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Seminar "Geometry, Topology and their Applications" by Novosibirsk State University

Proof of the Toponogov Conjecture on complete convex planes.


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Speaker: W. Klingenberg (Durham, England)


Time: December 5, 2022 (17:00 Beijing time, 16:00 Novosibirsk time).


Abstract: In joint work with B Guilfoyle we prove that complete convex embedded surfaces homeomorphic to R^2 need to have at least one umbilic point. Victor Andreevich Toponogov proved this under an additional assumption on the growth of the mean curvature of the surface. His ingenious argument uses comparison with round spheres. We deal with the problem in the quotient space of surfaces in R^3 modulo parallelism. It turns out that this reformulation allows for applying the Theorem of Riemann Roch, where the analytic index counts the number of umbilics in the equivalence class.



Magnetic Curvature and Closed Magnetic Geodesic.


Speaker: V. Assenza (Heidelberg, Germany)


Time: April 25, 2022 (17:00 Beijing time, 16:00 Novosibirsk time).


Abstract: A Magnetic System describes the motion of a charged particle moving on a Riemannian Manifold under the influence of a magnetic field. Trajectories for this dynamics are called Magnetic Geodesics and one of the main tasks in the theory is to investigate the existence of Magnetic Geodesic which are closed. In general, this depends on the magnetic system taken into account and on the topology of the base space. Inspired by the work of Bahri and Taimanov, I will introduce the notion of Magnetic Curvature which is a perturbation of the standard Riemannian curvature due to the magnetic interaction. We will see that Closed Magnetic Geodesic exists when the Magnetic Curvature is positive which happens, for instance, when the magnetic field is sufficiently strong.



Global surfaces of section for Reeb flows on closed 3-manifolds.


Speaker: Marco Mazzucchelli (Lyon, France)


Time: March 28, 2022 (17:00 Beijing time, 16:00 Novosibirsk time).


Abstract: In this talk, which is based on joint work with Gonzalo Contreras, I will sketch a proof of the existence of a global surface of section for any Reeb vector field of a closed 3-manifold satisfying the Kupka-Smale condition. This result implies the existence of global surfaces of section for the Reeb vector fields of C^\infty-generic contact forms on any closed 3-manifold, and for the geodesic vector fields of C^\infty-generic Riemannian metrics on any closed surface. I will discuss a few significant applications of this existence result, and in particular a confirmation of Palis-Smale's stability conjecture for geodesic flows of surfaces: any C^2-structurally stable Riemannian geodesic flow of a closed surface must be Anosov.


Bio: Marco Mazzucchelli is a CNRS researcher at the Unité de Mathématiques Pures et Appliquées of the École normale supérieure de Lyon. His main research interests are dynamical systems, Riemannian geometry, and symplectic topology. He is also co-coordinator of the ANR CoSyDy (Conformally Symplectic Dynamics beyond symplectic dynamics), and member of the ANR Cosy (New challenges in symplectic and contact topology). Mazzucchelli is an editorial board member of the Journal of Fixed Point Theory and Applications.



Quantum computing. Axiomatic model. Examples of algorithms.


Speaker: S.B. Tikhomirov (St. Petersburg State University, Russia).


Time: 2021-11-15 16:00 Novosibirsk time (12:00 Moscow time, 17:00 Beijing time).



Laguerre geometry and its applications in computer numerically controlled machining.


Speaker: M.B. Skopenkov (NRU Higher School of Economics, Institute for Information Transmission Problems RAS, Moscow, Russia).


Time: 2021-11-01 16:00 Novosibirsk time (12:00 Moscow time, 17:00 Beijing time).


Abstract: Motivated by applications in engineering, we provide characterizations of surfaces which are enveloped by a one-parametric family of congruent cones. As limit cases we also address developable surfaces and ruled surfaces. The characterizations are higher-order nonlinear PDEs generalizing the ones by Gauss and Monge. In the process we reconstruct the positions of the cones for a given envelope. The methodology is based on a model of Laguerre geometry, which transforms an envelope to a surface containing a special conic through each point. Most of the talk is explained in figures and is accessible to undergraduate students.

    This is joint work with Pengbo Bo, Michael Barton, Helmut Pottmann.




Old and new on the (non)-existence of complex structures on S^6.


Speaker: I. Agricola (Marburg, Germany)


Time: 2021-04-12 17:00 Beijing time



Non-Abelian generalizations of integrable PDEs and ODEs.


Speaker: V.V. Sokolov (Moscow)


Time: 2021-04-06 21:00 Beijing time


Abstract: A general procedure for non-abelinization of given integrable polynomial differential equation is described. We are considering the NLS type equations as an example. We also find non-abelinating Euler's top.
Results related to the Painlevé-2 and Painlevé-4 equations are discussed.



Fibrations with torus fibres and the toral rank conjecture.


Speaker: M. Amann (Augsburg, Germany)


Time: 2021-01-18 17:00



Foliations on closed 3-dimensional Riemannian manifolds with small modulus of mean curvature of the leaves.


Speaker: D.V. Bolotov (Verkin Institute for Low Temperature Physics and Engineering, Kharkiv, Ukraine)


Time: 2020-12-21 15:00 Beijing time (14:00 Novosibirsk time)


Abstract: We prove that a foliation on a closed 3-dimensional Riemannian manifold must be taut if the modulus of the mean curvature of the leaves is less than a certain constant depending on the volume, injectivity radius, and the maximum value of the sectional curvature of the manifold.



On the spectral problem of a three term difference operator.


Speaker: Rinat Kashaev (University of Geneva, Switzerland; St. Petersburg Department of V.A. Steklov Mathematical Institute, St. Petersburg, Russia)


Time: 2020-06-29 15:00 Beijing time


Abstract: We address the spectral problem of the quantum mechanical operator associated to the quantized mirror curve of the toric (almost) del Pezzo Calabi—Yau threefold called local P^2 in the case of complex values of Planck’s constant. This is joint work with Sergey Sergeev.



Geometric facets of quantization.


Speaker: Leonid Polterovich (Tel Aviv University, Tel Aviv, Israel)


Time: 2020-06-22 15:00 Beijing time


Abstract: I will explain why compatible almost-complex structures on symplectic manifolds correspond to optimal positive quantizations, and discuss classification of geometric quantizations up to conjugation and small error. Joint work with Louis Ioos and David Kazhdan.



Braided Descriptions of Homotopy Groups of the 2-Sphere.


Speaker: Wu Jie (Hebei Normal University, Shijiazhuang, China)


Time: 2020-06-08 15:00 Beijing time


Abstract: In this talk, we will discuss combinatorial and braided descriptions of homotopy groups of the 2-sphere. We will also discuss the connections between relative Lie algebras of Brunnian braid groups over the 2-sphere and the unstable Adams spectral sequence.



Infinite excitation limit: horocyclic chaos.


Speaker: M. Dubashinskiy (Chebyshev Laboratory, St. Petersburg State University)


Time: 2020-06-01 15:00 Beijing time


Abstract: What will be if, given a pure stationary state at a compact hyperbolic surface, we start applying creation operator every $\hbar$ "adiabatic" second? It turns that during adiabatic time comparable to 1 wavefunction will change as a wave traveling with a finite speed (with respect to the adiabatic time), whereas the semiclassical measure of the system will undergo a controllable transformation admitting a description in the spirit of geometric optics. If adiabatic time goes to infinity then, by quantization of Furstenberg Theorem, the system will become quantum uniquely ergodic. Thus, infinite excitation of a closed system leads to quantum chaos.



Action-minimizing methods and invariant Lagrangian Graphs.


Speaker: Alfonso Sorrentino (University of Rome Tor Vergata, Rome, Italy)


Time: 2020-05-25 16:00 Beijing time


Abstract: In this talk I would like to describe some properties of Hamiltonian and Lagrangian systems, with particular attention to the relation between their action-minimizing properties and their dynamics. More specifically, I shall illustrate what kind of information the principle of least Lagrangian action conveys into the study of the integrability of these systems, and, more generally, how this information relates to the existence or to the non-existence of invariant Lagrangian graphs.



Nonlinear Dirac equations on compact spin manifolds.


Speaker: Tian Xu (Tianjin University, Tianjin, China)


Time: 2020-05-18 15:00 Beijing time


Abstract: Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we shall consider analytic aspects of nonlinear Dirac equations on compact spin manifolds. Viavariational theory and blow-up analysis, our main target is to study the existence issue of a conformally invariant spinor field equation, which has a strong relationship with Spinorial Weierstraß representation.



Invariant geometry on nilmanifolds and narrow Lie algebras.


Speaker: D.V. Millionshchikov (Moscow State University, Steklov Mathematical Institute, Moscow, Russia)


Time: 2020-05-11 15:00 Beijing time


Abstract: The problems of invariant geometry on a nilmanifold G/Γ can be reduced to studying the corresponding algebraic structures on a nilpotent Lie algebra g, which corresponds to the one-connected nilpotent Lie group G. As nilpotent Lie algebras we will consider the so-called narrow positively graded Lie algebras g = ⊕ g_i , dim g_i ≤ 2. We will focus on left-invariant: 1) symplectic structures; b) affine structures and c) complex structures on G/Γ, and discuss the various classification lists of narrow positively graded Lie algebras that arose during such research.



Isoperimetric inequalities for Laplace eigenvalues on the sphere and the real projective plane.


Speaker: Alexei V. Penskoi (Moscow State University,  National Research University - Higher School of Economics, Independent University of Moscow, Interdisciplinary Scientific Center J.-V. Poncelet, Moscow, Russia)


Time: 2020-05-04 15:00 Beijing time


Abstract: This talk will be a review on results concerning sharp isoperimetric inequalities for Laplace eigenvalues on surfaces, mainly the sphere and the real projective plane.



A universal topological approach to data science.


Speaker: Wu Jie (Hebei Normal University, Shijiazhuang, China)


Time: 2020-04-27 15:00 Beijing time


Abstract: In this talk, we will report our current research for exploring topology of subgraphs and its applications. We introduce the notion of super-hypergraph, which can be briefly described as Delta set with missing faces, as a model for topological structure of subgraphs.
    The applications are given by providing a unified topological approach to data science. This is a joint work with Professor Jelena Grbic from Southampton.  



Matrix resolvent, Volterra lattice hierarchy, and the modified GUE partition function.


Speaker: Di Yang (University of Science and Technology of China)


Time: 2020-4-20 15:00 Beijing time