The Eighth Congress of Romanian Mathematicians

List of talks

I. Algebra and Number Theory

II. Algebraic, Complex and Differential Geometry and Topology

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

Algebra and Number Theory

(this list is in updating process)

1.
AGORE Ana
Vrije Universiteit Brussel, Belgium
Title: Jacobi and Poisson algebras (details)
Abstract:
Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra $A$ and Frobenius Jacobi algebras as symmetric objects in the category. A characterization theorem for Frobenius Jacobi algebras is given in terms of integrals on Jacobi algebras. For a vector space $V$ a non-abelian cohomological type object ${mathcal J}{mathcal H}^{2} , (V, , A)$ is constructed: it classifies all Jacobi algebras containing $A$ as a subalgebra of codimension equal to ${rm dim} (V)$. Representations of $A$ are used in order to give the decomposition of ${mathcal J}{mathcal H}^{2} , (V, , A)$ as a coproduct over all Jacobi $A$-module structures on $V$. The bicrossed product $P bowtie Q$ of two Poisson algebras recently introduced by Ni and Bai appears as a special case of our construction. A new type of deformations of a given Poisson algebra $Q$ is introduced and a cohomological type object $mathcal{H}mathcal{A}^{2} bigl(P,, Q ~|~ (triangleleft, , triangleright, , leftharpoonup, , rightharpoonup)bigl)$ is explicitly constructed as a classifying set for the bicrossed descent problem for extensions of Poisson algebras. Several examples and applications are provided.
2.
ANTON Marian
CCSU and IMAR, USA and Romania
Title: From class field to arithmetic group cohomology (details)
Abstract:
There are a few known examples of arithmetic groups for which the mod p cohomology is a free module over the ring of Chern classes. A. D. Rahm and M. Wendt have recently conjectured that this property is true for a class of arithmetic groups if the rank of the group is smaller than p and each cohomology class is detected on some finite subgroup. In this talk we present a preliminary report on the current status of their conjecture.
3.
"Simion Stoilow" Institute of Mathematics of Romanian Academy, and University Politehnica of Bucharest, Romania
Title: When Hopf monads are Frobenius (details)
Abstract:
Under suitable exactness assumptions, a Hopf monad $T$ on a monoidal category $\C$ having as right adjoint a Hopf comonad $G$ is shown to be also a Frobenius monad, if $T\I$ and $G\I$ are isomorphic (right) Hopf $T$-modules (in particular, $T\I$ is a Frobenius algebra), where $\I$ denotes the unit object of $\C$. If additionally the underlying base category is autonomous, then a Hopf monad $T$ becomes also a Frobenius monoidal functor.
4.
BONCIOCAT Nicolae Ciprian
Simion Stoilow Institute of Mathematics of the Romanian Academy, Romania
Title: Some applications of the resultant to factorization problems (details)
Abstract:
We present a method to obtain information on the factorization of two polynomials using the canonical decomposition of their resultant. In particular we obtain irreducibility criteria for pairs of polynomials whose resultant is a prime number. As another application we provide irreducibility conditions for polynomials that take a prime value, and for polynomials obtained by expressing prime numbers by quadratic forms. The use of the resultant in the study of linear combinations of relatively prime polynomials is also discussed. Similar results will be provided for multivariate polynomials over an arbitrary field. We will finally give a method to compute the resultant using linear recurrence sequences.
5.
BOTNARU Dumitru
State University from Tiraspol, Moldova
Title: The duality (σ,τ) (details)
Abstract:
---------------------------------------------------------------- % Article Class (This is a LaTeX2e document) ******************** % ---------------------------------------------------------------- \documentclass[12pt]{article} \usepackage[english]{babel} \usepackage{amsmath,amsthm} \usepackage{amsfonts} % THEOREMS ------------------------------------------------------- \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \numberwithin{equation}{section} % ---------------------------------------------------------------- \begin{document} \begin{center} \textbf{The duality $(\sigma,\tau)$ } \href{ }{} Botnaru D. State University from Tiraspol dumitru.botnaru@gmail.com \end{center} %\address{}% %\thanks{}% %\date{}% % ---------------------------------------------------------------- %\begin{abstract} In the category $\mathcal{C}_{2}\mathcal{V}$ of the locally convex topological vector Hausdorff spaces we denote by $\mathcal{B}$ the class of bijective morphisms $b:(E,u)\longrightarrow(F,v)$ for which $(E,u)\prime=(F,v)\prime$ and $\mathbb{R}_{\varepsilon}^{\varepsilon}(\mathcal{S})$ the class of all reflective subcategories $\mathcal{R}$ is closed under $\mathcal{B}$-subobjects and $\mathcal{B}$-factorobjects. Let $\mathcal{S}$ be the subcategory of the spaces with weak topology, $\Gamma_{0}$ - the subcategory of locally complete spaces, and $\mathbb{R}$ the lattice of all nonzero reflective subcategories. \textbf{Theorem 1.} For any element $\mathcal{R}\in\mathbb{R}_{\varepsilon}^{\varepsilon}(\mathcal{S})$ there is an element $\Gamma\in\mathbb{R}$ so that $\Gamma_{0}\subset\Gamma$ and $\mathcal{R}=\mathcal{S}\ast_{sr}\Gamma_{0}$, where $\mathcal{S}\ast_{sr}\Gamma_{0}$ is the semireflexive product of the elements $\mathcal{S}$ and $\Gamma$ (see [1]. For any morphism $f:(E,u)\longrightarrow(F,v)$ we take in correspondence the morphism $f\prime:F_{\tau}\prime \longrightarrow E_{\tau}$, where the dual spaces possess the Mackey topology. There was defined a contravariant functor $d_{\tau}:\mathcal{C}_{2}\mathcal{V}\longrightarrow \mathcal{C}_{2}\mathcal{V}$. \textbf{Theorem 2.} The functor $d_{\tau}$ is right exact and transfers the products into sums. Denote by $\widetilde{\mathcal{M}}$ the coreflective subcategory of the spaces with Mackey topology, $\mathbb{K}(\widetilde{\mathcal{M}})$ the class of the coreflective subcategories that is contained in the $\widetilde{\mathcal{M}}$ subcategory. For any $\mathcal{A}\subset\mathcal{C}_{2}\mathcal{V}$ subcategory we denote by $\delta(\mathcal{A})$ the full subcategory from $\mathcal{C}_{2}\mathcal{V}$ defined on the class of object $\{d_{\tau}(X)\mid X\in\mid\mathcal{A}\mid\}$. If $\mathcal{A}\in\mathbb{K}(\widetilde{\mathcal{M}})$ denote by $\delta^{-1}(\mathcal{A})$ the full subcategory from $\mathcal{C}_{2}\mathcal{V}$ defined by the class of objects $\{X\in\mid\mathcal{C}_{2}\mathcal{V}\mid, d_{\tau}(X)\in\mid\mathcal{A}\mid\}$. \textbf{Theorem 3.} 1. If $\mathcal{R}\in\mathbb{R}$, then $\delta(\mathcal{R})\in \mathbb{K}(\widetilde{\mathcal{M}})$. 2. If} $\mathcal{A}\in\mathbb{K}(\widetilde{\mathcal{M}})$, then $\delta^{-1}(\mathcal{A})\in \mathbb{R}_{\varepsilon}^{\varepsilon}(\mathcal{S})$. 3. Let $\mathcal{C}\subset\mathcal{R}$ and $\mathcal{R}\in\mathbb{R}$. Then $\delta(\mathcal{R})=\widetilde{\mathcal{M}}$. 4. Let $\mathcal{R}\in\mathbb{R}$. Then $\widetilde{\mathcal{M}}\ast_{d}\mathcal{R}=\delta^{-1}\delta(\mathcal{R})$, where $\widetilde{\mathcal{M}}\ast_{d}\mathcal{R}$ is the right product of the $\widetilde{\mathcal{M}}$ and $\mathcal{R}$ elements (see [2]). 5. Let $\mathcal{R}\in\mathbb{R}$. Then $\delta^{-1}\delta(\mathcal{R})$ is the first element of the class $\mathbb{R}_{\varepsilon}^{\varepsilon}(\mathcal{S})$ that contains the $\mathcal{R}$ element . 6. The maple $\delta$ sets an isomorphism of the $\mathbb{R}_{\varepsilon}^{\varepsilon}(\mathcal{S})$ and $\mathbb{K}(\widetilde{\mathcal{M}}) - \delta:\mathbb{R}_{\varepsilon}^{\varepsilon}(\mathcal{S})\longrightarrow \mathbb{K}(\widetilde{\mathcal{M}})$ lattices. 7. The lattices $\mathbb{R}_{\varepsilon}^{\varepsilon}(\mathcal{S})$ and $\mathbb{\widetilde{\mathcal{M}}}$ contains a proper class of elements. \textbf{Bibliography} 1. Botnaru D., Cerbu O., \textit{Semireflexive product of two subcategories}, Proc. of the $6^{th}$ Congress of Romanian Math., Bucharest, 2007, v.1, p. 5-19. 2. Botnaru D., Turcanu A., \textit{ The factorization of the right product of two subcategories}, ROMAI J., 2010, v.VI, Nr.2, p. 41-53. \end{document} % ----------------------------------------------------------------
6.
BREAZ Simion
Babes-Bolyai University, Romania
Title: Pure semisimple rings and direct products (details)
Abstract:
We present some characterizations for pure-semisimple rings which involve direct products of modules. One of them depends on the (non-)existence of some large cardinals: Let R be a ring and let W be the direct sum of all finitely presented right R-modules. Under the set theoretic hypothesis (V = L), the ring R is right pure semisimple if and only if there exists a cardinal $lambda$ such that $mathrm{Add}(W)subseteq mathrm{Prod}(W^{(lambda)})$. Moreover, there is a set theoretic model such that for every ring R there exists a cardinal $lambda$ such that $mathrm{Add}(W)subseteq mathrm{Prod}(W^{(lambda)})$. On the other side, we will see that a left pure-semisimple ring R is of finite representation type (i.e. it is right pure-semisimple) if and only if for every finitely presented left R-module M the right R-module $mathrm{Hom}_{mathbb{Z}}(M, mathbb{Q}/Z)$ is Mittag-Leffler.
7.
BULACU Daniel
University of Bucharest, Romania
Title: Frobenius and separable functors for the category of generalized entwined modules (details)
Abstract:
The explicit structure of a cowreath in a monoidal category $\mathcal{C}$ leads to the notion of generalized entwined module in a $\mathcal{C}$-category. A cowreath can be identified with a coalgebra $X$ in the Eilenberg-Moore category $EM(\mathcal{C})(A)$, for some algebra $A$ in $\mathcal{C}$, and the Frobenius or separable property of the forgetful functor from the category of generalized entwined modules to the category of representations over $A$ is transferred to the coalgebra $X$ and vice-versa.
8.
BURCIU Sebastian
"Simion Stoilow" Institute of Mathematics of Romanian Academy, Romania
Title: On the irreducible representations of Drinfeld doubles (details)
Abstract:
A description of all the irreducible representations of generalized quantum doubles associated to skew pairings of semisimple Hopf algebras is given. In particular, a description of the irreducible representations of semisimple Drinfeld doubles is obtained in this way. We also give a formula for the tensor product of any two such irreducible representations. Using this formula new information on the structure of the Grothendieck rings of these generalized quantum doubles is obtained.
9.
CAENEPEEL Stefaan
Vrije Universiteit Brussel, VUB, Belgium
Title: Hopf Categories (details)
Abstract:
We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories.
10.
CERBU Olga
Moldova State University, Moldova
Title: About B-inductive semireflexive spaces (details)
Abstract:
About B-inductive semireflexive spaces ---------------------------------------------------------------- % Article Class (This is a LaTeX2e document) ******************** % ---------------------------------------------------------------- \documentclass[12pt]{article} \usepackage[english]{babel} \usepackage{amsmath,amsthm} \usepackage{amsfonts} % THEOREMS ------------------------------------------------------- \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \numberwithin{equation}{section} % ---------------------------------------------------------------- \begin{document} \begin{center} \textbf{About $\mathcal{B}$-inductive semireflexive spaces } \href{ }{} Botnaru D., Cerbu O., \c{T}urcanu A. State University from Tiraspol, State University of Moldova % Technical University from Moldova dumitru.botnaru@gmail.com, olga.cerbu@gmail.com, alina.turcanu@gmail.com, \end{center} %\address{}% %\thanks{}% %\date{}% % ---------------------------------------------------------------- %\begin{abstract} Let $\mathcal{C}_{2}\mathcal{V}$ be the category of the locally convex topological vector Hausdorff spaces. We denote by $\widetilde{\mathcal{M}}$ the subcategory of the spaces with Mackey topology, $\mathcal{N}orm$ - the subcategory of normed spaces, $\Gamma_{0}$ - the subcategory of complete spaces, $l\Gamma_{0}$ - the subcategory of locally complete spaces (D. Ra\"{i}cov) or $b$-complete (W. Slovikovski), and $\mathcal{S}$ - the subcategory of the spaces with weak topology. For an object $(E,t)$ the absolute convex and bounded set ${A}$ is defined as a Banach sphere, if the normed space $(E_{A},n_{A})$ is the Banach space, where $E_{A}$ is the linear coverage of the set ${A}$, and $n_{A}$ - the Minkowski functional of the set ${A}$. We denote with $\mathcal{B }$ the set of all Banach spheres in the space $E^{\prime}_{\beta}$ ($\beta$ - the topology of the uniform convergence on all the bounded sets from $(E,t)$). The inductive topology $j(t)$ on $E^{\prime}$ is defined as the most fine locally convex topology for which the following applications $j_{A}:(E^{\prime}_{A},n_{A})\longrightarrow(E^{\prime},j(t))$ are continuous, $A\in\mathcal{B}$. \textbf{Definition} (V. Sekevanov ). The space $(E,t)$ is called semireflexive $\mathcal{B}$ - inductive if $(E^{\prime},j(t))\prime=E$. Let $i\mathcal{R}$ be the subcategory of the semireflexive inductive spaces [1], and $\mathcal{B}-i\mathcal{R}$ - of the semireflexive $\mathcal{B}$-inductive spaces. Then $i\mathcal{R}\subset \mathcal{B}-i\mathcal{R}$ (V. Sekevanov). We denote $\mathcal{A}$ the class of all bijective morphisms $b:(E,u)\longrightarrow(F,v)\in\mathcal{C}_{2}\mathcal{V}$ for which $(E,u)\prime=(F,v)\prime$. For a subcategory $\mathcal{C}\subset\mathcal{C}_{2}\mathcal{V}$ we denote by $Q_{A}(\mathcal{C})$ the subcategory $\mathcal{A}$-factorobjects of the objects from $\mathcal{C}$. \textbf{Theorem 1.} Let be $\mathcal{L}$ and $\Gamma$ two reflective subcategories $\mathcal{S}\subset \mathcal{L}$, $\Gamma_{0}\subset\Gamma$. Further let be $\mathcal{R}$ the semireflexive product of the subcategories $\mathcal{L}$ and $\Gamma$: $\mathcal{R}=\mathcal{L}\ast_{sr}\Gamma$ [2]. Then: 1. $Q_{\mathcal{A}}(\widetilde{\mathcal{M}}\cap \mathcal{R})$ is a reflective subcategory.\\ 2. $\mathcal{R}\subset Q_{\mathcal{A}}(\widetilde{\mathcal{M}}\cap \mathcal{R})$. 3. The subcategory $Q_{\mathcal{A}}(\widetilde{\mathcal{M}}\cap \mathcal{R})$ is closed under the $\mathcal{A}$-subobjects and $\mathcal{A}$-factorobjects. \textbf{Theorem 2.} 1. $\mathcal{B}-i\mathcal{R}=Q_{\mathcal{A}}(\widetilde{\mathcal{M}}\cap i\mathcal{R})$. 2. $l\Gamma_{0}=Q_{\mathcal{A}}(\widetilde{\mathcal{M}}\cap \Gamma_{0})=Q_{\mathcal{A}}(\mathcal{N}orm)$. %\end{abstract} %\maketitle % ---------------------------------------------------------------- %\section{} %\vspace(0.2cm) \textbf{Bibliography} 1.Berezanschi I.A., \textit{The inductive reflexive locally convex spaces}, DAN SSSR,1968, T.182, Nr.1, p.20-22. 2.Botnaru D., Cerbu O., \textit{Semireflexive product of two subcategories}, Proc. of the Sixth Congress of Romanian Math., Bucharest, 2007, v.1, p.5-19. \end{document} % ----------------------------------------------------------------
11.
CHINDRIS Calin
University of Missouri-Columbia, United States
Title: On the invariant theory of string algebras (details)
Abstract:
This talk is based on joint work with Andy Carroll. It is about studying the module category of a finite-dimensional algebra within the general framework of invariant theory. Our objective is to describe the tameness of an algebra in terms of its moduli spaces of modules. Specifically, we will show that for an acyclic string algebra, the irreducible components of any moduli space of modules are just products of projective spaces. Along the way, we will describe a decomposition result for moduli spaces of modules of arbitrary finite-dimensional algebras.
12.
CHIS Mihai
West University of Timisoara, Romania
Title: Some properties of autocommutator subgroups of certain p-groups (details)
Abstract:
We investigate some properties of autocommutator subgroups of certain classes of p-groups.
13.
CIMPOEAS Mircea
I.M.A.R., Romania
Title: On intersections of complete intersection ideals (details)
Abstract:
We present a class of complete intersection toric ideals whose intersection is a complete intersection, too.
14.
CIPU Mihai
Simion Stoilow Institute of Mathematics of the Romanian Academy Bucharest, Romania
Title: Recent advances in the study of Diophantine quintuples (details)
Abstract:
Apparently motivated by Heron's formula for triangle area, Diophantus has asked to find sets of numbers with the property that increasing by one the product of any two elements results in a perfect square. Such sets are called nowadays Diophantine or $D(1)$-sets. A lot of work has been prompted by the conjecture (put forward in 1978 by P. E. Gibbs and independently by J. Arkin, V. E. Hoggatt, and E. G. Strauss) that any Diophantine triple has a unique extension to a Diophantine quadruple. Clearly, this implies a weaker conjecture, predicting that there exists no Diophantine quintuple. The talk will contain a survey of very recent ideas and results, many of them still unpublished, which bring us closer to solution of these problems. Several results are obtained in common with A. Filipin (Croatia), Y. Fujita (Japonia), M. Mignotte (Franta), T. Trudgian (Australia).
15.
COBELI Cristian
"Simion Stoilow" Institute of Mathematics of the Romanian Academy, Romania
Title: On the Dew Line in Circle Packings (details)
Abstract:
Let $A$ be a fixed arc of a selected circle in a circle packings $\mathcal{P}$. The \textit{dew line} associated to $A$ is a curve $D_A(h)$, which is parallel to $A$ and lies at a distance $h> 0$ away from $A$, on the same side with the other circles of $\mathcal{P}$. % Denote by $\mathcal{C}_A$ the set of circles in $\mathcal{P}$ that are tangent to A and let $P_A(h)$ be the probability that a point on the dew line $D_A(h)$ is inside a circle of $\mathcal{C}_A$. % We present a few problems and results concerning the following questions: \textit{Is there a limit probability $\lim\limits_{h\to 0}P_A(h)$? If the answer is positive, does this limit depends on the arc and on the packing?}
16.
COCONET Tiberiu
UBB, Romania
Title: Module covers and the Green correspondence (details)
Abstract:
The Green correspondence can be expressed as an equivalence between certain quotient categories of modules over group algebras. M.E. Harris combined this categorical version with the Nagao-Green theorem on block induction, obtaining a version with blocks of the mentioned equivalence. We investigate this approach with respect to module covers and block covers and discover more general results that imply well-known correspondences.
17.
COJOCARU Alina Carmen
University of Illinois at Chicago, Institutul de Matematica al Academiei Romane, USA, Romania
Title: Arithmetic properties of the Frobenius traces of an abelian variety (details)
Abstract:
Given an abelian variety A/Q, with a trivial endomorphism ring (over the algebraic closure of Q), we investigate the arithmetic properties of the coefficients of the p-Weil polynomials of A, as p varies.
18.
COJUHARI Elena
Technical University of Moldova, Republic of Moldova
Title: Skew ring extensions and generalized monoid rings (details)
Abstract:
Given a ring $A$ with identity and a multiplicative monoid $G$, a $D$-structure is defined as a collection $sigma$ of self-mappings of $A$ indexed by elements of $G$ satisfying certain demanding but quite natural conditions [1, 2]. $D$-structures are used to define various skew and also twisted monoid rings which in turn being confined in a general construction of a ring $A langle G, sigma rangle$ named as a generalized monoid ring (e.g. [2]). Weyl algebras, skew polynomial rings and others related to them [3] become special concrete realizations of such monoid rings. Among many others we examine the relationships between generalized monoid rings, especially skew monoid rings, and normalizing and subnormalizing extensions. Relations between the existence of a $D$-structure and gradability of the ring by a cyclic group are also studied. The talk is based on joint work with Barry J. Gardner. begin{thebibliography}{99} bibitem{1} Cojuhari E.P., Gardner B.J. textit{ Generalized Higher Derivations}, Bull. Aust. Math. Soc. 86, no. 2 (2012), 266-281. bibitem{2} Cojuhari E.P. textit{ Monoid algebras over non-commutative rings,} Int. Electron. J. Algebra, 2 (2007), 28-53. bibitem{3} Cohn P.M. {em Free rings and their relations.} London Mathematical Society Monographs, No. 2. Academic Press, London-New York, 1971, 346 pp. end{thebibliography}
19.
"Simion Stoilow" Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Title: Towards longer-range topological properties for finite generation of subalgebras (details)
Abstract:
\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}
20.
CONSTANTINESCU Alexandru
Freie Universitaet Berlin, Germany
Title: Castelnuovo-Mumford regularity and triangulations of manifolds (details)
Abstract:
We show that for every positive integer r there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to r. For Gorenstein ideals we prove that the regularity of their quotients can not exceed four, thus showing that for d > 4 every triangulation of a d-manifold has a hollow square or simplex. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity is O(log(log(n)), where n is the number of variables.
21.
DEACONESCU Marian
Kuwait University, Kuwait
Title: OPERATOR THEORY FOR FINITE GROUPS (details)
Abstract:
My talk will present a handful of recent results obtained jointly with G.L. Walls. These results are quite general since they are related to the following situation: an arbitrary finite group $G$ is operated (acted) upon by an arbitrary finite group $A$. In older terminology, $A$ is " a group of operators of $G$". The newer terminology is that $A$ " acts on $G$ via automorphisms". This kind of an action, as general as it is (no other conditions are imposed here) comes with a set of "invariants" attached to it. The first is the subgroup $F$ of all of the fixed points of $A$ in $G$. The second is the "autocommutator subgroup" $[G, A]$, which is the subgroup of $G$ generated by the elements $g^{-1}g^{\alpha}$ for $g\in G, \alpha \in A$. Finally, the orbits of the elements in $G$ under the action of $A$ are also of interest. \bigskip Particular cases are important, of course; we where able to solve, among other things, the old well-known problem of characterizing (via a simple, compact, alternative group-theoretical condition) those finite groups whose automorphism group is abelian. The first example of a finite non-abelian group $G$ whose group $Aut(G)$ of automorphisms is abelian was given by G.A. Miller in 1913. Infinitely many more examples were produced since. \bigskip Whenever the subgroup $F$ is nontrivial it turns out that the sequence of the lengths of the orbits of $A$ in $G$ behaves in a very orderly manner. In particular, it is true that if $p$ is a prime dividing the order of $F$, then the number of orbits of $A$ in $G$ whose length is co-prime to $p$ must be a multiple of $p$. \bigskip When $H$ is an $A$-invariant subgroup of $G$, then we can determine the number of pairs $(g, \alpha)$ with $g\in G$ and $\alpha \in A$ such that $g^{-1}g^{\alpha}\in H$. This is a far reaching extension of a classic result of Frobenius (who determined this number when $A=G$ acts on $G$ via conjugation and when $H=1$ is the trivial subgroup of $G$) and it has several important consequences.
22.
UNIVERSITY OF MINNESOTA, USA
Title: Non-vanishing of quadratic twists of automorphic L-functions (details)
Abstract:
In this talk, I will discuss a novel approach in understanding the important problem of the non-vanishing of some of the quadratic twists of an L-function attached to a fixed cuspidal automorphic representation on GL(n).
23.
ENE Viviana
Universitatea Ovidius din Constanta, Romania
Title: Ideals of 2-minors (details)
Abstract:
In this talk we survey recent results on binomial edge ideals defined on generic (Hankel) matrices. Given a simple graph $G$ on the vertex set $[n],$ one may associate with it a binomial ideal $J_G$ in the polynomial ring $K[X]$ over a field $K,$ where $X= \left( \begin{array}{llll} x_1 & x_2 &\ldots& x_n\\ y_1 & y_2 &\ldots& y_n \end{array}\right).$ The ideal $J_G$ is generated by maximal minors of $X,$ $f_{ij}=x_iy_j-x_jy_i$ with $\{i,j\}$ edge of $G,$ and is called the {\em binomial edge ideal} of $G.$ Later on, the notion of binomial edge ideal was generalized to a pair of graphs. The interest in studying (generalized) binomial edge ideals partially comes from the fact that they turned out to have applications in statistics. In our talk, we discuss various algebraic and homological properties of binomial edge ideals. Similar constructions can be done by considering binomial edge ideals on Hankel matrices associated with (pairs of) graphs. They generalize the well known defining ideals of rational normal curves. We mainly focus on some recent results obtained in joint papers with F. Chaudhry, A. Dokuyucu, J. Herzog, T. Hibi, A. Qureshi, A. Zarojanu.
24.
ENESCU Florian
Georgia State University, United States
Title: The Frobenius complexity of a local ring (details)
Abstract:
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.
25.
GROZA Ghiocel
Technical University of Civil Engineering Bucharest, Romania
Title: On the analytic functions with p-adic coefficients (details)
Abstract:
\begin{document} \title{On the analytic functions with $p$-adic coefficients} \author{Ghiocel Groza} \date{} \maketitle \begin{center} \noindent {\it Technical University of Civil Engineering Bucharest, \\ Romania, E-mail: grozag@utcb.ro}\\ \end{center} \indent Let $p$ be a fixed prime and $|\;|$ the normalized $p$-adic absolute value defined on $\mathbb{Q}$, that is $|p|=\frac{1}{p}$. If $R$ is a positive real number and ${\mathbb Q}_p$ is the completion of ${\mathbb Q}$ with respect to $|\;|$, we denote by $B(R)=\{x\in \mathbb{Q}_p:|x|\le R\}$ and $S(R)=\{x\in \mathbb{Q}_p:|x|=R\}$ the ball with circumference and the sphere, with center $0$ and radius $R$, respectively. \\ \indent For a fixed non-negative integer $t$ let $$\label{eq1} f=\sum\limits_{i=0}^{\infty}{c_i X^i},\;c_i\in \mathbb{Q}_p,$$ be a convergent series on $B(p^{-t})$. We study the analytic functions of the form (\ref{eq1}) which define a mapping from $S(p^{-t})$ into $S(1)$. Hence we get a result concerning entire functions with $p$-adic coefficients which are bounded on ${\mathbb Q}_p$. Finally we study infinite interpolation by means of entire functions with $p$-adic coefficients. \end{document}
26.
IACOB Alina
Georgia Southern University, USA
Title: Gorenstein projective precovers (details)
Abstract:
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.
27.
ICHIM Bogdan
Simion Stoilow Institute of Mathematics, Romania
Title: How to compute the Stanley depth of a module (details)
Abstract:
We introduce an algorithm for computing the Stanley depth of a finitely generated multigraded module $M$ over the polynomial ring $K[X_1,ldots,X_n]$. As an application, we give an example of a module whose Stanley depth is strictly greater than the depth of its syzygy module. In particular, we obtain complete answers for two open questions raised by Herzog.
28.
JONES Nathan
University of Illinois at Chicago, USA
Title: The distribution of class groups of imaginary quadratic fields (details)
Abstract:
Which abelian groups occur as the class group of some imaginary quadratic field? Inspecting tables of M. Watkins on imaginary quadratic fields of class number up to 100, one finds that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance (Z/3Z)^3 does not). In this talk, I will combine heuristics of Cohen-Lenstra together with a refinement of a conjecture of Soundararajan to make precise predictions about the asymptotic distribution of imaginary quadratic class groups, partially addressing the above question. I will also present some numerical evidence of the resulting conjectures.
29.
LENART Cristian
State University of New York at Albany, USA
Title: A combinatorial model for Kirillov-Reshetikhin crystals and applications (details)
Abstract:
Crystals are colored directed graphs encoding information about Lie algebra representations. Kirillov-Reshetikhin (KR) crystals correspond to certain finite-dimensional representations of affine Lie algebras. I will present a combinatorial model which realizes tensor products of (column shape) KR crystals uniformly across untwisted affine types. Some computational applications are discussed. A corollary states that the Macdonald polynomials (which generalize the irreducible characters of semisimple Lie algebras), upon a certain specialization, coincide with the graded characters of tensor products of KR modules.
30.
MILITARU Gigel
Universitatea din Bucuresti, Romania
Title: The factorization problem and related questions (details)
Abstract:
Let $A leq G$ be a subgroup of a group $G$. An $A$-complement of $G$ is a subgroup $H$ of $G$ such that $G = A H$ and $A cap H = { 1}$. The emph{classifying complements problem} asks for the description and classification of all $A$-complements of $G$. We shall give the answer to this problem in three steps. Let $H$ be a given $A$-complement of $G$ and $(triangleright, triangleleft)$ the canonical left/right actions associated to the factorization $G = A H$. To start with, $H$ is deformed to a new $A$-complement of $G$, denoted by $H_r$, using a certain map $r: H to A$ called a deformation map of the matched pair $(A, H, triangleright, triangleleft)$. Then the description of all complements is given: ${mathbb H}$ is an $A$-complement of $G$ if and only if ${mathbb H}$ is isomorphic to $H_{r}$, for some deformation map $r: H to A$. Finally, the classification of complements proves that there exists a bijection between the isomorphism classes of all $A$-complements of $G$ and a cohomological object ${mathcal D} , (H, A , | , (triangleright, triangleleft) )$. As an application we show that the theoretical formula for computing the number of isomorphism types of all groups of order $n$ arises only from the factorization $S_n = S_{n-1} C_n$.
31.
NASTASESCU Constantin
"Simion Stoilow" Institute of Mathematics of the Romanian Academy, Romania
Title: Are graded semisimple algebras symmetric? (details)
Abstract:
We study graded symmetric algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded division algebra whose dimension is not divisible by the characteristic of the base field is graded symmetric. Using the structure of graded simple(semisimple) algebras,we extend the results to these classes. In particular, in characteristic zero any graded semisimple algebra is graded symmetric. We show that the center of a finite dimensional graded division algebra is often symmetric.
32.
NICHITA Florin
IMAR, ROMANIA
Title: Nonassociative Structures, Yang-Baxter Equations and Applications (details)
Abstract:
Several of our books, papers, talks and posters (since 1979 until now) treated topics on Jordan Algebras, Nonassociative Structures, Yang-Baxter Equations, Hopf Algebras and Quantum Groups. For example, we list the papers presented at previous congresses: [1] F.F. Nichita, Lie algebras and Yang-Baxter equations, Bull. Trans. Univ. Brasov, Series III, Vol. 5 (54), 2012, Special Issue: Proceedings of the 7-th Congress of Romanian Mathematicians, Brasov, 2011, 195-208. [2] F.F. Nichita and D. Parashar, Coloured bialgebras and nonlinear equations, Proceedings of the 6-th Congress of Romanian Mathematicians, Bucharest, 2007, Editura Academiei, vol. 1, 65-70, 2009. Recently, we published some joint works on the above mentioned topics: [3] Radu Iordanescu, Florin F. Nichita, Ion M. Nichita, The Yang-Baxter equation, (quantum) computers and unifying theories, Axioms, 2014; 3(4):360-368. [4] Radu Iordanescu, Florin F. Nichita, Ion M. Nichita, Non-associative algebras, Yang-Baxter equations, and quantum computers, Bulg. J. Phys., vol.41, n.2, 2014, 71-76. Motivated by the above achievements we would like to present new results and directions of study.
33.
OLTEANU Anda - Georgiana
University Politehnica of Bucharest & "Simion Stoilow" Institute of Mathematics of the Romanian Academy, Romania
Title: Classes of path ideals and their algebraic properties (details)
Abstract:
Given a directed graph $G$, the path ideal of the graph $G$ (of length $t\geq 2$) is the monomial ideal $I_t(G)$ generated by the squarefree monomials which correspond to the directed paths of length $t$ in $G$. Classes of directed graphs arise from posets. We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen--Macaulay in terms of the underlying poset. For Alexander dual of cycle posets, we compute the Castelnuovo--Mumford regularity and, as a consequence, we get the projective dimension of path ideals of cycle posets. We also pay attention to path ideals of powers of the line graph and study the property of being sequentially Cohen--Macaulay and having a linear resolution. The results are expressed in terms of the combinatorics of the underlying poset.
34.
PANAITE Florin
Institute of Mathematics of the Romanian Academy, Romania
Title: Hom-structures (details)
Abstract:
Hom-structures (Hom-associative algebras, Hom-Lie algebras etc) are generalizations of classical algebraic structures in which the defining identities are twisted by certain homomorphisms. We will present some recently introduced concepts, constructions and properties involving Hom-structures (such as twisted tensor products, smash products etc).
35.
PASOL Vicentiu
IMAR, Romania
Title: p-adic Analytic Functions from Recurrence Sequences (details)
Abstract:
B. Berndt, S. Kim and A. Zaharescu, in their study of the diophantine approximation of $e^{2/a}$ have constructed certain $p$-adic functions, naturally arising from the sequence of convergents of $e$. They prove that for certain primes $p$, these functions are continuous. They raised the question if those functions are in fact rigid analytic. We prove that in fact this question has a positive answer for all primes $p$.
36.
POP Horia
Mt San Antonio College, USA
Title: Heisenberg algebras and coefficient rings (details)
Abstract:
37.
POPA Alexandru Anton
IMAR, Bucuresti, Romania
Title: On the trace formula for Hecke operators (details)
Abstract:
We present a new, simple proof of the trace formula for Hecke operators on modular forms for congruence subgroups. It is based on an approach for the full modular group sketched by Don Zagier more than 20 years ago, by computing the trace of Hecke operators on the space of period polynomials associated with modular forms. This algebraic proof has been recently sharpened in a joint work with Zagier, and we show that it generalizes to congruence subgroups as well. We use the theory of period polynomials for congruence subgroups, developed jointly with Vicentiu Pasol.
38.
POPESCU Dorin
Institute of Mathematics IMAR, Romania
Title: A theorem of Ploski's type (details)
Abstract:
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)}$. 39. POPESCU Sever - Angel Technical University of Civil Engineering Bucharest, ROMANIA Title: On the v-extensions of a valued field, by Victor Alexandru and Sever Angel Popescu (details) Abstract: On the v-extensions of a valued field by Victor Alexandru and Sever Angel Popescu Abstract. Let (K,v) be a perfect nontrivial Krull valued field of rank 1 and let w be an extension of v to a fixed algebraic closure Ω of K. An intermediate valued field (L,w) is called a v-extension of (K,v) if v does not split in L. If (L,v) is maximal with this property, we say that it is a v-maximal extension of (K,v). For instance, if (K,v) is a henselian field, then the only v-maximal extension of (K,v) is L = Ω. We prove in this note that if (K,v) is a (finite) algebraic number field, then any v-maximal extension of it cannot be a normal extension of K. On the other hand, in the case of the rational function field, with coefficients in a field k of characteristic zero, endowed with the X-adic valuation, we give a constructive example of an X-adic maximal extension L of K which is also a normal extension of K. 40. RAIANU Serban California State University, Dominguez Hills, USA Title: A Coring Version of External Homogenization for Hopf Algebras (details) Abstract: We give a coring version for the external homogenization for Hopf algebras, which is a generalization of a construction from graded rings, called the group ring of a graded ting. We also provide a coring version of a Maschke-type theorem. 41. RAICU Claudiu University of Notre Dame, USA Title: The syzygies of some thickenings of determinantal varieties (details) Abstract: The space of mxn matrices admits a natural action of the group GL_m x GL_n via row and column operations on the matrix entries. The invariant closed subsets are the determinantal varieties defined by the (reduced) ideals of minors of the generic mxn matrix. The minimal free resolutions for these ideals are well-understood by work of Lascoux and others. There are however many more invariant ideals which are non-reduced, and whose syzygies are quite mysterious. These ideals correspond to nilpotent structures on the determinantal varieties, and they have been completely classified by De Concini, Eisenbud and Procesi. In my talk I will recall the classical description of syzygies of determinantal varieties, and explain how this can be extended to a large collection of their thickenings. 42. RUDEANU Sergiu Faculty of Mathematics and Computer Science, University of Bucharest, Romania Title: Most general forms in the study of Boolean equations (details) Abstract: TBA 43. SECELEANU Alexandra University of Nebraska-Lincoln, USA Title: Polynomial growth for Betti numbers (details) Abstract: It is well known that the asymptotic patterns of the Betti sequences of the finitely generated modules over a local ring R reflect the structure of R. For instance, these sequences are eventually zero if and only if R is regular (Auslander and Buchsbaum, Serre) and they are eventually constant if and only if R is a hypersurface (Shamash, Gulliksen, Eisenbud). We consider the problem of characterizing the rings R such that every R-module has Betti numbers eventually given by some polynomial. We give necessary and sufficient conditions for R to have this property. In some important cases, for example when R is homogeneous, these conditions coincide and therefore characterize R. 44. STAIC Mihai Bowling Green State University, USA Title: Operations on the Secondary Hochschild Cohomology (details) Abstract: Secondary cohomology is associated to a triple$(A, B, varepsilon)$, and was introduced in order to describe all the$B$-algebra structures on$A[[t]]$at the same time. We present some results related to this cohomology: the cup and bracket product, the Hodge decomposition, the bar complex, and the secondary cyclic cohomology associated to the triple$(A, B, varepsilon)$. 45. STAMATE Dumitru University of Bucharest, Romania Title: Ungraded strongly Koszul rings (details) Abstract: Various methods have been designed for checking that a standard graded algebra is Koszul, some being more efficient than the others. We are interested in semigroup rings R= K[H], which are not usually standard graded. In this context we introduce the strongly Koszul property, extending in a natural way the similar concept of Herzog, Hibi and Restuccia for standard graded K-algebras. We show that if K[H] is strongly Koszul, then its associated graded ring grK[H] is a Koszul ring in the classical sense and that the two rings have the same Poincare series. Our toolbox includes sequentially Cohen- Macaulayness and shellability for posets. This is a preliminary report on work in progress with Juergen Herzog, Essen, Germany. 46. STANCU Radu LAMFA, Universite de Picardie, France Title: Extentions of cohomological Mackey functors (details) Abstract: Let$k$be a field of characteristic$p$and$G$a finite group. The cohomological Mackey functors for$G$over$k$are modules over a specific finitely generated algebra$co\mu_k(G)$, called the cohomological Mackey algebra. This algebra shares many properties with the usual group algebra, and most questions about modules over the group algebra and methods used for them can be extended to Mackey functors : e.g. relative projectivity, vertex and source theory, Green correspondence, the central role played by the elementary abelian$p$-groups. These resemblances raise some natural questions, whether, a given theorem on$kG$admits an analogue for$co\mu_k(G)$. This was the main motivation in a previous work of Serge Bouc, where the question of complexity of cohomological Mackey functors was solved (in the only non-trivial case where$p$divides the order of$G$). It was also shown there how this question can be reduced to the consideration of elementary abelian$p$-groups$E$appearing as subquotients of$G$, and to the knowledge of enough information on the algebra$\Ee=\Ext^*_{co\mu_k(E)}(S_1^E,S_1^E)$of self-extensions of a particular simple functor$S_1^E$for these groups. The aim of the talk I give - which is based on a joint work with Serge Bouc - is to recall the basic properties of cohomological Mackey functors and give insight into how one can get an explicit presentation of the algebra$\Ext^*_{co\mu_k(G)}(S_1^G,S_1^G)$, when$G$is an elementary abelian$p$-group. 47. STEFANESCU Doru University of Bucharest, Romania Title: Irreducibility criteria for polynomials over discrete valuation domains (details) Abstract: \begin{abstract} We study properties of the Newton polygon of a product of two polynomials over a discrete valuation domain$(A,v)$and we establish corresponding properties of the Newton index of a polynomial in$A[X]\,$. There are deduced factorization properties of polynomials over$A$and there are obtained new irreducibility criteria. \smallskip The results are used for generating classes of irreducible polynomials over various discrete valuation domains. In particular we obtain criteria for quasi-generalized difference polynomials, for univariate polynomials over$\Z\,$and for polynomials over formal power series. \end{abstract} 48. SZOLLOSI Istvan Babes-Bolyai University, Romania Title: Computation of Hall polynomials in the Euclidean case (details) Abstract: Let$kQ$be the path algebra of the acyclic quiver$Q=(Q_{0},Q_{1})$over the finite field$k$(here$Q_{0}$is the set of vertices and$Q_{1}$the set of arrows). We will consider the category$\mbox{mod-}kQ$of finite$k$-dimensional right modules over$kQ$, which can be identified with the category$\mbox{rep-}kQ$of the finite dimensional$k$-representations of the quiver$Q$. Denote by$[X]$the isomorphism class of a module$X$in$\mbox{mod-}kQ$. The Ringel-Hall algebra$H(kQ)$associated to the algebra$kQ$is the rational space having as basis the isomorphism classes in$\mbox{mod-}kQ$together with a multiplication defined by$[N_{1}][N_{2}]=\sum_{[M]}F_{N_{1}N_{2}}^{M}[M]$, where the structure constant$F_{N_{1}N_{2}}^{M}$is the number of submodules$U$of$M$such that$U$is isomorphic to$N_{2}$and$M/U$is isomorphic to$N_{1}$. These structure constants are also called Ringel-Hall numbers. One can see that$H(kQ)$is an associative rational algebra with identity$[0]$. In case of Dynkin and Euclidean (tame) quivers the Ringel-Hall numbers are polynomials in the number of elements of the base field. These are the Hall polynomials, which appear in various contexts: they are the structure constants of quantum groups, they are used in the theory of cluster algebras and they can also be used successfully to investigate the structure of the module category. Apart from Ringel's famous list of Hall polynomials in the Dynkin case and a limited number of special cases, our knowledge on Hall polynomials is scarce. We present some of the theoretical and computational challenges one has to deal with, when trying to compute Hall polynomials. Deep theoretical results, unusual techniques, complex algorithms and huge computing power are all required in the process of obtaining these polynomials. We focus on the computational aspect of the problem and also present the first results of our quest. 49. TODEA Constantin - Cosmin Technical University of Cluj-Napoca, Romania Title: Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups (details) Abstract: The Bockstein homomorphism in group cohomology is the connecting homomorphism in the long exact sequence associated to some short exact sequence of coefficients. It appears in the Bockstein spectral sequence, which is a tool for comparing integral and mod$p$cohomology ($p$is a prime), and has applications for Steenrod operations. We will define the Bockstein homomorphisms for the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use and additive decomposition and a product formula for the Hochschild cohomology$HH^*(kG)$, given by Siegel and Witherspoon in 1999, where$G$is a finite group and$k$is an algebraically closed field of characteristic$p$. We obtain similar results for the cohomology algebra of a defect group of$B$with coefficients in the source algebra of a block algebra$B$of$kG$. 50. URSU Vasile Institute of Matematics Simion Stoilow of the Romanian Academy, Technical University of Moldova, Moldova Title: Commutators theory in language congruences for modular algebraic system (details) Abstract: In [1] V.A. Gorbunov asked a different definition of congruence on an algebraic system that is determined before. In this definition, congruence is associated not only with the basic operations and the basic relationships that would greatly extend the results and methods for universal algebra in the theory of algebraic systems. Following Gunma [2], in this work we were able to describe the theory of the switches in the language of the congruence of the algebraic system. It is possible to introduce the notion of Abelian, nilpotent and solvable algebraic systems which generalize concepts in universal algebra. References [1] V.А. Gorbunov. Algebraic theory of quasivarieties, Siberian school of algebra and logic, Novosibirsk "Science Book", 1999. [2] H.P. Gumm. An easy way to the commutator in modular varieties, Arch. Math., 1980, 34, 220-228. 51. VELICHE Oana Notheastern University, USA Title: Intersections and Sums of Gorenstein ideals (details) Abstract: 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. 52. VLADOIU Marius University of Bucharest, Romania Title: Bouquet Algebra of Toric Ideals (details) Abstract: To any toric ideal (encoded by an integer matrix A) we associate a matroid structure called the bouquet graph of A, and introduce another toric ideal called the bouquet ideal of A, which captures the essential combinatorics of the initial toric ideal. The new bouquet framework allows us to answer some open questions about toric ideals. For example, we provide a characterization of toric ideals forwhich the following sets are equal: the Graver basis, the universal Groebner basis,any reduced Groebner basis and any minimal generating set. Moreover, we show that toric ideals of hypergraphs encode all toric ideals. 53. VRACIU Adela University of South Carolina, U.S.A. Title: Totally reflexive modules for Stanley-Reisner rings of graphs (details) Abstract: 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. 54. WELKER Volkmar Universtaet Marburg, Germany Title: Ideals of orthogonal graph representations (details) Abstract: We describe algebraic properties of an ideal associated to an undirected graph by Lovasz, Saks and Schrijver. Over the reals it describes orthogonal representations of graphs in Euclidian space. 55. ZAROJANU Andrei IMAR, Romania Title: On the Stanley Depth (details) Abstract: Let$I \supset J$be two squarefree monomial ideals of a polynomial algebra over a field generated in degree$\geq d$, resp.$\geq d+1$. If the Stanley depth of$I/J$is$\leq d+1$then the usual depth of$I/J$is$\leq d+1$if$I$has at most four generators of degree$d\$.