The talks are held on every Friday from 3:00 to 4:00 PM in South Hall 6635 (unless otherwise noted).
Friday April 5: Lawrence Mouille (UC Riverside)
Maximal symmetry-rank for positive intermediate Ricci curvature
The Grove-Searle maximal symmetry-rank theorem provides an upper bound for the ranks of the isometry groups of positively curved manifolds. In this talk, I will present a generalized bound for manifolds which have positive intermediate Ricci curvature, a condition that interpolates between having positive sectional curvature and having positive Ricci curvature. The argument for this generalized bound is local in nature, and I will show that metrics that achieve the upper bound in the local sense are dense in the $C^1$-topology.
Friday April 12: Sebastian Goette (University of Freiburg)
Extra twisted connected sums and their nu invariants
Joyce’s orbifold construction and the twisted connected sums by Kovalev and Corti-Haskins-Nordström-Pacini provide many examples of compact Riemannian 7-manifolds with holonomy G_2 (i.e., G_2-manifolds). We would like to use this wealth of examples to guess further properties of G_2-manifolds and to find obstructions against holonomy G_2, taking into account the underlying topological G_2-structures.
The Crowley-Nordström v-invariant distinguishes topological G_2-structures. It vanishes for all twisted connected sums. By adding an extra twist to this construction, we show that the v-invariant can assume all of its 48 possible values. This shows that G_2-bordism presents no obstruction against holonomy G_2. We also exhibit examples of 7-manifolds with disconnected G_2-moduli space. Our computation of the v-invariants involves integration of the Bismut-Cheeger η-forms for families of tori, which can be done either by elementary hyperbolic geometry, or using modular properties of the Dedekind η-function.
Friday April 19: Jiayin Pan (UCSB)
Semi-local simple connectedness of non-collapsing Ricci limit spaces
We prove that any non-collapsing Ricci limit space is semi-locally simply connected. This is joint work with Guofang Wei.
Friday April 26: Yongjia Zhang (UCSD)
On the curvature bounds for some Ricci flow singularity models in dimension four
When a Ricci flow on a closed manifold develops finite-time singularity, a blow-up limit around the singular time is called a singularity model. Such process of taking blow-up limit must imply that the topology of the singularity model is related to that of a closed manifold. We make use of such relation and prove: (1) If a four-dimensional steady gradient Ricci soliton is a singularity model, then it has bounded curvature. (2) If a four-dimensional shrinking gradient Ricci soliton is a singularity model, then its curvature has at most quadratic growth rate. This is a joint work with Bennett Chow, Michael Freedman, and Henry Shin.
Friday May 10: Antonio De Rosa (NYU)
Elliptic integrands in geometric analysis
We present our extension of Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded anisotropic first variation. We identify a necessary and sufficient condition on the integrand for its validity and we discuss the connections of this condition to Almgren's ellipticity. We apply this result to the set-theoretic anisotropic Plateau problem, obtaining solutions to three different formulations: one introduced by Reifenberg, one proposed by Harrison and Pugh and another one studied by David. Moreover, we apply the rectifiability theorem to prove an anisotropic counterpart of Allard's compactness result for integral varifolds. Some of the presented theorems are joint works with De Lellis, De Philippis, Ghiraldin and Kolasiński.
Friday May 17: Junrong Yan (UCSB)
Witten Deformation on Non-compact Manifolds
In an extremely influential paper, Witten introduced a deformation of the de Rham complex by considering the new differential d + df, where f is a Morse function. Then this new differential still computes the De Rham cohomology of a compact manifold. Here we study the case of non-compact manifolds. With some curvature assumptions on manifolds and growth assumptions on Morse functions, we are able to prove that the cohomology of the Witten deformation acting on the complex of smooth L^2 forms is isomorphic to cohomology of Thom-Smale complex and relative cohomology of the pair (M,U) for some open set U. In the end, we will apply our main results to the case of compact manifolds with boundaries. This is joint work with Xianzhe Dai.
Friday May 31: Bo Guan (Ohio State University)
Fully nonlinear elliptic equations for conformal deformation of Chern-Ricci curvatures
There are several ways to define Chern-Ricci curvatures for the Chern connection on non-Kaehler Hermitian manifolds. We introduce a notion of mixed-Chern-Ricci forms, which naturally occur in geometric problems and seem interesting to study, and consider fully nonlinear elliptic equations for their conformal deformation. We establish a priori estimates and prove existence results for these equations under very general structure conditions.
Our work is motivated by the close connections of these equations to problems in non-Kahler complex geometry, and the fact that there have been increasing interests in fully nonlinear pde's beyond the complex Monge-Ampere equation from complex Geometry. This talk is based on work with Chunhui Qiu and Rirong Yuan.
Friday June 7: Michael Freedman (UCSB/Station Q)
Families of Jacobi Equations: correlated but not coupled
I’ll explain a new (?) technique in ODE, “coefficient shuffling”, and how it can be applied to improve the Bishop-Gromov volume estimates.
Winter 2019
The talks are held on every Friday from 3:00 to 4:00 PM in South Hall 6635 (unless otherwise noted).
Friday January 11: Davit Harutyunyan (UCSB)
On geometric rigidity of the rotation group and thin domains
The rotation group $SO(n)$ is known to be rigid in the following sense: A famous theorem by Reshetnyak states that if the
gradient of a $W^{1,2}$ function belongs to the rotation group a.e. in an open and connected set $\Omega\subset R^n$, then in fact
the gradient must be a constant rotation. For a connected open subset $\Omega$ of $R^n$, and given two matrices $A, B\in M^{n\times n}$,
there exists a Lipschitz map $u\colon\Omega\to R^n$ such that $\nabla u=A$ or $B$ a.e. in $\Omega,$ if and only if $rank(A-B)\leq 1.$
Similar result is true for 3 and 4 matrices, but fails to hold for 5 matrices. This shows the speciality of the rotation group in
some sense. In this talk we will discuss stronger versions of Reshetnyak's theorem as well as some optimal forms in the
case of thin domain $\Omega$, which is applicable in nonlinear elasticity and plasticity.
Wednesday January 23: Burkhard Wilking (University of Münster)
On Ricci flow on manifolds with integral lower curvature bounds
TBA
Friday January 25: Yuchin Sun (University of Chicago)
Morse Index Bound for Min-Max Two Spheres
We prove that given a Riemannian manifold of dimension at least three, with a generic metric and nontrivial homotopy group \pi_3, there exists a collection of finitely many harmonic spheres whose sum of areas realizes the width with Morse index bound one.
Friday February 15: Guofang Wei (UCSB)
Volume entropy estimate for integral Ricci curvature
We give an optimal estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate of Aubry and give an application on the algebraic entropy of its fundamental group. We also extend the quantitative almost maximal volume entropy rigidity of Chen-Rong-Xu and almost minimal volume rigidity of Bessieres-Besson-Courtois-Gallot to integral Ricci curvature. This is a joint work with Lina Chen.
Friday February 22: Xiaolong Li (UC, Irvine)
Ancient Solutions to the Ricci Flow in Higher Dimensions
It is well-known that the Ricci flow will generally develop singularities if one flows an arbitrary initial metric. Ancient solutions arise as limits of suitable blow-ups as the time approaches the singular time and thus play a central role in understanding the formation of singularities. By the work of Hamilton, Perelman, Brendle, and many others, ancient solutions are now well-understood in two and three dimensions. In higher dimensions, only a few
classification results were obtained and many examples were constructed. In this talk, we show that for any dimension $n \geq 4$, every noncompact rotationally symmetric ancient $kappa$-solution to the Ricci flow with bounded positive curvature operator must be the Bryant soliton, extending a recent result of Brendle to higher dimensions. This is joint work with Yongjia Zhang.
Friday March 8: Fred Wilhelm (UC Riverside)
Stability, Finiteness, and Dimension 4
I will discuss the history and proof of the following result.
Theorem. For any $k\in \mathbb{R}$, $v>0$, and $D>0$, there are only finitely many diffeomorphism types of Riemannian 4-manifolds with sectional curvature $\geq k$, volume $\geq v$, and diameter $\leq D$.
Fall 2018
The talks are held on every Friday from 3:00 to 4:00 PM in South Hall 6635 (unless otherwise noted).
Tuesday September 4: Jesse Ratzkin (University of Wuerzburg), SH4607
On the rate of change of the first eigenvalue of a moving domain
Extremal Sobolev functions on a bounded domain solve a nonlinear eigenvalue problem, and there is a large literature examining the relation between eigenvalues, the extremal functions, and the geometry of the underlying domain. We examine the case of an expanding domain, and in this case estimate the rate of change of the best constant in the Poincare-Sobolev inequality. Our estimates are isoperimetric, and along the way we prove an inequality which reverses the usual Holder inequality and may be of independent interest.
Friday October 5: Guofang Wei (UCSB)
Eigenvalue Estimates for Integral Curvature
Studying the eigenvalues of the Laplacian is both important in mathematics and physics. Some classical results are Lichnerowicz and Zhong-Yang estimates for the first nonzero eigenvalue of the Laplacian on closed manifolds with positive and zero Ricci curvature lower bounds. We will discuss extensions of these results to manifolds with integral Ricci curvature lower bound, which is a much weaker condition. This is a joint work with Xavier Ramos Oliver, Shoo Seto and Qi Zhang.
Friday October 12: Guangbo Xu (Stony Brook University)
Bershadsky--Cecotti--Ooguri--Vafa torsion in Landau--Ginzburg models
In the celebrated work of Bershadsky--Cecotti--Ooguri--Vafa
the genus one string amplitude in the B-model is identified with certain
analytic torsion of the Hodge Laplacian on a K\"ahler manifold. In a
joint work with Shu Shen (IMJ-PRG) and Jianqing Yu (USTC) we study the
analogous torsion in Landau--Ginzburg models. I will explain the
corresponding index theorem based on the asymptotic expansion of the
heat kernel of the Schr\"odinger operator. I will also explain the
rigorous definition of the BCOV torsion for homogeneous polynomials on
${\mathbb C}^N$. Lastly I will explain the conjecture stating that in
the Calabi--Yau case the BCOV torsion solves the holomorphic anomaly
equation for marginal deformations.
Friday October 19: Yousef Chahine (UCSB)
Volume estimates for tubes around submanifolds using integral curvature bounds
We generalize an inequality of E. Heintze and H. Karcher for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for k-Ricci curvature. Even in the case of a pointwise bound this generalizes the classical inequality by replacing a sectional curvature bound with a k-Ricci bound. This work is motivated by the estimates of Petersen-Shteingold-Wei for the volume of tubes around a geodesic and generalizes their result. Using similar ideas we also prove a Hessian comparison theorem for k-Ricci curvature which generalizes the usual Hessian and Laplacian comparison for distance functions from a point and give several applications.
Friday October 26: Jiayin Pan (UCSB)
Nonnegative Ricci curvature, stability at infinity, and structure of fundamental groups
We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We show that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a priorly fixed metric space, then $\pi_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\widetilde{M}$ has Euclidean volume growth, then we can further bound the index of that abelian subgroup in terms of $n$ and the volume growth constant.
Friday November 30: Junrong Yan (UCSB)
Positive Scalar Curvature and Index Theorem
A fundamental problem in differential geometry concerns the relationship between local geometry and global topology. In particular, how curvatures control topology of manifolds. In this talk, we will review several such results with respect to scalar curvature. We are going to answer two questions: What kinds of manifolds admit metrics of positive scalar curvature? Can we say something about the fundamental group of such manifolds?
We will see that index theorem plays a fundamental role in addressing these questions.
Friday December 7: Jesús Núñez-Zimbrón (Centro de Ciencias Matemáticas UNAM)
On the Borel conjecture for Alexandrov 3-spaces
The Borel conjecture (BC) states that if two closed, aspherical
n-manifolds are homotopy equivalent then they are homeomorphic. The validity
of this conjecture for n=3 follows from Perelman's resolution of the
Geometrization Conjecture. Generalizations of the BC outside of the manifold
category have been obtained, for example, for CAT(0)-spaces and certain
classes of topological orbifolds. It is therefore natural to inquire whether
the BC holds for the class of Alexandrov 3-spaces (with curvature bounded
below). I will speak about work in progress in this direction which shows that
two aspherical, irreducible Alexandrov 3-spaces which are sufficiently
collapsed with respect to their diameters satisfy the BC. The results
presented here are joint with Noé Bárcenas.
PAST TALKS
Winter and Spring 2018
1/19 Hanming Zhou, UCSB "Lens
Rigidity for a Particle in a Yang-Mills Field"
Abstract: In this talk, we consider an inverse problem related to
the motion of a classical colored spinless particle under the influence
of an external Yang-Mills potential $A$ on a compact manifold with
boundary of dimension $\geq 3$. We show that under suitable convexity
assumptions, one can recover the potential $A$, up to gauge
transformations, from the lens data of the system, namely, scattering
data plus travel times between boundary points. The talk is based on
joint work with Gabriel Paternain and Gunther Uhlmann.
1/26 Changliang Wang, McMaster University,"Perelman's functionals on compact manifolds with
isolated conical singularities"
Abstract: We extend the theory of the Perelman's functionals on compact
smooth manifolds to compact manifolds with isolated conical
singularities. For the lambda-functional, this is essentially an
eigenvalue problem for a Schrodinger operator with singular potential.
We obtain a certain asymptotic behavior of eigenfunctions near the
singularities. This asymptotic behavior plays an important role for
deriving the variation formulas of the lambda-functional and other
applications. Moreover, we show that the infimum of the W-functional
over a suitable weighted Sobolev space on compact manifolds with
isolated conical singularities is finite, and the minimizing function
exists. We also obtain a certain asymptotic behavior for the minimizing
function near the singularities. This is a joint work with Professor
Xianzhe Dai.
2/2 Sho Seto, UCSB "Learning
seminar on
needle decomposition"
2/23 Sho Seto, UCSB "Learning
seminar on
needle decomposition"
3/2 Nicholas Brubaker, California State University,
Fullerton, "A numerical method for
computing constant mean curvature surfaces with boundary"
Abstract: Constant mean curvatures (CMC) surfaces, defined as critical
points of surface area subject to a volume constraint, describe
mathematical idealizations of physical interfaces occurring between two
immiscible fluids. Accordingly, their predicted shapes give insight
into the behavior of many micro-scale systems, such as beading or
stiction in microelectromechanical system devices. However, explicitly
computing such shapes is often impossible, especially when the boundary
of the interface is fixed and/or parameters vary. In this talk, we will
propose a robust novel numerical method for computing families of
discrete versions of CMC surfaces that is based on solving a partial
differential equation (PDE) via arc-length continuation. The method
computes both stable and unstable surfaces, unlike many direct
optimization methods, and naturally identifies bifurcations. Multiple
examples will be presented to highlight the efficacy and accuracy of
the proposed approach, including the reconstruction of a branch of
asymmetric surfaces appearing from a symmetry-breaking bifurcation.
3/9 Ruobing Zhang, Stony Brook University "Nilpotent Structure and Examples of Collapsed
Einstein Spaces"
Abstract: In this talk, we will focus on the construction of new
examples of collapsed Einstein spaces. A special case is to construct a
family of hyperkähler metrics on a K3 surface which are collapsing
to a closed interval [0,1]. Geometrically, the regular collapsing
fibers in our example are 3D-Heisenberg (nilpotent) manifolds with
almost flat metrics, while the singular collapsing fibers are singular
circle fibrations over a flat torus. Moreover, there is a natural
constant mean curvature foliation from the regular fiber to the
singular fiber. Compared with the known examples of codim-1 and codim-2
collapsed Ricci-flat hyperkähler spaces, the collapsing fiber in
our example are non-abelian. We will also see how such a collapsing
phenomenon is related to a general regularity theorem.
Abstract: Given a closed Riemann surface and a Lie group G, the
non-abelian Hodge theory gives a correspondence between the space of
representations of the surface group into G with the moduli space of
G-Higgs bundles. The correspondence is through looking for an
equivariant harmonic map to the symmetric space associated to G, to a
given representation or a given Higgs bundle. We derive a maximum
principle for a type of elliptic systems and apply it to study cyclic
Higgs bundles. We show several domination results on the pullback
metric of the (possibly branched) minimal immersion associated to
cyclic Higgs bundles. Also, we obtain a lower and upper bound of the
extrinsic curvature of the image of the minimal immersion.
4/13 Shoo Seto, UCSB, "The
first eigenvalue of the p-Laplacian on Riemann and Kahler manifolds"
4/20 Katy Craig, UCSB "Gradient
flow in the Wasserstein Metric"
Abstract: For a range of partial differential
equations–including the porous medium equation, the Fokker-Planck
equation, and the Keller-Segel equation—solutions of the equations can
be characterized as gradient flows with respect to the Wasserstein
metric on the space of probability measures. This gradient flow
structure lies at the heart of many recent analytic and numerical
results regarding questions of stability, uniqueness, and singular
limits.
Gradient flows with respect to Hilbert space norms are a classical tool
in the study of partial differential equations, but the geometry of the
Wasserstein metric presents new challenges. First, even for probability
measures on Euclidean space, the Wasserstein metric it is positively
curved in dimensions higher than one. Second, the metric lacks a
rigorous Riemannian structure, which one would normally use to make
sense of the “gradient” in a “gradient flow”. In this talk, I will
introduce a time discretization of the gradient flow problem, due to
Jordan, Kinderlehrer, and Otto, by which these problems can be overcome
and present new results which extend the convergence of the time
discrete scheme to a new class of partial differential equations of
applied interest.
5/11 Li Ma, University of Science and Technology
Beijing "Results of Lichnerowicz equations
on manifolds"
5/18 Jeffrey A Viaclovsky, UCI "Nilpotent structures and collapsing Ricci-flat
metrics on K3 surfaces"
ABSTRACT: I will discuss a new construction of families of
Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval,
with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a
corresponding singular fibration from the K3 surface to the interval,
with regular fibers diffeomorphic to either 3-tori or Heisenberg
nilmanifolds. This is joint work with Hans-Joachim Hein, Song Sun, and
Ruobing Zhang.
6/1 Luca Spolaor, MIT & Princeton "(Log-)epiperimetric inequality and regularity at
isolated singularities for almost Area-Minimizing currents"
Abstract: The uniqueness of blow-up and regularity of multiplicity-one
minimal surfaces at isolated singularities has been successfully
investigated by Allard-Almgren [Ann. of Math. '81], in the integrable
case, and by L. Simon [Ann. of Math. '83], in its full generality.
In this talk I will present a simple and completely variational
approach to this problem, achieved by proving a new logarithmic
epiperimetric inequality for multiplicity-one stationary cones with
isolated singularity. In contrast to classical epiperimetric
inequalities by Reifenberg [Ann. of Math. '64], Taylor [Invent. Math.
'73, Ann. of Math. '76] and White [Duke '83], we require no a priori
assumptions on the structure of the cone (e.g. integrability). If the
cone is integrable (not only through rotations), we recover the
classical epiperimetric inequality. Epiperimetric inequalities of
logarithmic type were first introduced by M.Colombo, B. Velichkov and
myself in the context of the obstacle and thin-obstacle problems.
As a consequence of our analysis we give a new proof of Allard-Almgren
and Simon results in the case of minimizers and we deduce a new
epsilon-regularity result for almost area-minimizing currents at
isolated singularities.
This is joint work with M. Engelstein (MIT) and B. Velichkov (Grenoble).
6/8 Xavier Ramos Olive, UCR "Li-Yau Gradient Estimate under Integral Ricci
Curvature Bounds"
Abstract: Li-Yau gradient estimates are one of the key
ingredients in many results in Geometric Analysis. To obtain them, one
usually needs to assume a lower bound on the Ricci curvature and some
convexity condition on the boundary. Following the work of Q.S.Zhang
and M.Zhu, we will describe a new Li-Yau gradient estimate under
integral Ricci curvature conditions for the Neumann heat kernel. We
will show how we can get an estimate on domains that are not
necessarily convex, but that satisfy the interior rolling $R-$ball
condition introduced by J.Wang and R.Chen.