Abstract: Recently Chen and Cheng have made surprising breakthrough on existence of Kahler metrics of constant scalar curvature, where they are able to solve the long standing problem of Calabi-Donaldson program. Technically they solved a fourth order nonlinear elliptic equation of scalar curvature type, assuming necessary properness condition; the key is to derive a priori estimates of scalar curvature type equations (fourth order on Kahler manifolds). We will report their achievements and extend their results to Calabi’s extremal metric.

Abstract: Einstein’s equations can be considered as a particular geometric evolution system where the underlying PDEs are essentially of nonlinear wave equation type. Similar to other famous geometric evolution problems, there has been a lot of effort over the last decades to study the formation of singularities of solutions. In general this turns out to be a formidable (and essentially unsolved) task due to the complexity of Einstein’s equations. In this talk I will discuss a recent result by P. LeFloch and myself (based on joint work with J. Isenberg and E. Ames) regarding singular solutions of the coupled Einstein-Euler equations. I will also explain the underlying notion of a "singular initial value problem".

Abstract: In this talk, we calculate the dimension of the J-anti-invariant cohomology subgroup on torus T^4. Inspired by this concrete example, we get that: On a closed symplectic 4-dimensional manifold (M,\omega),dimension of the J-anti-invariant cohomology subgroup vanishes for generic \omega-compatible almost complex structures.

Abstract: Ancient solutions to the Ricci flow are very important for the understanding of singularity formation, and strongly utilized in Hamilton-Perelman`s proof of geometrization conjecture. In this talk, I will discuss my proof of the classification of three-dimensional Type I noncollapsed ancient solutions to the Ricci flow. If there were time, I will also talk about a similar rigidity result in dimension four.

Abstract: We consider the Hamiltonian stationary equation for all phases in dimension two. We show that solutions that are C^{1,1} will be smooth and we also derive a C^{2,\alpha} estimate for it.

Absract: We extend the Atiyah, Patodi, and Singer index theorem for Dirac operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. The index is expressed in terms of a new periodic eta-invariant that equals the classical eta-invariant in the cylindrical setting. We use this theorem to study Riemannian metrics of positive scalar curvature on some even-dimensional manifolfds, as well as the Seiberg-Witten invariants of 4-manifolds with integral homology of \(S^1 \times S^3\) This is a joint work with Jianfeng Lin, Tom Mrowka, and Daniel Ruberman. Abstract: TBA

Abstract: Nash`s classical bargaining solution suggests that n players should maximize a product of utility functions. We consider a special case: Suppose that the players are chosen from a continuum, distribution \( \mu \) and suppose they are to divide up a resource \( \nu \) that is also on a continuum, whose utility to each player is the inverse of exponential of a cost function. The maximization problem becomes an optimal type transport problem, where the target density is the minimizer to the functional \( F(\beta) = H_{\nu}(\beta) + W^2(\mu,\beta) \) where \( H \) is the entropy and W is the Wasserstein distance. One may recognize this problem from a single time step in the Jordan-Kinderlehrer-Otto scheme. Thanks to optimal transport theory, the solution may be described as by a potential that solves a fourth order nonlinear elliptic PDE.

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Abstract: A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local \( L^{\infty} \)-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as\( \epsilon\) -regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \( \epsilon\) -regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant.

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Title:Tangent cones of Yang-Mills connections with isolated singularities.

Abstract: Originally developed in the study of minimal surfaces Simon and Almgren, tangent cones have proved useful in applications to many geometric equations, including Yang-Mills connections. In this talk I will discuss how to uniquely identify the tangent cone of a Yang-Mills connection with isolated singularity in the complex setting, with an assumption on the complex structure of the bundle. This is joint work with H. Sa Earp and T. Walpuski.

Title:Affine symmetric spaces.

Title:Constant scalar curvature metric and its weak solution on compact Kahler manifolds.

AAbstract: We define a notion of weak solution for constant scalar curvature equation on Kahler manifolds for metrics with only bounded coefficients. We develop some linear theory on compact Kahler manifolds for elliptic equations with only bounded coefficients. Using the linear theory, we prove a weak solution is smooth. This in part confirms a conjecture of X.X. Chen, regarding the regularity of minimizers of K-energy. This is joint work with Yu Zeng from University of Rochester

Title:Radial solutions of a fourth order Hamiltonian stationary equation

We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded domains; moreover, near the origin, the gradient graph of such a solution is continuous if and only if the graph is special Lagrangian. Joint with Jingyi Chen.

Title:Gluing manifolds with boundary and bordisms of positive scalar curvature metrics

This thesis presents two main results on geometric and topological aspects of scalar curvature. The first is a gluing theorem for scalar-flat manifolds with vanishing mean curvature on the boundary. Our methods involve tools from conformal geometry and perturbation techniques for nonlinear elliptic PDE. The second part studies bordisms of positive scalar curvature metrics. We present a modification of the Schoen-Yau minimal hypersurface technique to manifolds with boundary which allows us to prove a hereditary property for bordisms of positive scalar curvature metrics. The main technical result is a convergence theorem for stable minimal hypersurfaces with free boundary in bordisms with long collars which may be of independent interest.

Abstract :We consider a C^{2,α} solution (u) of a fourth order non linear elliptic PDE. We show that the solution will be smooth if the PDE is regular. We define a regular PDE as a PDE whose coefficient matrix, a smooth function of the hessian of u, along with it’s derivative w.r.t the hessian satisfy the Legendre Hadamard (L.H) condition of ellipticity. This result can be applied to C^{2,α} minimizers of functionals defined on the hessian space satisfying certain convexity conditions, therefore showing that the minimizers are in fact smooth functions.

Title:The extension problem of the mean curvature flow

Abstract: We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in \( \mathbb{R}^3 \). This is a joint work with H.Z. Li.

Title: Regularity of weak solutions of scalar curvature equation.

We prove some regularity result for weak solutions of scalar curvature equation. This is a joint work with Yu Zeng

This is joint work with David Wraith. We study the space \( \Riem^{\Ric+}(M) \) of metrics with positive Ricci curvature on a closed spin manifold \(M\)of \( \dim M=d\). There is a natural map \(\iota: \Riem^{\Ric+}(M)\to \Riem^+(M) \)$ to the space of metrics with positive scalar curvature. Let \(g_0\in \Riem^+(M)\) be any metric, then there is the index-difference map \( \mathrm{inddiff}_{g_0}: \Riem^+(M)\to \Omega^{\infty+d+1}KO\) defined by Hitchin. Recently it was established by Botvinnik, Ebert and Randal-Williams that the index-difference map \(\mathrm{inddiff}_{g_0}\) detects non-trivial homotopy groups \(\pi_q\Riem^+(M) \) for all $q \) such that \(KO_{d+q+1}\neq 0 \), where \(d\geq 6\). We show that for any \(\ell\geq 1\) and even integer \(d\geq 6\), there exists a spin manifold \(W\), \(\dim W = d\), together with a metric \(g_0\in \Riem^{\Ric+}(W)\), such that the composition \(\mathsf{inddiff}_{g_0}: \Riem^{\Ric+}(W)\xrightarrow[ ]{\iota} \Riem^{+}(W) \xrightarrow[ ]{\mathrm{inddiff}_{g_0}} \Omega^{n+1} KO\) detects non-trivial homotopy groups \(\pi_q \Riem^{\Ric+}(W)\) for all\(q\leq \ell\) and such that \(KO_{d+q+1}\neq 0\).

Title: Compactness, finiteness properties of Lagrangian self-shrinkers in $\mathbb R^4$ and Piecewise mean curvature flow.

Abstract: In this talk, we discuss a compactness result on the space of compact Lagrangian self-shrinkers in $\mathbb R^4$. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a {\L}ojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in $\mathbb R^4$, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in $\mathbb R^3$ by Colding and Minicozzi. This is a joint work with Jingyi Chen.

“Neckpinches and Caps in Geometric Heat Flows"

Title: Increased Regularity for Hamiltonian Stationary submanifolds

Abstract: A Hamiltonian Stationary submanifold of complex space is a Lagrangian manifold whose volume is stationary under Hamiltonian variations. We consider gradient graphs \( (x,Du(x)) \) for a function \( u \). For a smooth \( u \), the Euler-Lagrange equation can be expressed as a fourth order nonlinear equation in \( u \) that can be locally linearized (using a change of tangent plane) to the bi-Laplace. The volume can be defined for lower regularity, however, and computing the Euler-Lagrange equation with less assumed regularity gives a "double divergence" equation of second order quantities. We show several results. First, there is a \( c_n \) so that if the Hessian \( D^2u \) is \(c_n\)close to a continuous matrix-valued function, then the potential must be smooth. Previously, Schoen and Wolfson showed that when the potential was \(C^{2,\alpha}\), then the potential \( u \)must be smooth. We are also able to show full regularity when the Hessian is bounded within certain ranges. This allows us to rule out conical solutions with mild singularities.

Title: Geodesics on locally homogeneous affine surfaces

Abstract: We examine questions of geodesic completeness in the context of locally homogeneous affine surfaces. Any locally homogeneous surface has a local model \( \mathcal{M} \) where either the Christoffel symbols take the form \(\Gamma_{ij}{}^k\) are constant and the underlying space is \(\mathbb{R}^2\) (Type-A) or the Christoffel symbols take the form \(\Gamma_{ij}{}^k=C_{ij}{}^k/x^1\) where the underlying space is \(\mathbb{R}^+\times\mathbb{R}\) (Type-B). The model space \(\mathcal{M}\) is said to be ESSENTIALLY GEODESICALLY COMPLETE if there does not exist a complete locally homogeneous surface modeled on \( \mathcal{M}\) which is geodesically complete. Up to linear equivalence, there are exactly 3 models which are geodesically incomplete but not essentially geodesically incomplete. We classify all the geodesically complete models of Type-A and present some partial results concerning Type-B models. This is joint work in progress with E. Puffini (Krill Institute, Islas Malvinas) and with Daniela Dascanio and Pablo Pisani (Universidad Nacional de La Plata, Argentina)

Title: Minimal hypersurfaces with free boundary and psc-bordism

Abstract: There is a well-known technique due to Schoen-Yau from the late 70s which uses (stable) minimal hypersurfaces to study the topological implications of a (closed) manifold`s ability to support positive scalar curvature metrics. In this talk, we describe a version of this technique for manifolds with boundary and discuss how it can be used to study bordisms of positive scalar curvature metrics.

Title: Perelman`s construction for gluing manifolds with positive Ricci curvature

Title: Kähler-Einstein metrics and higher alpha-invariants

Abstract: I will describe a condition on the Bergman metrics of a Fano manifold M, which guarantees the existence of a Kähler-Einstein metric on M. I will also discuss a conjectural relationship between this condition and \( M`s \) higher alpha-invariants \( \alpha_{m,k}(M) \), analogous to a 1991 theorem of Tian for \( \alpha_{m,2}(M). \)

Title:The Calabi flow with rough initial data.

Abstract: We prove the short time existence of the Calabi flow for continuous initial metric. This is the joint work with Yu Zeng.

Title: Solitons for the inverse mean curvature flow

Abstract: In this expository talk, we will describe some rigidity results for area-minimizing surfaces in 3-manifolds. For instance, Cai-Galloway (2000) show that a locally area-minimizing (2-sided) torus in a scalar-nonnegative 3-manifold is flat and in fact the ambient manifold is flat near the torus. I will highlight the more recent (2013) result of I. Nunes for hyperbolic surfaces which has a connection to Escobar`s Yamabe problem for manifolds with boundary.

Abstract: The classification of Riemannian manifolds with positive and non-negative sectional curvature is a long-standing problem in Riemannian geometry. In this talk I will summarize recent joint work with Catherine Searle on the classification of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, maximal torus action. This classification has many applications, in particular the Maximal Symmetry Rank conjecture holds for this class of manifolds.

This is joint work with E. Puffini, J.H. Park, E. Garcia-Rio, and M. Brozos-Vazquez.

Abstract: Study of the dynamic properties of the Einstein Field Equations has recently been dominated by local stability results: the stability of Minkowski space and the linear stability of Kerr. One may on the other hand study global stability questions, although this has only been possible under symmetry assumptions. I will survey such recent results and some current work with Jim Isenberg and Beverly Berger on future stable solutions of the Einstein Flow.

Abstract: On a closed K\"ahler manifold, people have been searching for canonical representative in each K\"ahler class. In 80`s, Calabi has proposed to look for the constant scalar curvature K\"ahler(cscK) metric, or more generally the extremal metric, the critical point of Calabi energy. The cscK metric satisfies a fourth order nonlinear PDE. Recently, X. Chen proposed a new continuity path which connects the cscK metric with the critical point of J-functional. In this talk, I will present various openness results about this path. Mostly, I will describe how to deform from the critical point of J-functional(second order) to a twisted cscK metric(fourth order).

abstract: We consider the convergence of Kahler metrics (in the same Kahler class) under various conditions. Compared with the general Reimannian setting, the convergence results of Kahler metrics exhibit many special properties with interesting applications.

Abstract: Compact 6-dimensional nearly Kähler manifolds are the cross-sections of Riemannian cones with G2 holonomy. Viewing Euclidean 7-space as the cone over the round 6-sphere endows the latter with a nearly Kähler structure which coincides with the standard G2-invariant almost complex structure induced by octonionic multiplication. A long-standing problem has been the question of existence of complete nearly Kähler 6-manifolds besides the four known homogeneous ones. We resolve this problem by proving the existence of an exotic (inhomogeneous) nearly Kähler structure on the 6-sphere and on the product of two 3-spheres. This is joint work with Mark Haskins, Imperial College London.

Abstract: For a positive measure set of Klein-Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein-Klein-Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein-Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein-Klein-Gordon equations. This is joint work with Otis Chodosh.

Abstract: We study the asymptotic behaviors of solutions of the Loewner-Nirenberg problem in singular domains and prove that the solutions are well approximated by the corresponding solutions in tangent cones at singular points on the boundary. The conformal structure of the underlying equation plays an essential role in the derivation of the optimal estimates.

A few years ago, Tony Ache and I started on a project to recover the Ricci curvature on a submanifold from a sample of points and their extrinsic distances. This led us to study the notion of Coarse Ricci curvature, which is a function on pairs of points. Using the Carre du Champ of Bakry-Emery, we followed this to a somewhat satisfying notion of Coarse Ricci curvature that can be defined on a large class of spaces. Unfortunately, this turns out to be something that does not work when considering an extrinsic distance function. Instead, we consider approximate Ricci curvature, which is really just an approximation of the classical Ricci tensor using Bakry-Emery, applied to the approximate tangent spaces that are obtained using PCA. We will discuss the pros and cons of both approaches.

title: Mean curvature flow of an entire graph evolving away from the heat flow

abstract: We present an initial graph over the entire plane for which the mean curvature flow behaves different from the heat flow. This is a joint work with Xuan Hien Nguyen.

Abstract: In this talk, I will discuss the rigidity problem of \( \kappa \)-noncollapsing steady solitons. First , we show that any \( \kappa \)-noncollapsing steady Kaehler-Ricc soliton with nonnegative sectional curvature must be falt. Secondly, we prove that any \( \kappa \)-noncollapsing steady Ricc soliton with nonnegative curvature operator and horizontally \( \epsilon \)-pinched Ricci curvature should be rotationally symmetric.

Abstract: In 1867, Riemann discovered a family of complete, embedded, singly periodic minimal surfaces (in the three dimensional Euclidean space) foliated by circles and lines. He also proved that his staircases, planes, catenoids, and helicoids are the only minimal surfaces fibered by circles or lines in parallel planes. We explicitly construct generalized helicoids in odd dimensional Euclidean space, and minimal cones in even dimensional Euclidean space. Our minimal varieties unify various interesting examples: helicoids foliated by straight lines, Choe-Hoppe’s minimal hypersurfaces foliated by Clifford cones, Barbosa-Dajczer-Jorge’s ruled minimal submanifolds, and Harvey-Lawson’s twisted normal cone over Clifford torus. This is joint work with Eunjoo Lee.

Abstract: The Calabi conjecture, proved by Yau in 1976, says that the complex Monge-Ampere equation on a compact Kahler manifold is always solvable, and this can be interpreted geometrically as constructing Kahler metrics with prescribed volume form (or equivalently prescribed Ricci curvature). In recent years there has been much interest in extending this theory to general compact Hermitian manifolds, which may not admit any Kahler metric. I will describe several results in this direction (joint with B. Weinkove) as well as a recent solution of a conjecture of Gauduchon (joint with G. Szekelyhidi and B. Weinkove).

This is joint work with M. Bozos-Vazquez, E. Garcia-Rio, and E. Puffini.

Abstract: Suppose that \( L \) is a Hamiltonian stationary Lagrangian submanifold submanifold of complex space, that is, a manifold which is a critical point for volume under Hamiltonian variations. Suppose that \( L \) can be locally represented by the graph of a \(C^{1,\alpha} \) function over its tangent plane on a ball of radius \(r\). Then then manifold enjoys a priori interior derivative estimates of all orders based on \(r\) and the \(C^{1,\alpha} \) bound. While the above result follows almost immediately by Schauder theory, we prove a stronger result that holds for any modulus of continuity on the first derivatives of the graph.

Abstract: We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path-connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using results of Maximo, Nunes, and Smith (2013), we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary. This is joint work with Antonio Ache (Princeton) and Davi Maximo (Stanford).

Abstract: The skew mean curvature flow(SMCF) or binormal flow, which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. In this talk, I will show the basic properties of the SMCF and prove the existence of a short-time solution to the SMCF of surfaces in Euclidean space R^4 . If time permits, I will also talk about a moving frame method which transforms the SMCF to a non-linear Schrödinger system.

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P. Gilkey, C. Y. Kim, J. H. Park, and E. Puffini.

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Abstract: Given two Yamabe-null manifolds with boundary (i.e. scalar and boundary mean curvatures vanish) our goal is to glue them along a common submanifold \(K \), producing a Yamabe-null metric on the resulting manifold under suitable geometric conditions. Our construction is based on L. Mazzieri`s work on gluing solutions to the classical (closed manifold) Yamabe problem. When $K$ intersects the boundaries of the original manifolds, complications occur and some new estimates are needed to carry out the construction. We will focus on this case in the talk.

Abstract: In this talk, I will discuss global smoothness of the eigenfunctions of the Monge-Ampere operator on smooth, bounded and uniformly convex domains in all dimensions. A key ingredient in our analysis is boundary Schauder estimates for certain degenerate Monge-Ampere equations. This is joint work with Ovidiu Savin.

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Abstract: : A one-parameter family of hypersurfaces in Euclidean space evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector. It can be viewed as a geometric heat equation, i.e., it is locally moving in the direction of steepest descent for the volume element, deforming surfaces towards optimal ones (minimal surfaces). In this talk we will discuss some recent work on the local curvature estimate and convexity estimate for the star-shaped mean curvature flow and the consequences. In particular, star-shaped MCF is generic in the sense of Colding-Minicozzi. This is joint work with Robert Haslhofer.

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Abstract: It is a basic fact that the Riemannian curvature becomes unbounded at every finite-time singularity of the Ricci flow. Sesum showed that, more precisely, even the Ricci curvature becomes unbounded at every such singularity. Whether the same can be said about the scalar curvature has since remained a conjecture, which has resisted several attempts of resolution. In this talk, I will present a new result that partially confirms this conjecture in dimension 4 and motivates some interesting questions in 4 dimensional Ricci flow. Its proof relies on a combination of multi-scale arguments and Perelman`s Harnack inequality on the conjugate heat equation. a byproduct, we obtain an unconventional backwards pseudolocality theorem, which holds in any dimension. This project is joint work with Qi Zhang.

We show that if a figure 8 bounds convex symmetric regions, then the curve shortening flow shrinks the curve to a point, whose blow-up is a double line. This is an initial step in solving a conjecture of Grayson. The is work in progress with Gregory Drugan and Weiyong He.