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

Local rings and homological algebra. Special session dedicated to Prof. Nicolae Radu (special session) 

(this list is in updating process)

"Simion Stoilow" Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Title: Towards longer-range topological properties for finite generation of subalgebras (details)
\documentstyle{amsart} \begin{document} Let $A$ be a reduced subalgebra of an algebra $A'$ of finite type over a field $k$. The problem of the finite generation of $A$ is a restatement of the renowned 14-th Hilbert Problem, representing an interplay of Algebra with Geometry and Topology. According to some author's results, there exists a complete topological control about the finite generation of such a subalgebra $A$ when $k = \mathbb C$, as well when $k$ is arbitrary and $A$ is Noetherian. Passing to the associated geometric objects $X^{*} = Spec A$, resp. $X = Spec A'$ ([2]), we have a canonical dominant morphism $f: X \rightarrow X^{*}$ of affine $k$-schemes with $X$ an algebraic $k$-variety and then we are naturally guided to the more general situation of a similar dominant morphism $f: X \rightarrow X^{*}$ of arbitrary ( not necessarily affine ) $k$-schemes. The problem of the algebraization of the $k$-scheme $X^*$ ( i.e. $X^*$ to be exactly an algebraic $k$-variety ) is close related to the "good'' topological properties of the $k$-schemes morphism $f$. In this talk we review a class of such topological properties and center on a possible new situation, suggested by the central Hilbert-Mumford-Nagata Theorem of the Invariant Theory ([3]), as by a topological result due to Prof. M.Ciobanu : namely the case when $f$ is a universally topological quotient morphism. \smallskip References \smallskip 1. A. Constantinescu, {\it Schemes dominated by algebraic varieties and some classes of scheme morphisms.I.II,III}: I, Acta Univ. Apulensis, Math.-Info., {\bf 16}\,(2008), 37 - 51; II, Preprint Ser. in Math., IMAR, Bucharest, ISSN 0250 - 3638, {\bf 8}\,(2010), 36 p. ; III, to appear 2. A. Grothendieck, {\it Elements de geometrie algebrique. I,II}, Publ. Math. IHES, {\bf 4}\,(1960); {\bf 8}\,(1961). 3. D. Mumford, {\it Geometric Invariant Theory}, Springer, 1965. \end{document}
ENESCU Florian
Georgia State University, United States
Title: The Frobenius complexity of a local ring (details)
The talk will outline the notion of Frobenius complexity of a local ring of prime characteristic and discuss various examples. This is joint work with Yongwei Yao.
Georgia Southern University, USA
Title: Gorenstein projective precovers (details)
We consider a right coherent and left n-perfect ring R. We prove that the class of Gorenstein projective complexes is special precovering in the category of unbounded complexes, Ch(R). As a corollary, we show that the class of Gorenstein projective modules is special precovering over such a ring. This is joint work with Sergio Estrada and Sinem Odabasi.
Institute of Mathematics IMAR, Romania
Title: A theorem of Ploski's type (details)
Let ${\bf C}{x}$, $x=(x_1,ldots,x_n)$, $f=(f_1,ldots,f_s)$ be some convergent power series from ${\bf C}{x,Y}$, $Y=(Y_1,ldots,Y_N)$ and $ y$ in {\bf C}[[x]]^N$ with $ y(0)=0$ be a solution of $f=0$. Then Ploski proved that the map $v:B={\bf C}{x,Y}/(f)\rightarrow {\bf C}[[x]]$ given by $Y\rightarrow y$ factors through an $A$-algebra of type $B'={\bf C}{x,Z}$ for some variables $Z=(Z_1,ldots,Z_s)$, that is $v$ is a composite map $B\rightarrow B'\rightarrow {\bf C}[[x]]$. Now, let $(A,m)$ be an excellent Henselian local ring, $ A'$ its completion, $B$ a finite type $A$-algebra and $v:B\rightarrow A'$ an $A$-morphism. Then we show that $v$ factors through an $A$-algebra of type $ A[Z]^h$ for some variables $Z=(Z_1,ldots,Z_s)$, where $A[Z]^h$ is the Henselization of $A[Z]_{(m,Z)}$.
Notheastern University, USA
Title: Intersections and Sums of Gorenstein ideals (details)
A complete local ring of embedding codepth $3$ has a minimal free resolution of length $3$ over a regular local ring. Such resolutions carry a differential graded algebra structure, based on which one can classify local rings of embedding codepth $3$. The Gorenstein rings of embedding codepth $3$ belong to the class called {bf G}$(r)$, which was conjectured not to contain any non Gorenstein rings. In a previous work with Lars W. Christensen and Jerzy Weyman we gave examples and constructed non Gorenstein rings in {bf G}$(r)$, for any $rgeq 2$. We show now that one can get such rings generically, from intersections of Gorenstein ideals. The class of the rings obtained from sums of such ideals will also be discussed.
University of South Carolina, U.S.A.
Title: Totally reflexive modules for Stanley-Reisner rings of graphs (details)
For a Cohen-Macaulay non-Gorenstein ring it is known that either there are infinitely many isomorphism classes of indecomposable totally reflexive modules, or else there are none except for the free modules. However it is not known how to determine which of this situations holds for a given ring. We investigate this question for the case of Stanley-Reisner rings of graphs.