The Eighth Congress of Romanian Mathematicians

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List of talks

I. Algebra and Number Theory

Special session: Local rings and homological algebra. Special session dedicated to Prof. Nicolae Radu

II. Algebraic, Complex and Differential Geometry and Topology

Special session: Geometry and Topology of Differentiable Manifolds and Algebraic Varieties

III. Real and Complex Analysis, Potential Theory

IV. Ordinary and Partial Differential Equations, Variational Methods, Optimal Control

Special session: Optimization and Games Theory

V. Functional Analysis, Operator Theory and Operator Algebras, Mathematical Physics

Special session: Spectral Theory and Applications in Mathematical Physics

Special session: Dynamical Systems and Ergodic Theory

VI. Probability, Stochastic Analysis, and Mathematical Statistics

VII. Mechanics, Numerical Analysis, Mathematical Models in Sciences

Special session: Mathematical Modeling of Some Medical and Biological Processes

Special session: Mathematical Models in Astronomy

VIII. Theoretical Computer Science, Operations Research and Mathematical Programming

Special session: Logic in Computer Science

IX. History and Philosophy of Mathematics

Algebraic, Complex and Differential Geometry and Topology 

(this list is in updating process)

Mathematical Institute "O.Mayer' Iasi and The University Alexandru Ioan Cuza from Iasi, Romania
Title: Some foliations on the cotangent bundle (details)
A Cartan space is a manifold whose cotangent bundle is endowed with a smooth function $K$ which is positively homogeneous of degree $1$ in momenta. Then the vertical distribution (the kernel of the differential of the projection of the cotangent bundle on its base manifold ) becomes a semi Riemannian foliation whose transversal distribution is completely determined by $K$ and is orthogonal on the vertical distribution with respect to a semi Riemannian metric of Sasaki type. In the same framework there exist and another foliations on the cotangent bundle. One is that defined by the level surfaces of the function $K$. One determines various connections associated to these two foliations and some properties of them are pointed out.
BEJAN Cornelia - Livia
Seminarul Matematic "Al. Myller", Romania
Title: Parallel second order tensors on Vaisman manifolds (details)
We present some aspects on Ricci solitons from our recent works.
BERCEANU Barbu Rudolf
Simion Stoilow Institute of Mathematics of the Romanian Academy, Romania
Title: De la alfabetul Artin la alfabetul Garside (details)
Schimbind generatorii Artin $\{x_1,x_2,\ldots,x_{n-1}\}$ ai grupului braid $B_n$ cu generatorii $\{\delta |\Delta_n\}$ se obtine un sistem finit de relatii derivate ale lui $B_n$.
Universitatea Alexandru Ioan Cuza, Romania
Title: Projective deformations for Finsler functions (details)
The geometry of a systems of second order ordinary differential equations (SODE) $$ \frac{d^2 x^i}{dt^2}+ 2G^i\left( x, \frac{dx}{dt}\right)=0,$$ on some configuration manifold $M$, is determined by the (differential) properties of functions $G^i$, and it can be: affine, Riemannian, Finslerian or Lagrangian. A Finsler function, $F(x, \dot{x})$, is given by a family of Minkowski norms in each tangent space of the manifold. In the Finslerian case, the functions $G^i$ are $2$-homogeneous in $\dot{x}=dx/dt$, and this property allows for reparameterizations of the system. Such reparameterization (\emph{projective deformation}) can change substantially the geometry of the system. In this talk, I will discuss the behaviour of a (SODE) under projective deformations, regarding some geometric properties: Finsler metrizability, curvature and isotropy. A special attention will pe paid to Hilbert's fourth problem, which asks to determine and study all Finsler metrics that are projectively equivalent to the standard flat metric.
Ohio State University, United States
Title: Refinements of homology provided by a real or angle valued map (details)
To any pair (X,f), X compact ANR and f a real (angle) valued map defined on X and any nonnegative integer $r$ we assign: (1) a finite configuration of points ''z'' with multiplicities $delta^f_r(z)$ located in the complex plane and (2) a finite configuration of vector spaces $hat delta^f_r(z)$ indexed by the same $z'$ s in analogy with (1) the configuration of eigenvalues and of (2)generalized eigenspaces of a linear operator in a finite dimensional complex vector space. The analogy goes quite far as long as the formal properties are concerned and becomes particularly subtle in the case of an angle valued map (involving L-2 topology). The basic properties /implications are discussed. The configurations $delta^f_r$'s are effectively computable in case that $X$ is a finite simplicial complex and $f$ a simplicial map and enjoy remarkable properties promising application in and outside mathematics.
Ohio State University, U.S.A.
Title: Monodromy / Alexander rational function of a circle valued map (details)
Abstract: I will provide an alternative presentation of the monodromy of $(X; xiin H^1(X;mathbb Z)$ based on the linear algebra of "linear relations".This presentation is a source of new invariants derived from any homology/ cohomology type of vector valued homotopy functor. The Alexander polynomial of a knot is a particular example.
University of Wisconsin--Madison, USA
Title: Towards a new algebraic proof of the Barannikov-Kontsevich theorem (details)
We present a new approach to an algebraic proof of a claim of Barannikov-Kontsevich, which was first proved with analytic methods by Sabbah. This result is conceptually the analogue of the Hodge-de Rham degeneration statement (which applies for complex Kahler manifolds), but applied to a dg category of matrix factorizations. Our proof relies on reducing to positive characteristic and then applying our earlier results on formality of derived intersections in Azumaya spaces (spaces endowed with an Azumaya algebra).
CIOBAN Mitrofan
Tiraspol State University, Republic of Moldova
We consider general distance $d$ on a space $X$ with the condition: $d(x,y) + d(y,x) = 0$ if and only if $x = y$. The distance $d$ is called: an H-distance if any convergent sequence has a unique limit; a wH-distance if any Cauchy convergent sequence has a unique limit. If $g$ is a mapping of $X$ into itself, the distance $d$ is $g$-bounded if for any point $x$ from $X$ there exists a number $k(x) > 0$ such that $d(x, g^n(x)) + d( g^n(x),x)< k(x)$ for any natural number $n$. Let $g$ be a contraction mapping into itself of a distance space $(X, d)$. Then: if a distance space $(X, d)$ is $g$-bounded, then any Picard sequence is Cauchy; if $(X, d)$ is a complete $g$-bounded wH-distance space, then $g$ has a unique fixed point; if $(X, d)$ is a complete $b$-metric space, then the space $(X, d)$ is $g$-bounded and $g$ has a unique fixed point. The S. Banach contraction principle and the theorems of I. A. Bakhtin, J. Matkowski, M. Jleli and B. Samet are extended. Some examples are constructed.
"Al. I. Cuza" University, Iasi, Romania
Title: Geometric inverse problems in Lagrangian mechanics (details)
The classic inverse problem of Lagrangian mechanics requires to find the necessary and sufficient conditions, which are called emph{Helmholtz conditions}, such that a given system of second order ordinary differential equations (SODE) is equivalent to the Euler-Lagrange equations of some regular Lagrangian function. In this talk we discuss the inverse problem of Lagrangian systems with non-conservative forces. Locally, the problem can be formulated as follows. We consider a SODE in normal form begin{eqnarray} frac{d^2x^i}{dt^2} + 2G^ileft(x, dot{x}right)=0 label{sode} end{eqnarray} and an arbitrary covariant force field $sigma_i(x, dot{x})dx^i$. We will provide necessary and sufficient conditions, which we will call emph{generalized Helmholtz conditions}, for the existence of a Lagrangian $L$ such that the system eqref{sode} is equivalent to the Lagrange equations begin{eqnarray} frac{d}{dt}left(frac{partial L}{partial dot{x}^i}right) - frac{partial L}{partial x^i} = sigma_i(x,dot{x}). label{lagrange_eq} end{eqnarray} The general theory is applied to some particular cases, for dissipative and respectively gyroscopic forces. One main result is that any SODE on a $2$-dimensional manifold is of dissipative type. We provide examples where the proposed generalized Helmholtz conditions, expressed in terms of a semi-basic $1$-form, can be integrated and the corresponding Lagrangian and Lagrange equations can be found.
Technical University of Civil Engineering Bucharest, Romania
Title: An equivariant generalization of the Segal's finiteness theorem (details)
\documentclass[a4paper,12pt]{article} \usepackage{amsthm} \newtheorem*{theorem}{ Theorem} %opening \title{AN EQUIVARIANT GENERALIZATION OF THE SEGAL'S FINITENESS THEOREM } \author{ Cristian Costinescu} \date{} \begin{document} \maketitle \begin{abstract} A very useful result for calculations in equivariant K-theory is the following (due to Segal): \begin{theorem} If X is a locally G – contractible compact G – space such that the orbit space X/G has finite covering dimension, then $ K^*_G(X) $ is a finite R(G)-module; here G is a compact Lie group and by R(G) one denotes the representation ring of G. \end{theorem} In this paper we consider a generalization of the Segal's finiteness theorem to G-cohomology theories defined on a suitable category of G-spaces. For obtaining that we are led to consider G-cohomology theories $h^*_G$ which are "complete" with respect to a family S of closed subgroups of G. The completeness allows us to induct up from conditions on the associated H-cohomology theories (where $H\in S$), to obtain conclusions about $h^*_G$ . The appropriated generalization of the Segal's finiteness theorem can then be stated in terms of conditions concerning the associated cohomology theories $ h^*_H$. The tool used is the generalized Atiyah-Hirzebruch spectral sequence. \end{abstract} \end{document}
Moldova State University, Republic of Moldova
Title: Hyperbolic manifolds and their representations by lens polytopes (details)
In topology a three-dimensional ($n$-dimensional) manifold is often given by the indication of the way how to identify pairwise faces of polytopes of some homogeneous complex. Poincare noticed that one polytope is sufficient. In [1] and in the present work, we discuss an ''intermediate'' way to represent the manifold by lens polytopes. We start with cells complexes over regular (semiregular or $k$-regular) maps on totally geodesic hyperbolic submanifolds, named compact lens polytopes, and indicate the pairwise faces of lens polytope that lead to hyperbolic manifolds. This geometric construction will be illustrated by examples. 1. F.~L.~Damian, V.~S.~Makarov. {\it On lens polytopes.} International Seminar on Discrete Geometry. State Univ. Moldova, Chisinau. P. 32--35, 2002.
Title: (Multi)nets and monodromy (details)
The existence of non-trivial monodromy for the comomology of the Milnor fiber F associated to a complex hyperplane arrangement seems to be connected to the existence of a symmetric structure on the intersection lattice of the arrangement. We present instances of this occurence and describe the (combinatorial) monodromy action on $H^1(F)$, in relation to Aomoto-Betti numbers.
MAXIM Laurentiu
University of Wisconsin-Madison, USA
Title: Motivic infinite cyclic covers (details)
To an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold we associate (assuming certain finiteness conditions are satisfied) an element in the equivariant Grothendieck ring of varieties, called motivic infinite cyclic cover, which satisfies birational invariance. Our construction provides a unifying approach for the Denef-Loeser motivic Milnor fibre of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively. This is joint work with M. Gonzalez Villa and A. Libgober.
Mathematical Institute O. Mayer, Romanian Academy, Romania
Title: Lagrangian and Hamiltonian Geometries. Applications to Analytical Mechanics (details)
The purpose of this talk is to provide a short presentation of the geometrical theory of Lagrange and Hamilton spaces as well as to define and investigate some new Analytical Mechanics. It is largely recognized that a rigorous geometrical theory of conservative and nonconservative mechanical systems can not be constructed without the use of the geometry of the tangent and cotangent bundle of the configuration space. Such a theory can be raised based on the Lagrangian and Hamiltonian geometries. And conversely, the construction of these geometries relies on the mechanical principle and on the Legendre transformation. In the last thirty five years, geometers, mechanicians and physicists from all over the world worked in the field of Lagrange or Hamilton geometries and their applications. We mention only few names: P.L. Antonelli , M. Anastasiei , G.~S. Asanov , A. Bejancu , I. Buc\u{a}taru , M. Crampin , R. S. Ingarden , S. Ikeda , M. de Leon , M. Matsumoto , H. Rund , H. Shimada , P. Stavrinos , L. Tamassy . The Lagrangian and Hamiltonian geometries are useful also for applications in Variational calculus, Mechanics, Physics, Biology etc. Finsler geometry as well as the Riemannian geometry are the geometries of particular Lagrangians whose dual by the Legendre transformation define interesting geometries on the cotangent bundle. The following topics are reviewed: \begin{itemize} \item[$1^\circ$] A solution of the problem of geometrization of the classical nonconservative mechanical systems, whose external forces depend on velocities, based on the differential geometry of velocity space. \item[$2^\circ$] The introduction of the notion of Finslerian mechanical system. \item[$3^\circ$] The definition of Cartan mechanical system. \item[$4^\circ$] The study of theory of Lagrangian and Hamiltonian mechanical systems by means of the geometry of tangent and cotangent bundles. \item[$5^\circ$] The geometrization of the higher order Lagrangian and Hamiltonian mechanical systems. \item[$6^\circ$] The determination of the fundamental equations of the Riemannian mechanical systems whose external forces depend on the higher order accelerations. \end{itemize}
The Ohio State University, USA
Title: Modular geometry on noncommutative tori (details)
The concept of intrinsic curvature, which lies at the very core of classical geometry, has only lately begun to be understood in the noncommutative framework. I will present recent results in this direction for noncommutative tori, obtained in joint works with A. Connes and with M. Lesch, which illustrate both the challenges and the rewards of doing geometry on noncommutative spaces.
University of Connecticut, USA
Title: Four dimensional Ricci solitons (details)
Shrinking Ricci solitons are self similar solutions of the Ricci flow and arise as Type I singularities of the flow. They are classified in dimension two and three, by Hamilton, Ivey and Perelman's work. I will present some recent results about the asymptotic geometry of four dimensional complete noncompact shrinking Ricci solitons. This is based on joint work with Jiaping Wang.
University of Notre Dame, USA
Title: A stochastic Gauss-Bonnet-Chern formula. (details)
A Gaussian ensemble of smooth sections of a smooth vector bundle E determines a metric and a compatible connection on E. If the bundle is oriented, and the base manifold M is compact and oriented, then the zero locus of a random section in the ensemble is a random current in M and we prove that the expectation of this current is equal to the current determined by the Euler form associated to the above connection by the Chern-Weil construction.
PAUNESCU Laurentiu
The University of Sydney, Australia
Title: Proof of Whitney fibering conjecture (details)
In this paper we show Whitney fibering conjecture in the real and complex, local analytic and global algebraic cases. For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of Zariski equisingularity, we show the existence of Whitney stratified fibration, satisfying the conditions (b) of Whitney and (w) of Verdier. Our construction is based on Puiseux with parameter theorem and a generalization of Whitney interpolation. For algebraic sets our construction gives a global stratification.
POPESCU Clement Radu
Institutul de Matematica "Simion Stoilow" al Academiei Romane, Romania
Title: Flat connections and resonance varieties of rank larger than 1 (details)
A way of studying the topological and geometrical properties of a connected CW-complex X, is to study the representation variety of the fundamental group $\pi_1(X)$ into a linear algebraic group G. The set of $\underline{g}$ - valued flat connections, $\underline{g}$ - being the Lie algebra of the group G, an infinitesimal version of the representation variety has a filtration by resonance varieties associated to a representation of $\underline{g}$. I present results concerning these resonance varieties of rank larger than 1.
Vanderbilt University, SUA
Title: Counting real rational curves on K3 surfaces (details)
Real enumerative invariants of real algebraic manifolds are derived from counting curves with suitable signs. Based on a joint work with V. Kharlamov, I will discuss the case of counting real rational curves on simply connected complex projective surfaces with zero first Chern class (K3 surfaces), equipped with an anti-holomorphic involution. An adaptation to the real setting of a formula due to Yau and Zaslow will be presented. The proof passes through results of independent interest: a new insight into the signed counting, and a formula computing the Euler characteristic of the real Hilbert scheme of points on a $K3$ surface, the real version of a result due to G"ottsche.
Tokai University, Japan
Title: Convexity on Finsler manifolds (details)
We will discuss some convexity related problems on Finsler manifolds. In special we will focus on the geometrical and topological information provided by a convex function defined on a Finsler manifold.
University of Nebraska-Lincoln, USA
Title: Symbolic powers and line arrangements (details)
Symbolic powers of ideals play a significant part in algebraic geometry and in commutative algebra, where containment relations between symbolic powers and ordinary powers of ideals have become a focus of interest. In this talk, we consider new algebraic invariants that measure this containment. Examples will focus on the case of ideals of points arising as the singular locus of a planar line arrangement.
SUCIU Alexandru
Northeastern University, USA
Title: Topology of complex line arrangements (details)
I will discuss some recent advances in our understanding of the relationship between the topology, group theory, and combinatorics of an arrangement of lines in the complex plane.
Vanderbilt University, USA
Title: Asymptotically Locally Euclidean Complex Surfaces (details)
Asymptotically locally Euclidean (ALE) scalar flat Kahler surfaces play an important role in the study of the moduli space of constant scalar curvature Kahler metrics on compact complex surfaces. In this talk, we present the classification of ALE Ricci-flat Kahler surfaces, and we also discuss the classification of ALE scalar flat Kahler surfaces.
Université de Lille 1, France
Title: Topology of real polynomial maps (details)
The topology of fibres of a real polynomial function may change due to the behavior at infinity. We focus on the detection of those fibres which are asymptotically atypical.
Yaroslavl State University, Russian Federation
Title: A compactification of moduli of stable vector bundles on a surface by locally free sheaves (details)
The compactification mentioned is obtained when families of Gieseker-stable vector bundles on the surface $S$ are comleted by Gieseker-semistable vector bundles satisfying some additional requirement, on projective schemes of some certain class. We give functorial interpretation of this compactification as moduli space of objects we call semistable admissible pairs. The target result of the talk is the isomorphism of (main components of) the functor of moduli of semistable admissible pairs and (main conponents of) the classical functor of semistable torsion-free coherent sheaves on the surface $S$ which leads to the Gieseker -- Maruyama compactification obtained by adding nonlocally free semistable torsion-free sheaves. This implies the isomorphism of corresponding moduli schemes with possibly nonreduced scheme structures and interprets Gieseker -- Maruyama compactification as a compactification of moduli of semistable vector bundles by locally free sheaves only.
University of Haifa, Israel
Title: Generalized para-Kähler manifolds (details)
We define a generalized almost para-Hermitian structure to be a commuting pair $(mathcal{F},mathcal{J})$ of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-K"ahler structure. This class of structures contains both the classical para-K"ahler structure and the classical K"ahler structure. We show that a generalized almost para-Hermitian structure is equivalent to a triple $(gamma,psi,F)$, where $gamma$ is a (pseudo) Riemannian metric, $psi$ is a $2$-form and $F$ is a complex $(1,1)$-tensor field such that $F^2=Id,gamma(FX,Y)+gamma(X,FY)=0$. We deduce integrability conditions similar to those of the generalized K"ahler structures and give several examples of generalized para-K"ahler manifolds. We discuss submanifolds that bear induced para-K"ahler structures and, on the other hand, we define a reduction process of para-K"ahler structures.
VILCU Costin
IMAR, Romania
Title: Baire categories for Alexandrov surfaces (details)
An {\it Alexandrov surface} is a compact 2-dimensional Alexandrov space with curvature bounded below, without boundary, as defined in \cite{B}. It is known that these surfaces are 2-dimensional topological manifolds. The set $\mathcal{A}(\kappa)$ of all Alexandrov surfaces with curvature bounded below by $\kappa$ is a Baire space, and it has a dense subset of Riemannian surfaces, and a dense subset of $\kappa$-polyhedra \cite{A}. The talk is mainly based on joint works with Jo\"el Rouyer, and will present properties of most surfaces in $\mathcal{A}(\kappa)$, see \cite{Z}, \cite{C}, \cite{D}, \cite{E}, \cite{F}. Here {\it most} means ``all, except those in a first category set''. \begin{thebibliography}{9} \bibitem {Z}K. Adiprasito and T. Zamfirescu, {\it Few Alexandrov surfaces are Riemann}, J. Nonlinar Convex Anal., to appear \bibitem {A}A.D. Aleksandrov and V.A. Zalgaller, {\sl Intrinsic geometry of surfaces}, Transl. Math. Monographs, Providence, RI, Amer. Math. Soc., 1967 \bibitem {B}Y. Burago, M. Gromov and G. Perelman, {\it A. D. Alexandrov spaces with curvature bounded below}, Russian Math. Surveys 47 (1992), 1-58 \bibitem {C}J. Itoh, J. Rouyer and C. V\^{\i}lcu, {\it Moderate smoothness of most Alexandrov surfaces}, Int. J. Math., to appear \bibitem {D}J. Rouyer and C. V\^{\i}lcu, {\it The connected components of the space of Alexandrov surfaces}, in D. Ibadula and W. Veys (eds.), {\sl Bridging Algebra, Geometry and Topology}, Springer Proc. Math. Stat. 96 (2014), 249-254 \bibitem {E}J. Rouyer and C. V\^{\i}lcu, {\it Simple closed geodesics on most Alexandrov surfaces}, Adv. Math., to appear \bibitem {F}J. Rouyer and C. V\^{\i}lcu, {\it Farthest points on most Alexandrov surfaces}, arXiv:1412.1465 [math.MG] \end{thebibliography}