Speaker: Luigi Vezzoni - Torino
Date: Friday, 28/06/2024 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
The talk focuses on geometric flows of Hermitian metrics on non-Kähler manifolds, paying particular attention to the family of Hermitian curvature flows introduced by Streets and Tian. It will be shown that, under suitable assumptions, a Hermitian Curvature flow starting from a left-invariant Hermitian metric on a Lie group has a long time solution converging to a soliton, up to renormalization. The study of solitons and static solutions of geometric flows on Lie groups will be also addressed. The last part of the talk is about a work in progress on the Second Chern-Ricci flow on complex parallelizable manifolds. The results are in collaboration with Lucio Bedulli, Nicola Enrietti, Anna Fino, Ramiro Lafuente and Mattia Pujia.
Speaker: Yamile Godoy - UNCordoba
Date: Friday, 14/06/2024 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
Given a smooth closed strictly convex curve $\gamma$ in the plane and a point $x$ outside of $\gamma$, there are two tangent lines to $\gamma$ through $x$; choose one of them consistently, say, the right one from the viewpoint of $x$, and the outer billiard map $B$ is defined by reflecting $x$ about the point of tangency. We observe that the good definition and the injectivity of the plane outer billiard map is a consequence of the fact that the tangent rays associated to both tangent vectors to $\gamma$ determine foliations of the exterior of the curve. In this talk, we will present the results obtained from a generalization of the problem of defining outer billiards in higher dimensions. Let $v$ be a smooth unit vector field on a complete, umbilic (but not totally geodesic) hypersurface $N$ in a space form; for example on the unit sphere $S^{2k-1} \subset \mathbb{R}^{2k}$, or on a horosphere in hyperbolic space. We give necessary and sufficient conditions on $v$ for the rays with initial velocities $v$ (and $-v$) to foliate the exterior $U$ of $N$. We find and explore relationships among these vector fields and geodesic vector fields on $N$. When the rays corresponding to each of $\pm v$ foliate $U$, $v$ induces an outer billiard map whose billiard table is $U$. We describe the unit vector fields on $N$ whose associated outer billiard map is volume preserving. This is a joint work with Michael Harrison (Institute for Advanced Study, Princeton) and Marcos Salvai (UNC, Argentina).
Speaker: Francisco Vittone - UNRosario
Date: Friday, 31/05/2024 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In any Riemannian manifold one can define two natural subspaces of each tangent space. The first is given by the nullity of the curvature tensor, and the second is given by the parallel Killing vector fields at a point (transvections). In a homogeneous spaces, both subspaces allow to define invariant distributions, called the nullity distribution and the distribution of symmetry, which are related to each other. We present some recent works which study the restrictions that the existence of nullity imposes in the Lie algebra of the whole isometry group of a Riemannian homogeneous space and its relation to the distribution of symmetry. We finally introduce some work in progress on the extension of these concepts to Lorentzian homogeneous spaces.
Speaker: Jakob Stein - Unicamp
Date: Friday, 03/05/2024 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
Asymptotically locally conical (ALC) metrics can be viewed as higher-dimensional analogues of ALF gravitational instantons, such as the Taub-NUT metric. In the setting of special holonomy, families of Yang-Mills instantons on ALC G2-metrics are expected to display some of the same features as the families of instantons on ALF spaces, studied recently by Cherkis-Larrain-Hubach-Stern. We will demonstrate this relationship explicitly in the cohomogeneity one setting, and study the behaviour of Yang-Mills instantons as the underlying geometry varies in a one-parameter family. This talk features two ongoing joint works, one with Matt Turner, and one with Lorenzo Foscolo and Calum Ross.
Speaker: Alejandro Tolcachier - UNC
Date: Friday, 19/04/2024 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
The canonical bundle of a complex manifold $(M,J)$, with $\operatorname{dim}_{\mathbb{C}} M=n$, is defined as the $n$-th exterior power of its holomorphic tangent bundle and it is a holomorphic line bundle over $M$. Complex manifolds with holomorphically trivial canonical bundle are important in differential, complex, and algebraic geometry and also have relations with theoretical physics. It is well known that every nilmanifold $\Gamma\backslash G$ equipped with an invariant complex structure has (holomorphically) trivial canonical bundle, due to the existence of an invariant (holomorphic) trivializing section. For complex solvmanifolds such a section may or may not exist. In this talk, we will see an example of a complex solvmanifold with a non-invariant trivializing holomorphic section of its canonical bundle. This new phenomenon lead us to study the existence of holomorphic trivializing sections in two stages. In the invariant case, we will characterize this existence in terms of the 1-form $\psi$ naturally defined in terms of the Lie algebra of $G$ and $J$ by $\psi(x)=\operatorname{Tr} (J\operatorname{ad} x)-\operatorname{Tr} \operatorname{ad} (Jx)$. For the non-invariant case, we will provide an algebraic obstruction for a solvmanifold to have a trivial canonical bundle (or, more generally, holomorphically torsion) and we will explicitly construct, in certain examples, a trivializing section of the canonical bundle that is non-invariant. We will apply this construction to hypercomplex geometry to provide a negative answer to a question posed by M. Verbitsky. Based on joint work with Adrián Andrada.
Speaker: Ernani Ribeiro Jr (UFC)
Date: Friday, 05/04/2024 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In this talk, we discuss the geometry of compact quasi-Einstein manifolds with boundary. This topic is directly related to warped product Einstein metrics, static spaces and smooth metric measure spaces. We show that a 3-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature must be isometric to either the standard hemisphere $S^3_{+},$ or the cylinder $I\times S^2$ with product metric. For dimension n=4, we prove that a 4-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric to either the standard hemisphere $S^4_{+},$ or the cylinder $I\times S^3$ with product metric, or the product space $S^2_{+}\times S^2$ with the doubly warped product metric. Other related results for arbitrary dimensions are also discussed. This is a joint work with J. Costa and D. Zhou.
Speaker: João Henrique Santos de Andrade (USP)
Date: Friday, 22/03/2024 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
We study some compactness properties of the set of conformally flat singular metrics with constant positive Q-curvature (integer or fractional) on a finitely punctured sphere. Based on some recent classification results, we focus on some cases of integer Q-curvature. We introduce a notion of necksize for these metrics in our moduli space, which we use to characterize compactness. More precisely, we prove that if the punctures remain separated and the necksize at each puncture is bounded away from zero along a sequence of metrics, then a subsequence converges with respect to the Gromov-Hausdorff metric. Our proof relies on an upper bound estimate which is proved using moving planes and a blow-up argument. This is combined with a lower bound estimate which is a consequence of a removable singularity theorem. We also introduce a homological invariant which may be of independent interest for upcoming research.
Speaker: Andrei Moroianu - Paris-Saclay/CNRS
Date: Friday, 08/03/2024 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
We consider compact conformal manifolds $(M,[g])$ endowed with a closed Weyl structure $\nabla$, i.e. a torsion-free connection preserving the conformal structure, which is locally but not globally the Levi-Civita connection of a metric in $[g]$. Our aim is to classify all such structures when both $\nabla$ and $\nabla^g$, the Levi-Civita connection of $g$, have special holonomy. In such a setting, $(M,[g],\nabla)$ is either flat, or irreducible, or carries a locally conformally product (LCP) structure. Since the flat case is already completely classified, we focus on the last two cases. When $\nabla$ has irreducible holonomy we prove that $(M,g)$ is either Vaisman, or a mapping torus of an isometry of a compact nearly Kähler or nearly parallel $\mathrm{G}_2$ manifold, while in the LCP case we prove that $g$ is neither Kähler nor Einstein, thus reducible by the Berger-Simons Theorem, and we obtain the local classification of such structures in terms of adapted metrics. This is joint work with Florin Belgun and Brice Flamencourt.
Speaker: Eveline Legendre - U. Lyon
Date: Friday, 01/12/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In this talk I will present a recent joint work with Abdellah Lahdilli and Carlo Scarpa where, given a polarised Kähler manifold $(M,L)$, we consider the circle bundle associated to the polarization with the induced transversal holomorphic structure. The space of contact structures compatible with this transversal structure is naturally identified with a bundle, of infinite rank, over the space of Kähler metrics in the first Chern class of $L$. We show that the Einstein--Hilbert functional of the associated Tanaka--Webster connections is a functional on this bundle, whose critical points are constant scalar curvature Sasaki structures. In particular, when the group of automorphisms of $(M,L)$ is discrete, these critical points correspond to constant scalar curvature Kähler metrics in the first Chern class of $L$. If time permits, I will explain how we associate a two real parameters family of these contact structures to any ample test configuration and relate the limit, on the central fibre, to a primitive of the Donaldson-Futaki invariant. As a by-product, we show that the existence of cscK metrics on a polarized manifold implies K-semistability
Speaker: Viviana Del Barco - Unicamp
Date: Friday, 17/11/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In this talk we will discuss recent advancements on G$_2$-instantons on 7-dimensional 2-step nilpotent Lie groups endowed with a left-invariant coclosed G$_2$-structures. I will present necessary and sufficient conditions for the characteristic connection of the G$_2$-structure to be an instanton, in terms of the torsion of the G$_2$-structure, the torsion of the connection and the Lie group structure. These conditions allow to show that the metrics corresponding to the G$_2$-instantons define a naturally reductive structure on the simply connected 2-step nilpotent Lie group with left-invariant Riemannian metric. This is a joint work with Andrew Clarke and Andrés Moreno.
Speaker: Michael Jablonski - University of Oklahoma
Date: Friday, 03/11/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
Real semi-simple Lie algebras arise naturally both algebraically, in the study of Lie theory, and geometrically, in the study of symmetric spaces. After recalling why these algebras are of interest, we will investigate their uniqueness properties through the lens of special subalgebras, the so-called Iwasawa subalgebras. While the results are algebraic, the tools to obtain them come from the Riemannian geometry of solvmanifolds. We will finish the talk with a quick discussion of the complex setting and how it differs from the real setting. This is joint work with Jon Epstein (McDaniel College).
Speaker: Rayssa Caju - PUC Chile
Date: Friday, 20/10/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
Over the past few decades, there has been significant exploration of the interplay between geometry and partial differential equations. In particular, some problems arising in conformal geometry, such as the classical Yamabe problem, can be reduced to the study of PDEs with critical exponent on manifolds. More recently, the so-called Q-curvature equation, a fourth-order elliptic PDE with critical exponent, is another class of conformal equations that has drawn considerable attention by its relation with a natural concept of curvature. In this talk, I would like to motivate these problems from a geometric and analytic perspective, and discuss some recent developments in the area, in particular regarding the singular Q-curvature problem.
Speaker: Pedro Gaspar - PUC Chile
Date: Friday, 06/10/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
The Allen–Cahn equation is a semilinear parabolic partial differential equation that models phase-transition and phase-separation phenomena and which provides a regularization for the mean curvature flow (MCF), one of the most studied extrinsic geometric flows. In this talk, we employ Morse-theoretical considerations to construct eternal solutions of the Allen–Cahn equation that connect unstable equilibria in compact manifolds. We describe the space of such solutions in a round 3-sphere under a low-energy assumption, and indicate how these solutions could be used to produce geometrically interesting MCFs. This is joint work with Jingwen Chen (University of Pennsylvania).
Speaker: Rafael Montezuma - UFC
Date: Friday, 22/09/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In this talk we develop a Morse-Lusternik-Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical definition of the width of plane curves. Pairs of points of the circle realising the width bound one or more minimising geodesics that intersect the curve in special configurations. When the circle bounds a totally convex disc, we classify the possible configurations under a further geometric condition. We also present properties and characterisations of curves that can be regarded as the Riemannian analogues of plane curves of constant width. This talk is based on a joint work with Lucas Ambrozio (IMPA) and Roney Santos (UFC).
Speaker: Mauro Sibils - UN Rosario
Date: Friday, 25/08/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
A magnetic trajectory is a curve $\gamma$ on a Riemannian manifold $(M, g)$ satisfying the equation: $$\nabla_{\gamma'}{\gamma'}= q F\gamma'$$ where $\nabla$ is the corresponding Levi-Civita connection and $F$ is a skew-symmetric $(1,1)$-tensor such that the corresponding 2-form $g(F\cdot ,\cdot)$ is closed. In this talk we are going to describe all magnetic trajectories on the Heisenberg Lie group of dimension three $H_3$ for any invariant Lorentz force. We will write explicitly the magnetic equations and show that the solutions are described by Jacobi's elliptic functions. As a consequence, we will prove the existence and characterize the periodic magnetic trajectories. Then we will induce the Lorentz force to a compact quotient $H_3/\Gamma$ and study the periodic magnetic trajectories there, proving its existence for any energy level when $F$ is non-exact. This is a joint work with Gabriela Ovando (UNR).
Speaker: Marcos Salvai - UNC
Date: Friday, 11/07/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
Let $M$ be an oriented three dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle $SO(M) \to M$ of all its positively oriented orthonormal tangent frames. When $M$ is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on $\operatorname{Iso}_o (M) \equiv SO(M)$: A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases. M. Salvai, A split special Lagrangian calibration associated with frame vorticity, accepted for publication in Adv. Calc. Var.
Speaker: Giulia Dileo - University of Bari
Date: Friday, 23/06/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
I will discuss some special classes of almost contact metric manifolds $(M,\varphi,\xi,\eta,g)$ such that the structure $(\varphi,g)$ is projectable along the 1-dimensional foliation generated by $\xi$, and the transverse geometry is given by a Kähler structure. I will focus on quasi-Sasakian manifolds and the new class of anti-quasi-Sasakian manifolds. In this case, the transverse geometry is given by a Kähler structure endowed with a closed 2-form of type (2,0), as for instance hyperkähler structures. I will describe examples of anti-quasi-Sasakian manifolds, including compact nilmanifolds and principal circle bundles, investigate Riemannian curvature properties, and the existence of connections with torsion preserving the structure. This is a joint work with Dario Di Pinto (Bari).
Speaker: Maria Laura Barberis - UNC
Date: Friday, 09/06/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
There is a notion of nilpotent complex structures on nilpotent Lie algebras introduced by Cordero-Fernández-Gray-Ugarte (2000). Not every complex structure on a nilpotent Lie algebra $\mathfrak{n}$ is nilpotent, but when $\mathfrak{n}$ is $2$-step nilpotent any complex structure on $\mathfrak{n}$ is nilpotent of step either $2$ or $3$ (a fact proved by J. Zhang in 2022). The class of nilpotent complex structures of step $2$ strictly contains the space of abelian and bi-invariant complex structures on a $2$-step nilpotent Lie algebra. In this work in progress, we obtain a characterization of the $2$-step nilpotent Lie algebras whose corresponding Lie groups admit a left invariant complex structure. We consider separately the cases when the complex structure is nilpotent of step $2$ or $3$. Some applications of our results to Hermitian geometry are discussed, for instance, it turns out that the $2$-step nilpotent Lie algebras constructed by Tamaru from Hermitian symmetric spaces admit pluriclosed (or SKT) metrics. We also show that abelian complex structures are frequent on naturally reductive $2$-step nilmanifolds, while it is known (Del Barco-Moroianu) that these do not admit orthogonal bi-invariant complex structures.
Speaker: Yang-Hui He - LIMS
Date: Friday, 26/05/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
The search for the Theory of Everything has led to superstring theory, which then led physics, first to algebraic/differential geometry/topology, and then to computational geometry, and now to data science. With a concrete playground of the geometric landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of interest to theoretical physics and to pure mathematics. At the core of our programme is the question: how can AI help us with mathematics?
Speaker: Da Rong Cheng - University of Miami
Date: Friday, 12/05/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
Given a surface S in $\mathbb{R}^3$, a classical problem is to find disk-type surfaces with prescribed constant mean curvature whose boundary meets S orthogonally. When S is diffeomorphic to a sphere, direct minimization could lead to trivial solutions and hence min-max constructions are needed. Among the earliest such constructions is the work of Struwe, who produced the desired free boundary CMC disks for almost every mean curvature value up to that of the smallest round sphere enclosing S. In a previous joint work with Xin Zhou (Cornell), we combined Struwe's method with other techniques to obtain an analogous result for CMC 2-spheres in Riemannian 3-spheres and were able to remove the "almost every" restriction in the presence of positive ambient curvature. In this talk, I will report on more recent progress where the ideas in that work are applied back to the free boundary problem to refine and improve Struwe's result.
Speaker: Daniel Fadel - IMPA
Date: Friday, 28/04/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In this talk, I will report on recent results of an ongoing collaboration with Éric Loubeau, Andrés Moreno and Henrique Sá Earp on the study of the harmonic flow of $H$-structures. This is the negative gradient flow of a natural Dirichlet-type energy functional on an isometric class of $H$-structures on a closed Riemannian $n$-manifold, where $H$ is the stabilizer in $\mathrm{SO}(n)$ of a finite collection of tensors in $\mathbb{R}^n$. Using general Bianchi-type identities of $H$-structures, we are able to prove monotonicity formulas for scale-invariant local versions of the energy, similar to the classic formulas proved by Struwe and Chen (1988-89) in the theory of harmonic map heat flow. We then deduce a general epsilon-regularity result along the harmonic flow and, more importantly, we get long-time existence and finite-time singularity results in parallel to the classical results proved by Chen-Ding (1990) in harmonic map theory. In particular, we show that if the energy of the initial $H$-structure is small enough, depending on the $C^0$-norm of its torsion, then the harmonic flow exists for all time and converges to a torsion-free $H$-structure. Moreover, we prove that the harmonic flow of $H$-structures develops a finite time singularity if the initial energy is sufficiently small but there is no torsion-free $H$-structure in the homotopy class of the initial $H$-structure. Finally, based on the analogous work of He-Li (2021) for almost complex structures, we give a general construction of examples where the later finite-time singularity result applies on the flat $n$-torus, provided the $n$-th homotopy group of the quotient $\mathrm{SO}(n)/H$ is non-trivial; e.g. when $n=7$ and $H=\mathrm{G}_2$, or when $n=8$ and $H=\mathrm{Spin}(7)$.
Speaker: Lino Grama - Unicamp
Date: Friday, 14/04/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In this talk, we will discuss the curvature properties of invariant almost Hermitian geometry on generalized flag manifolds. Specifically, we will focus on the "Kähler-like scalar curvature metric" - that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_C$, where $s$ is the Riemannian scalar curvature and $s_C$ is the Chern scalar curvature. We will provide a classification of such metrics on generalized flag manifolds whose isotropy representation decomposes into two or three irreducible components. This is a joint work with A. Oliveira.
Speaker: Ruy Tojeiro - USP
Date: Friday, 31/03/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
The classical Bonnet problem is to classify all immersions $f\colon\,M^2\to\mathbb{R}^3$ into Euclidean three-space that are not determined, up to a rigid motion, by their induced metric and mean curvature function. The natural extension of Bonnet problem for Euclidean hypersurfaces of dimension $n\geq 3$ was studied by Kokubu. In this talk we report on joint work with M. Jimenez, in which we investigate an infinitesimal version of Bonnet problem for hypersurfaces with dimension $n\geq 3$ of any space form, namely, we classify the hypersurfaces $f\colon M^n\to\mathbb{Q}_c^{n+1}$, $n\geq 3$, of any space form $\mathbb{Q}_c^{n+1}$ of constant curvature $c$, for which there exists a (non-trivial) one-parameter family of immersions $f_t\colon M^n\to\mathbb{Q}_c^{n+1}$, with $f_0=f$, whose induced metrics $g_t$ and mean curvature functions $H_t$ coincide ``up to the first order", that is, $\partial/\partial t|_{t=0}g_t=0=\partial/\partial t|_{t=0}H_t.$
Speaker: Carlos Olmos - UNC
Date: Friday, 17/03/2023 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
A Hopf-Berger sphere of factor $\tau$ is a sphere which is the total space of a Hopf fibration and such that the Riemannian metric is rescaled by a factor $\tau\neq 1$ in the directions of the fibers. A Hopf-Berger sphere is the usual {\it Berger sphere} for the complex Hopf fibration. A Hopf-Berger sphere may be regarded as a geodesic sphere $\mathsf{S}_t^m(o)\subset\bar M$ of radius $t$ of a rank one symmetric space of non-constant curvature ($\bar M$ is compact if and only if $\tau <1$). A Hopf-Berger sphere has positive curvature if and only if $\tau <4/3$. A standard totally geodesic submanifold of $\mathsf{S}_t^m(o)$ is obtained as the intersection of the geodesic sphere with a totally geodesic submanifold of $\bar M$. We will speak about the classification of totally geodesic submanifolds of Hopf-Berger spheres. In particular, for quaternionic and octonionic fibrations, non-standard totally geodesic spheres with the same dimension of the fiber appear, for $\tau <1/2$. Moreover, there are totally geodesic $\mathbb RP^2$, and $\mathbb RP^3$ (with some restrictions on $\tau$, the dimension and the type of the fibration). On the one hand, as a consequence of the connectedness principle of Wilking, there does not exist a totally geodesic $\mathbb RP^4$ in a space of positive curvature which diffeomorphic to the sphere $S^7$. On the other hand, we construct an example of a totally geodesic $\mathbb RP^2$ in a Hopf-Berger sphere of dimension $7$ and positive curvature. Natural question: could there exist a totally geodesic $\mathbb RP^3$ in a space of positive curvature which diffeomorphic to $S^7$?. This talk is related to a joint work with Alberto Rodríguez-Vázquez.
Speaker: Ilka Agricola - Marburg
Date: Friday, 09/12/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
We consider $3$-$(\alpha, \delta)$-Sasaki manifolds, generalizing the classic 3-Sasaki case. We show how these are closely related to various types of quaternionic Kähler orbifolds via connections with skew-torsion and a canonical submersion. Making use of this relation we discuss curvature operators and show that in dimension 7 many such manifolds have strongly positive curvature. Joint work with Giulia Dileo (Bari) and Leander Stecker (Hamburg).
Speaker: Raquel Villacampa - CUD Zaragoza
Date: Friday, 18/11/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
Nilmanifolds are a special type of differentiable compact manifolds defined as the quotient of a nilpotent, simply connected Lie group by a lattice. Since Thurston used them in 1976 to show an example of a compact complex symplectic manifold being non-Kähler, many other topological and geometrical questions have been answered using nilmanifolds. In this talk we will show some of these problems such as the holonomy of certain metric connections, deformations of structures or spectral sequences.
Speaker: Guilhermo Henry - UBA
Date: Friday, 04/11/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
A foliation such that their regular leaves are parallel CMC hypersurfaces is called isoparametric. In this talk we are going to discuss some results on the existence of solutions of the Yamabe equation on compact Riemannian manifolds with boundary induced these type of foliations. Joint work with Juan Zuccotti.
Speaker: Marcos Origlia - CONICET
Date: Friday, 21/10/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
A differential $p$-form $\eta$ on a $n$-dimensional Riemannian manifold $(M,g)$ is called Conformal Killing Yano (CKY for short) if it satisfies for any vector field $X$ the following equation \[ \nabla_X \eta=\dfrac{1}{p+1}\iota_X\mathrm{d}\eta-\dfrac{1}{n-p+1}X^*\wedge \mathrm{d}^*\eta, \] where $X^*$ is the dual 1-form of $X$, $\mathrm{d}^*$ is the codifferential, $\nabla$ is the Levi-Civita connection associated to $g$ and $\iota_X$ is the interior product with $X$. If $\eta$ is coclosed ($\mathrm d^*\eta=0$) then $\eta$ is said to be a Killing-Yano $p$-form (KY for short). We study left invariant Conformal Killing Yano $2$-forms on Lie groups endowed with a left invariant metric. We determine, up to isometry, all $5$-dimensional metric Lie algebras under certain conditions, admitting a CKY $2$-form. Moreover, a characterization of all possible CKY tensors on those metric Lie algebras is exhibited.
Speaker: Mario Garcia-Fernández - ICMAT
Date: Friday, 23/09/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In this talk I will overview joint work with J. Jordan and J. Streets, in arXiv:2106.13716, about Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form. These metrics give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kählermanifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow, as introduced by Streets and Tian, implying new global existence results. In particular, on all complex non-Kähler surfaces of nonnegative Kodaira dimension. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric.
Speaker: Ivan Struchiner - USP
Date: Friday, 09/09/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
I will discuss some aspects of the interplay between Lie groupoids and singular Riemannian foliations. To each singular Riemannian foliation we associate a linear holonomy groupoid to a neighbourhood of each leaf. This groupoid is a dense subgroupoid of a proper Lie groupoid. On the other hand, Lie groupoids with compatible metrics give rise to singular Riemannian foliations. We discuss how far these groupoids are from being a dense subgroupoid of a proper Lie groupoid. The talk will be based on joint work with Marcos Alexandrino, Marcelo Inagaki and Mateus de Melo.
Speaker: Luis Florit - IMPA
Date: Friday, 26/08/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive scalar curvature metrics on closed simply connected manifolds in dimensions at least five appears on spin manifolds, and is given by the non-vanishing of the α-genus of Hitchin. When unobstructed we shall realise a positive scalar curvature metric by an immersion into Euclidean space whose dimension is uniformly close to the classical Whitney upper bound for smooth immersions, and it is in fact equal to the Whitney bound in most dimensions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure for constructing positive scalar curvature metrics. This is a joint work with B. Hanke published in Commun. Contemp. Math. 2022.
Speaker:Lucas Ambrozio - IMPA
Date: Friday, 19/08/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
a Zoll metric is a Riemannian metric g on a manifold such that all of its geodesics are periodic and have the same finite fundamental period. In particular, (M,g) is a compact manifold such that each tangent one-dimensional subspace of each one of its points is tangent to some closed geodesic. Since periodic geodesics are not only periodic orbits of a flow, but also closed curves that are critical points of the length functional, the notion of Zoll metrics admits natural generalisations in the context of minimal submanifold theory, that is, the theory of critical points of the area functional. In this talk, based on joint work with F. Codá (Princeton) and A. Neves (UChicago), I will discuss why these new, generalised notions seem relevant to me beyond its obvious geometric appeal, and discuss two different methods to obtain infinitely many such examples on spheres, with perhaps unexpected properties.
Speaker: Gregorio Pacelli Bessa - UFC
Date: Friday, 29/07/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
We show that the global mean exit time of cylindrically bounded submanifolds of $N\times \mathbb{R}$ is finite, where the sectional curvature $K_N\leq b\leq 0$.
Speaker: Eder Moraes Correa - UFMG
Date: Friday, 01/07/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
As shown in [2], the first Aeppli-Chern class of a compact Hermitian manifold can be represented by its first Levi-Civita Ricci curvature. From this, a natural question to ask (inspired by the Calabi-Yau theorem [3]) is the following: On a compact complex manifold with vanishing first Aeppli-Chern class, does there exist a smooth Levi-Civita Ricci-flat Hermitian metric? In general, it is particularly challenging to solve the Levi-Civita Ricci-flat equation, since there are non-elliptic terms involved in the underlying PDE problem. In this talk, we will investigate the above question in the setting of compact Hermitian Weyl-Einstein manifolds. The main purpose is to show that every compact Hermitian Weyl-Einstein manifold admits a Levi-Civita Ricci-flat Hermitian metric [1]. This result generalizes previous constructions on Hopf manifolds [2]. [1] Correa, E. M.; Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds, arxiv:2204.04824v3 (2022). [2] Liu, K.; Yang, X.; Ricci curvatures on Hermitian manifolds, Trans. Amer. Math. Soc. 369 (2017), no. 7, 5157-5196. [3] Yau, S.-T.; On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411. MR480350.
Speaker: Luis L. Alías - Murcia
Date: Friday, 17/06/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In this lecture we establish a natural framework for the study of mean curvature flow solitons in warped product spaces. Our approach allows us to identify some natural geometric quantities that satisfy elliptic equations or differential inequalities in a simple and manageable form for which the machinery of weak maximum principles is valid. The latter is one of the main tools we apply to derive several new characterizations and rigidity results for MCFS that extend to our general setting known properties, for instance, in Euclidean space. Besides, as in Euclidean space, MCFS are also stationary immersions for a weighted volume functional. Under this point of view, we are able to find geometric conditions for finiteness of the index and some characterizations of stable solitons. The results of this lecture have been obtained in collaboration with Jorge H. de Lira, from Universidade Federal do Ceará, and Marco Rigoli, from Università degli Study di Milano, and they can be found in the following papers:
Speaker: Gonçalo Oliveira - UFF
Date: Friday, 03/06/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
(joint work with Jason Lotay) Richard Thomas and Shing-Tung-Yau proposed two conjectures on the existence of special Lagrangian submanifolds and on the use of Lagrangian mean curvature flow to find them. In this talk, I will report on joint work with Jason Lotay to prove these on certain symmetric hyperKahler 4-manifolds. If time permits I may also comment on our work in progress to tackle more refined conjectures of Dominic Joyce regarding the existence of Bridgeland stability conditions on Fukaya categories and their interplay with Lagrangian mean curvature flow.
Speaker:Sueli I. R. Costa - Unicamp
Date: Friday, 06/05/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
In this talk it will be presented an introduction and some recent developments in two topics of Geometry we have been working which have applications in Communications: Lattices and Information Geometry. Lattices are discrete additive subgroups of the n-dimensional Euclidean space and have been used in coding for reliability and security in transmissions through different channels. Currently Lattice based cryptography is one of the main subareas of the so called Post-quantum Cryptography. Information Geometry is devoted to the study of statistical manifolds of probability distributions by considering different metrics and divergence measures and have been used in several applications related to data analysis. We will approach here particularly the space of multivariate normal distributions with the Fisher metric and some applications.
Speaker:Andrés Moreno - Unicamp
Date: Friday, 08/04/2022 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract:
A $G_2$-structure with free divergence torsion can be interpreted as a critical point of the energy functional, restricted to its isometric class. Hence, it represents the better $G_2$ -structure in a given family. These kinds of $G_2$-structures are an alternative for the study of $G_2$-geometry, in cases when the torsion free problem is either trivial or obstructed. In general, there are some known classes of $G_2$-structures with free-divergence torsion, namely closed and nearly parallel $G_2$-structures. In this talk, we are going to present some unknown classes of invariant $G_2$-structures with free divergence torsion, specifically in the context of the 7-sphere and of the solvable Lie groups with a codimension-one Abelian normal subgroup.
Speaker: Romina M. Arroyo - UNC
Date: Friday, 08/04/2022- 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: A $J$-Hermitian metric $g$ on a complex manifold $(M,J)$ is called strong Kähler with torsion (SKT for short) if its $2$-fundamental form $\omega:=g(J\cdot,\cdot)$ satisfies $\partial \bar \partial \omega =0$. The aim of this talk is to discuss the existence of invariant SKT structures on nilmanifolds. We will prove that any nilmanifold admitting an invariant SKT structure is either a torus or $2$-step nilpotent, and we will provide examples of invariant SKT structures on $2$-step nilmanifolds in arbitrary dimensions. This talk is based on a joint work with Marina Nicolini.
Speaker: Claudio Gorodski-USP
Date: Friday, 25/03/2022- 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: We will explain our proof of the existence of $\epsilon>0$ such that every quotient of the unit sphere $S^n$ ($n\geq2$) by a isometric group action has diameter zero or at least $\epsilon$. The novelty is the independence of $\epsilon$ from~$n$. The classification of finite simple groups is used in the proof. (Joint work with C. Lange, A. Lytchak and R. A. E. Mendes.)
Speaker: Jorge Lauret - UNC
Date: Friday, 03/12/2021- 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: Given a compact differentiable manifold M, the critical points of the total scalar curvature functional Sc on the space of all unit volume Riemannian metrics on M are called Einstein metrics and play a fundamental role in Differential Geometry and Physics. Among Einstein metrics with positive scalar curvature, those which are stable as critical points of Sc (i.e., negative definite Hessian) on the subspace of all constant scalar curvature metrics, and in particular local maxima, seems to be extremely rare. In this talk, after some general preliminaries, we will focus on the case when the metrics and the variations are considered to be G-invariant for some compact Lie group G acting transitively on M. The standard metrics (i.e., defined by minus the Killing form of G) which are Einstein will be specially treated. This is joint work with Emilio Lauret (Universidad Nacional del Sur and INMABB (CONICET) , Argentina) and Cynthia Will (Universidad Nacional de Córdoba and CIEM (CONICET), Argentina).
Speaker: Llohann Sperança - Unifesp
Date: Friday, 19/11/2021- 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: The Ricci flow dictates a natural way to deform a geometric space using a reaction-diffusion equation. It was introduced by Hamilton and gained its importance through the years, including several applications and the resolution of the Poincaré Conjecture. As the heat flow equation spreads the heat over a surface, the Ricci flow tries to spread the Ricci curvature, however, topology and geometry come into play and obstacles arise, leading us to partial or total collapses of the initial object. In this talk we study the particularly interesting case of homogeneous manifolds. We present a novel normalization of the flow with natural compactness properties, and study its collapses by presenting a characterization of Gromov-Hausdorff limits of homogeneous spaces. As an application, we present a detailed picture of the Ricci flow for a special class of homogeneous spaces: phase portraits, basins of attractions, conjugation classes and collapsing phenomena. Moreover, we achieve a full classification of the collapses in this class.
Speaker: Leonardo Cavenaghi - Fribourg
Date: Friday, 05/11/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: In a seminal work, Lawson and Yau showed that compact Riemannian manifolds with non-abelian symmetry admit metrics of positive scalar curvature. In 2018, Prof. Sperança and myself found a slight generalization of this result: every invariant metric by a compact non-abelian Lie gorup develops positive scalar curvature after a finite Cheeger deformation. This generalization is regarded with the following improvement: in the result by Lawson and Yau, the symmetry group needed to be reduced to ensure positivity of the scalar curvature for some Riemannian metric, moreover, such a metric has no relation to the initial one. To our result, however, the final metric is obtained by appropriate shrinking of the manifold in the direction of the group orbits, naturally preserving symmetry. Also, the fact that any invariant metric develops positive scalar curvature raised the conjecture that the space of invariant metrics of positive scalar curvature should be connected provided if the symmetry is non-abelian. In this talk we go beyond this conjecture and prove that such a space is contractible. The result follows from a deep analysis of a generalization of the Cheeger deformation we introduce. This is a joint work with Llohann Sperança.
Speaker: Gabriela Ovando - UNR
Date: Friday, 22/10/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: From the perspective of classical mechanics, a charged particle moving on a Riemannian manifold $M$ experiences a Lorentz force, and its trajectory is called a magnetic trajectory. The Lorentz force determines a magnetic field which is introduced as a closed 2-form on $M$. We focus on 2-step nilpotent Lie groups equipped with a left-invariant metric and a left-invariant magnetic field. The aim is to study magnetic fields, their corresponding magnetic equations and solutions.
Speaker: Udhav Fowdar-Unicamp
Date: Friday, 08/10/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: Quaternion-Kähler manifolds are examples of Einstein manifolds with positive or negative scalar curvature. In this talk I want to describe certain explicit examples of negative QK metrics arising on the total spaces of certain rank 4 bundles over hyperKähler manifolds. I will explain how these examples share remarkable similarities with the Swann's bundle (i.e. cones over 3-Sasakian manifolds). Time permitting, I will also mention examples of balanced Hermitian metrics and exceptional holonomy metrics arising on these bundles.
Speaker: Mihaela Pilca-Regensburg university
Date: Friday, 24/09/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: It is well known that on a compact Kähler manifold every conformal vector field is Killing and every Killing vector field is holomorphic. In this talk I will explain how these two results extend to compact locally conformally Kähler manifolds. More precisely, we will see that any conformal vector field on a compact lcK manifold is Killing with respect to a special metric of the conformal class, the so-called Gauduchon metric. Furthermore, any conformal vector field on a compact lcK manifold whose Kähler cover is neither flat, nor hyperkähler, is holomorphic. This talk is based on joint work with Andrei Moroianu.
Speaker: Keti Tenenblat-UNB
Date: Friday, 110/09/2021- 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: In this joint work with Fabio Nunes da Silva, we show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We prove that there are three classes of such solutions. We show that for each fixed vector there is a 2-parameter family of soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the whole real line, it is embedded and its geodesic curvature, at each end, converges to a constant.
Speaker: Vanderson Lima-UFRGS
Date: Friday, 13/08/2021- 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: In this talk, I will describe how to use weak versions of the mean curvature flow to show that certain hyperbolic 3-manifolds admit smooth entire foliations whose leaves are either minimal or have non-vanishing mean curvature. This is joint work with Marco Guaraco (Imperial College) and Franco Vargas Pallete (Yale University).
Speaker: Vinicius Ramos-IMPA
Date: Friday, 30/07/2021- 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: Symplectic embeddings have been a central subject in symplectic topology with a strong relationship to Reeb dynamics. Sharp embeddings of very simple domains turn out to be quite difficult to determine. More recently, we have been able to study a wide variety of 4-dimensional symplectic manifolds relating them to toric domains using integrable systems. From this description we can find study many interesting symplectic embedding problems. In this talk, I will give a survey of this subfield and explain some recent developments. This talk is based on joint works with Yaron Ostrover, Daniele Sepe and Brayan Ferreira.
Speaker: Rafael Montezuma-UFC
Date: Friday, 02/07/2021- 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: In this talk we discuss some central results in the min-max theory for the area functional, including Morse inequalities in codimension one assuming the ambient dimension between 3 and 7. In addition, we will present methods to compare the natural min-max invariants to other geometric quantities, such as volume and curvature. These min-max invariants are special critical values of the area functional, defined through the topology of the space of surfaces. This talk is based on joint works with Lucas Ambrozio (IMPA), Fernando Codá Marques (Princeton University) and André Neves (The University of Chicago).
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Speaker: Detang Zhou-IME-UFF
Date: Friday, 18/06/2021- 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: The discovery of variation structure of Hamilton's Ricci flow plays a key role in proving Poincare conjecture. To do this Perelman defined his W-functional and proved the entropy monotonicity formulae which enabled him to complete Hamilton's program in 3-dimensional case. The critical points of W-functional are shrinking gradient Ricci solitons(SGRS). It is well known that gradient Ricci solitons are generalizations of Einstein manifolds and basic models for smooth metric measure spaces. In this talk I will discuss the definition of W-functional and its variation properties. We will present examples, the basic properties of shrinking gradient Ricci solitons. One of the challenging problems is to classify all gradient Ricci solitons with constant scalar curvature. Recently in a joint work with X. Cheng, we prove that a 4-dimensional shrinking gradient Ricci soliton has constant scalar curvature if and only if it is either Einstein, or a finite quotient of cylinder shrinking solitons $\mathbb{R}^4$, $\mathbb{S}^2 × \mathbb{R}^2$, $\mathbb{S}^3×\mathbb{R}$.
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Speaker: Luigi Vezzoni-Torino
Date: Friday, 04/06/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: HKT Geometry (HyperKahler with torsion Geometry) is the Geometry of hyperHermitian manifolds equipped with a nondegenerate ∂-closed (2,0)-form Ω. The interest for this kind of manifolds is mainly motivated by theoretical and mathematical physics since they appear on supersymmetric sigma models with Wess-Zumino term and in supergravity theories. From the mathematic point of view HKT manifolds generalize HyperKahler manifolds. The talk mainly focuses on a Calabi-Yau type problem in HKT geometry introduced by Alesker and Verbiskiy in 2010. I will describe a recent study of the problem in some explicit examples and a new approach via a geometric flow. The talk is based on a work in progress with Lucio Bedulli and Giovanni Gentili.
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Speaker: Spiro Karigiannis-University of Waterloo
Date: Friday, 21/05/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: Let $(M, \omega)$ be a compact symplectic manifold. If we choose a compatible almost complex structure $J$ (which in general is not integrable) then we can study the space of $J$-holomorphic maps $u \colon \Sigma \to (M, J)$ from a compact Riemann surface into $M$. By appropriately “compactifying” the space of such maps, one can obtain powerful global symplectic invariants of $M$. At the heart of such a compactification procedure is understanding the ways in which sequences of such maps can degenerate, or develop singularities. Crucial ingredients are conformal invariance and an energy identity, which lead to to a plethora of analytic consequences, including: (i) a mean value inequality, (ii) interior regularity, (iii) a removable singularity theorem, (iv) an energy gap, and (v) compactness modulo bubbling. Riemannian manifolds with closed $G_2$ or $\mathrm{Spin}(7)$ structures share many similar properties to such almost Kahler manifolds. In particular, they admit analogues of $J$-holomorphic curves, called associative and Cayley submanifolds, respectively, which are calibrated and hence homologically volume-minimizing. A programme initiated by Donaldson-Thomas and Donaldson-Segal aims to construct similar such “counting invariants” in these cases. In 2011, a somewhat overlooked preprint of Aaron Smith demonstrated that such submanifolds can be exhibited as images of a class of maps $u \colon \Sigma \to M$ satisfying a conformally invariant first order nonlinear PDE analogous to the Cauchy-Riemann equation, which admits an energy identity involving the integral of higher powers of the pointwise norm $|du|$. I will discuss joint work (to appear in Asian J. Math.) with Da Rong Cheng (Waterloo) and Jesse Madnick (NCTS/NTU) in which we establish the analogous analytic results of (i)-(v) in this setting. arXiv:1909.03512
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Speaker: Patricia Marçal-IUPUI
Date: Friday, 07/05/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: We will discuss different notions of warped product for Finsler spaces and some special curvature properties for them. The talk will encompass the work of many mathematicians for the past couple of decades, including my joint work with Dr. Zhongmin Shen on the explicit construction of some Ricci-flat Finsler metrics with a specific type of warping.
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Speaker: Carlos Olmos-UNC
Date: Friday, 23/04/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: We will speak about results obtained in cooperation with Jürgen Berndt and with Berndt and Juan Sebastián Rodríguez. A submanifold $\Sigma$ of a Riemannian manifold $M$ is totally geodesic if every geodesic in $\Sigma$ is also a geodesic in $M$. The existence and classification of totally geodesic submanifolds are two fundamental problems in submanifold geometry. We will consider totally geodesic submanifolds of irreducible Riemannian symmetric spaces. No complete classifications are known for totally geodesic submanifolds in irreducible Riemannian symmetric spaces of rank greater than two. The classification problem is hopeless unless one restricts to maximal totally geodesic submanifolds. The classification of such maximal submanifolds is an open major question in this classical subject. In order to gain some insight into this problem there were proposed some simplifications. In 1980, Onishchik introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank less than or equal to $2$, but for higher rank it was unclear how to tackle the problem. Some years ago we proposed a new and geometric approach to the index problem based on the study of the slice representation. We found and application of Simons theorem on holonomy systems that was very important for our purposes. By means of this approach we were able, in several papers. to prove the so-called index conjecture which allowed us to determine the index of all symmetric spaces.
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Speaker: Miriam Telichevsky-UFRGS
Date: Friday, 09/04/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: Some results about the geodesic boundary of minimal surfaces in H^2×R are extended to CMC H surfaces, H between 0 and 1/2. This is a joint work with Félix Nieto Cacais.
Speaker: Luiz San Martin - UNICAMP
Date: Friday, 26/03/2021 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: The talk will describe some aspects of the symplectic geometry of hyperbolic orbitas in a noncompact semi-simple Lie algebra $g$. These orbits are diffeomorphic to the tangent and cotangent bundles of the flag manifolds of $g$. Lefschetz fibrations in the complex case will be discussed. Lagrangean submanifolds with respect to different symplectic forms are constructed. The vector space underlying g has also a structure of a semi-direct Lie algebra $g_s$. Coadjoint orbits of gs are related to the adjoint orbits of $g$.
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Speaker: Simon Salamon - King's College London
Date: Friday, 04/12/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: I shall describe the geometry underlying known examples of explicit metrics with holonomy $SU(2)$ (dimension 4) and $G_2$ (dimension 7), arising from the action of both nilpotent and simple Lie groups.
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Speaker: Paolo Piccione - Universidade de São Paulo
Date: Thursday, 26/11/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: In 1987, Yau posed the question of whether all minimal 2-spheres in a 3-dimensional ellipsoid inside $\mathbb{R}^4$ are planar, i.e., determined by the intersection with a hyperplane. While this is the case if the ellipsoid is nearly round, Haslhofer and Ketover have recently shown the existence of an embedded non-planar minimal $2$-sphere in sufficiently elongated ellipsoids, with min-max methods. Using bifurcation theory and the symmetries that arise in the case where at least two semi-axes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal $2$-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint work with R. G. Bettiol.
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Speaker: María Amelia Salazar - Universidade Federal Fluminense
Date: Friday, 20/11/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: The foundation of Lie theory is Lie's three theorems that provide a construction of the Lie algebra associated to any Lie group; the converses of Lie's theorems provide an integration, i.e. a mechanism for constructing a Lie group out of a Lie algebra. The Lie theory for groupoids and algebroids has many analogous results to those for Lie groups and Lie algebras, however, it differs in important respects: one of these aspects is that there are Lie algebroids which do not admit any integration by a Lie groupoid. In joint work with Cabrera and Marcut, we showed that the non-integrability issue can be overcome by considering local Lie groupoids instead. In this talk I will explain a construction of a local Lie groupoid integrating a given Lie algebroid and I will point out the similarities with the classical theory for Lie groups and Lie algebras.
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Speaker: Matheus Vieira - Universidade Federal do Espírito Santo
Date: Thursday, 12/11/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: We consider the Balmuş-Montaldo-Oniciuc's conjecture in the case of hemispheres. We prove that a compact non-minimal biharmonic hypersurface in a hemisphere of $S^{n+1}$ must be the small hypersphere $S^n (1/\sqrt{2})$, provided that $n^2-H^2$ does not change sign.
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Speaker: Asun Jiménez - Universidade Federal Fluminense
Date: Friday, 06/11/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: In this talk we will study isolated singularities of graphs whose mean and Gaussian curvature satisfy the elliptic linear relation $2\alpha H+\beta K=1$, $\alpha^2+\beta>0$. This family of surfaces includes convex and non-convex singular surfaces and also cusp-type surfaces. We determine in which cases the singularity is in fact removable, and classify non-removable isolated singularities in terms of regular analytic strictly convex curves in $S^2$. This is a joint work with João P. dos Santos.
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Speaker: Álvaro K. Ramos - Universidade Federal do Rio Grande do Sul
Date: Thursday, 29/10/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: Recall that $\mathbb{E}(-1,\tau)$ is a homogeneous space with four-dimensional isometry group which is given by the total space of a fibration over $\mathbb{H}^2$ with bundle curvature $\tau$. Given a finite collection of simple closed curves in $\partial_{\infty}\mathbb{E}(-1,\tau)$, we provide sufficient conditions on $\Gamma$ so that there exists an area minimizing surface $\Sigma$ in $\mathbb{E}(-1,\tau)$ with asymptotic boundary $\Gamma$. We also present necessary conditions for such a surface $\Sigma$ to exist. This is joint work with P. Klaser and A. Menezes.
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Speaker: Anna Fino - Università di Torino
Date: Friday, 23/10/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: A Hermitian metric on a complex manifold is balanced if its fundamental form is co-closed. An important tool for the study of balanced manifolds is the Hull-Strominger system. In the talk I will review some general results about balanced metrics and present new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. The talk is based on a joint work with G. Grantcharov and L. Vezzoni.
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Speaker: Daniel Fadel - Universidade Federal Fluminense
Date: Thursday, 15/10/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: G$_2$-geometry is a very rich and vast subject in Differential Geometry which has been seeing a lot of progress in the last two decades. There are by now very powerful methods that produce millions of examples of G$_2$ holonomy metrics on the compact setting and infinitely many on the non-compact setting. Besides these fruitful advances, at present, there is no systematic understanding of these metrics. In fact, a very important problem in G$_2$-geometry is to develop methods to distinguish G$_2$-manifolds. One approach intended at producing invariants of G$_2$-manifolds is by means of higher dimensional gauge theory. G$_2$-monopoles are solutions to a first order nonlinear PDE for pairs consisting of a connection on a principal bundle over a noncompact G$_2$-manifold and a section of the associated adjoint bundle. They arise as the dimensional reduction of the higher dimensional Spin$(7)$-instanton equation, and are special critical points of an intermediate energy functional related to the Yang-Mills-Higgs energy.
Donaldson-Segal (2009) suggested that one possible approach to produce an enumerative invariant of (noncompact) G$_2$-manifolds is by considering a ``count" of G$_2$-monopoles and this should be related to conjectural invariants ``counting" rigid coassociate (codimension 3 and calibrated) cycles. Oliveira (2014) started the study of G$_2$-monopoles providing the first concrete non-trivial examples and giving evidence supporting the Donaldson-Segal program by finding families of G$_2$-monopoles parametrized by a positive real number, called the mass, which in the limit when such parameter goes to infinity concentrate along a compact coassociative submanifold. In this talk I will explain some recent results, obtained in collaboration with Ákos Nagy and Gonçalo Oliveira, which show that the asymptotic behavior satisfied by the examples are in fact general phenomena which follows from natural assumptions such as the finiteness of the intermediate energy. This is a very much needed development in order to produce a satisfactory moduli theory and making progress towards a rigorous definition of the putative invariant. Time permitting, I will mention some interesting open problems and possible future directions in this theory.
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Speaker: Ivan Struchiner - Universidade de São Paulo
Date: Friday, 09/10/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: I will discuss the problem of obtaining a "Holonomy Groupoid" for a singular Riemannian foliation (SRF). Throughout the talk I will try to explain why we want to obtain such a Lie groupoid by stating results which are valid for regular foliations and how they can be obtained from the Holonomy groupoid of the foliation. Although we do not yet know how to associate a holonomy groupoid to any SRF, we can obtain the holonomy groupoid of the linearization of the SRF in a tubular neighbourhood of (the closure of) a leaf. I will explain this construction. I will not assume that the audience has prior knowledge of Singular Riemannian Foliations or of Lie Groupoids and will try to make the talk accessible to a broad audience.
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Speaker: Mariel Sáez Trumper - Pontificia Universidad Católica de Chile
Date: Thursday, 01/10/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: The network flow is a system of parabolic differential equations that describes the motion of a family of curves in which each of them evolves under curve-shortening flow. This problem arises naturally in physical phenomena and its solutions present a rich variety of behaviors. The goal of this talk is to describe some properties of this geometric flow and to discuss an alternative proof of short-time existence for non-regular initial conditions. The methods of our proof are based on techniques of geometric microlocal analysis that have been used to understand parabolic problems on spaces with conic singularities. This is joint work with Jorge Lira, Rafe Mazzeo, and Alessandra Pluda. arXiv:2008.07004.
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Speaker: Mario Garcia Fernandez - Universidad Autónoma de Madrid
Date: Friday, 25/09/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: The generalized Ricci flow equation is a geometric evolution equation which has recently emerged from investigations into mathematical physics, Hitchin’s generalized geometry program, and complex geometry. The generalized Ricci flow can regarded as a tool for constructing canonical metrics in generalized geometry and complex non-Kähler geometry, and extends the fundamental Hamilton/Perelman theory of Ricci flow. In this talk I will give an introduction to this topic, with a special emphasis on examples and geometric aspects of the theory. Based on joint work with Jeffrey Streets (UC Irvine).
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Speaker: Lino Grama - Universidade Estadual de Campinas
Date: Thursday, 17/09/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: In this talk we will discuss the classification of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible sub-representations. We also discuss some phenomena in real flag manifolds that can not happen in complex flag manifolds. This includes the non-existence of invariant Einstein metric and examples of non-diagonal Einstein metrics. This is a joint work with Brian Grajales.
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Speaker: Elizabeth Gasparim - Universidad Católica del Norte
Date: Friday, 11/09/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: I will explain the new concepts of graft surgeries which allow us to modify surfaces, Calabi-Yau threefolds and vector bundles over them, producing a variety of ways to describe local characteristic classes. In particular, we generalize the construction of conifold transition presented by Smith-Thomas-Yau. This is joint work with Bruno Suzuki.
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Speaker: Jason Lotay - University of Oxford
Date: Thursday, 03/09/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
Abstract: Deformed G$_2$-instantons are special connections occurring in G$_2$ geometry in $7$ dimensions. They arise as “mirrors” to certain calibrated cycles, providing an analogue to deformed Hermitian-Yang-Mills connections, and are critical points of a Chern-Simons-type functional. I will describe an elementary construction of the first non-trivial examples of deformed G$_2$-instantons, and their relation to $3$-Sasakian geometry, nearly parallel G$_2$-structures, isometric G$_2$-structures, obstructions in deformation theory, the topology of the moduli space, and the Chern-Simons-type functional.
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Speaker: Daniel J. Pons - Universidad Andrés Bello, Santiago
Date: Thursday, 20/08/2020, 14h00 GMT-3 (Brasilia - Buenos Aires)
Abstract: We review some of V.I. Arnold’s ideas on diffeomorphism groups on manifolds. When the underlying manifold is the circle, we study the geometry of such a group endowed with some metrics.
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Speaker: Eduardo Rosinato Longa - Universidade de São Paulo
Date: Friday, 14/08/2020, 14h00 GMT-3 (Brasilia - Buenos Aires)
Abstract: Systolic Geometry dates back to the late 1940s, with the work of Loewner and his student, Pu. This branch of differential geometry received more attention after the seminal work of Gromov, where he proved his famous systolic inequality and introduced many important concepts. In this talk I will recall the notion of systole and present some sharp systolic inequalities for free boundary surfaces in 3-manifolds.
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Speaker: Nicolau S. Aiex - University of Auckland
Date: Thursday, 06/08/2020, 14h00 GMT-3 (Brasilia - Buenos Aires)
Abstract: We will talk about the compactness of the space of CMC surfaces on ambient manifolds with positive Ricci curvature and convex boundary. We characterize compactness based on geometric information on the surface. This is analogous to a result of Fraser-Li on free boundary minimal surfaces, however, the lack of a Steklov eigenvalue lower bound makes the proof fairly different. The proof is an adaptation of White's proof of the compactness of stationary surfaces of parametric elliptic functionals. This is a joint work with Han Hong.
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Speaker: Raquel Perales - Universidad Nacional Autónoma de México
Date: Friday, 31/07/2020, 14h00 GMT-3 (Brasilia - Buenos Aires)
Abstract: [Based on join work with Allen-Sormani and Cabrera Pacheco-Ketterer] Given a Riemannian manifold $M$ and a pair of Riemannian tensors $g_0 \leq g_j$ on $M$ it follows that $\vol_0(M) \leq \vol_j(M)$. Furthermore, the volumes are equal if and only if $g_0=g_j$. In this talk I will show that for a sequence of Riemannian metrics $g_j$ defined on $M$ that satisfy $g_0\leq g_j$, $\diam (M_j) \leq D$ and $\vol(M_j)\to \vol(M_0)$ then $(M,g_j)$ converge to $(M,g_0)$ in the volume preserving intrinsic flat sense. I will present examples demonstrating that under these conditions we do not necessarily obtain smooth, $C^0$ or Gromov-Hausdorff convergence. Furthermore, this result can be applied to show the stability of graphical tori.
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Speaker: Romina M. Arroyo - Universidad Nacional de Córdoba - CONICET
Date: Thursday, 23/07/2020, 14h00 GMT-3 (Brasilia - Buenos Aires)
Abstract: One of the most important challenges of Riemannian geometry is to understand the Ricci curvature tensor. An interesting open problem related with it is to find a Riemannian metric whose Ricci curvature is prescribed, that is, a Riemannian metric $g$ and a real number $c>0$ satisfying \[ \operatorname{Ric} (g) = c T, \] for some fixed symmetric $(0, 2)$-tensor field $T$ on a manifold $M,$ where $\operatorname{Ric} (g)$ denotes the Ricci curvature of $g.$ The aim of this talk is to discuss this problem within the class of naturally reductive metrics when $M$ is a simple Lie group, and present recently obtained results in this setting. This talk is based on joint works with Mark Gould (The University of Queensland) Artem Pulemotov (The University of Queensland) and Wolfgang Ziller (University of Pennsylvania).
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Speaker: Gregório Pacelli Bessa - Universidade Federal do Ceara
Date: Friday, 17/07/2020, 14h00 GMT-3 (Brasilia - Buenos Aires)
Abstract: I will talk about a stochastic half-space theorem for minimal surfaces of $\mathbb{R}^{3}$ . More precisely; Thm. $\Sigma$ be a complete minimal surface with bounded curvature in $\mathbb{R}^{3}$ and $M$ be a complete, parabolic (recurrent) minimal surface immersed in $\mathbb{R}^{3}$. Then $\Sigma \cap M \neq \emptyset$ unless they are parallel planes. This is a work in progress with Luquesio Jorge and Leandro Pessoa.
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Speaker: Emilio Lauret - Universidad Nacional del Sur
Date: Thursday, 09/07/2020, 14h00 GMT-3 (Brasilia - Buenos Aires)
Abstract: Given $G$ a compact Lie group and $K$ a closed subgroup of it, we will study whether the functional $\lambda_1(G/K,g) \textrm{diam}(G/K,g)^2$ is bounded by above among $G$-invariant metrics $g$ on the (compact) homogeneous space $G/K$. Here, $\textrm{diam}(G/K,g)$ and $\lambda_1(G/K,g)$ denote the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to $(G/K,g)$
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Speaker: Umberto Hryniewicz - RWTH Aachen University
Date: Friday, 03/07/2020, 14h00 GMT-3 (Brasilia - Buenos Aires)
Abstract: This talk is intended to survey applications of pseudo-holomorphic curves to Reeb flows in dimension three, with an eye towards geometry. For the geometer the interest stems from the fact that geodesic flows are particular examples of Reeb flows. I will discuss characterizations of lens spaces, existence/non-existence of closed geodesics with a given knot type under pinching conditions on the curvature, sharp systolic inequalities, existence of elliptic dynamics (in relation to an old conjecture of Poincaré), and generalizations of Birkhoff's annular global surface of section for positively curved 2-spheres.
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Speaker: Marcos Petrúcio Cavalcante - Universidade Federal de Alagoas
Abstract: Click here.
Date: Thursday, 25/06/2020 - 14h:00 GMT-3 (Brasilia - Buenos Aires)
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Speaker: Viviana del Barco - UPSaclay / UNR
Abstract: Click here.
Date: Friday, 19/06/2020 - 14h:00 GMT 3 (Brasilia - Buenos Aires)
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Speaker: Mircea Petrache - Pontificia Universidad Católica de Chile
Date: Thursday, 11/06/2020 - 14h:00 GMT 3 (Brasilia - Buenos Aires)
Abstract: In his fundamental 1987 paper on the geometry of measures, Preiss posed the problem of classifying uniform measures in $d$-dimensional Euclidean space, a question at the interface of measure theory and differential geometry. A uniform measure is a positive measure such that for all $r>0$, all balls of radius r with center in the support of the measure, are given equal masses. It was proved by Kirchheim-Preiss that a uniform measure in $\mathbb{R}^d$ is a multiple of the k-dimensional Hausdorff measure restricted to a $k$-dimensional analytic variety. This establishes the link to differential geometry. An important class of uniform measures are G-invariant measures, for $G$ any subgroup of isometries of Euclidean space. These are called homogeneous measures. Intriguing examples of non-homogeneous uniform measures do exist (the surface area of the 3D cone $x^2=y^2+w^2+z^2$ in $\mathbb{R}^4$ is one), but they are not well understood, making Preiss' classification question is still widely open. After a historical survey, I will describe a recent joint paper with Paul Laurain, about uniform measures of dimension $1$ in $d$-dimensional Euclidean space: we prove by a direct approach that these are all given by at most countable unions of congruent helices or of congruent toric knots. In particular, $1$-dimensional uniform measures with connected support are homogeneous.
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Speaker: John Alexander Cruz - Universidad Nacional de Colombia
Date: Friday, 05/06/2020 - 14h:00 GMT 3 (Brasilia - Buenos Aires)
Abstract: We will revisit the computations of Stokes matrices for tt*-structures done by Cecotti and Vafa in the 90's in the context of Frobenius manifolds and the so-called monodromy identity. We will argue that those cases provide examples of non-commutative Hodge structures of exponential type in the sense of Katzarkov, Kontsevich and Pantev.
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Speaker: Lucas Ambrozio - University of Warwick
Date: Thursday, 28/05/2020 - 14h:00 GMT 3 (Brasilia - Buenos Aires)
Abstract: The word "systole" is commonly used in Geometry to denote the infimum of the length of homotopically non-trivial loops in a compact Riemmanian manifold M. In a generalised sense, we may use it also to refer to the infimum of the k-dimensional volume of a class of k-dimensional submanifolds that represent some non-trivial topology of M. In this talk, we will discuss some inequalities comparing the systole to other geometric invariants, e.g. the total volume of M. After reviewing in details the celebrated inequality of Pu regarding the systole of Riemannian projective planes, we will discuss its generalisations to higher dimensions. This is joint work with Rafael Montezuma.
Speaker: Adrián Andrada - Universidad Nacional de Córdoba
Date: Friday, 22/05/2020 - 14h:00 GMT 3 (Brasilia - Buenos Aires)
Abstract: Abelian complex structures on Lie groups have proved to be very useful in several areas of differential and complex geometry. In particular, an abelian hypercomplex structure on a Lie group G (that is, a pair of anticommuting abelian complex structures), together with a compatible inner product, gives rise to an invariant hyperKähler with torsion (HKT) structure on G. This means that G admits a (unique) metric connection with skew-symmetric torsion (called the Bismut connection) which parallelizes the hypercomplex structure. In this talk we move to the odd-dimensional case and we introduce the notion of abelian almost contact structures on Lie groups. We study their properties and their relations with compatible metrics. Next we consider almost 3-contact Lie groups where each almost contact structure is abelian. We study their main properties and we give their classification in dimension 7. After adding compatible Riemannian metrics, we study the existence of a certain type of metric connections with skew symmetric torsion, introduced recently by Agricola and Dileo and called canonical connections. We provide examples of such groups in each dimension 4n+3 and show that they admit co-compact discrete subgroups, which give rise to compact almost 3-contact metric manifolds equipped with canonical connections.
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Speaker: Gonçalo Oliveira - Universidade Federal Fluminense
Date: Thursday, 14/05/2020 - 14h:00 GMT 3 (Brasilia - Buenos Aires)
Abstract: (All is joint work with Jason Lotay) Some years ago Dominic Joyce set up a conjectural program to better understand Fukaya categories in terms of Bridgeland stability conditions and Lagrangian mean curvature flow. In this talk, I will explain how Jason Lotay and myself have proved/are proving (circle invariant) versions of some of Joyce's conjectures by drawing some pictures on the plane.
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Speaker: Guillermo Henry - Universidad de Buenos Aires
Date: Friday, 08/05/2020 - 14h:00 GMT 3 (Brasilia - Buenos Aires)
Abstract: In this talk we will discuss the relationship between isoparametric functions on closed Riemannian manifolds and solutions of the Yamabe equation. I will show some results on the existence and multiplicity of positive and nodal solutions (i.e., alternating sign solutions) of the Yamabe equation that have the property of being constant along the leaves on an isoparametric foliation.
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Speaker: Henrique N. Sá Earp - Universidade estadual de Campinas
Date: Friday, 24/04/2020 - 14h:00 GMT 3 (Brasilia)
Abstract: Arxiv
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