This project has as main aim to approach actual problems of high-level mathematical research. The branches of mathematics are diversifying and specializing continuously. However, mathematics becomes more and more unitary. This is realized by using structures. The mathematical domains are communicating very well. Algebraic objects get topological structures, geometric objects get algebraic structures, new "mixed" concepts are developping. This project has as characteristic the use of algebraic methods in the study and classification of structures (Hopf algebras, classes of graded algebras, near-rings, moduli spaces etc.), of topological methods in group theory and in the study of families of geometric objects, in the intersection theory in algebraic geometry, in the study of subalgebras of affine algebras, combinatorial and topological methods in the study of standard graded algebras, p-adic analysis, factorization problems.The goal of the present project is to use and develop these methods in the following seven research topics:

**1.**
The Lefschetz Property of some graded standard algebras and
algorithms for determining the state polytope for some classes of
rings.

**2.**
Homological methods in commutative algebra: MCM module and classes of
arithmetical rings.

**3.**
Algebraic and toplogical approaches in group theory and in the study
of some moduli spaces.

**4.**
Special classes of Hopf algebras, near-rings and intersection theory
on algebraic stacks.

**5**.
Algorithms for determining the Newton polyhedron and classes of MCM
modules.

**6.
**p-adic analysis**,
**factorization of polynomial
convolutions, constructions of near-rings**. **

**7.
**Subalgebras of affine algebras,
polilogarithms, moduli of some classes of abelian surfaces.

Nicolae Manolache, Senior Researcher I in the Institute of Mathematics "Simion Stoilow" of the Romanian Academy

Nicolae Manolache

Constantin Nastasescu

Nicolae Popescu

Vasile Brinzanescu

Dorin Mihail Popescu

Stefan Papadima

Serban Basarab

Barbu Berceanu

Iustin Coanda

Mircea Eugen Craioveanu

Mircea Puta

Razvan Dinu Litcanu

Cristodor Ionescu

Mihai Cipu

Marian Vajaitu

Nicolae Ciprian Bonciocat

Tiberiu Dumitrescu

Florin Panaite

Marian Aprodu

Serban Barcanescu

Adrian Constantinescu

Nicusor Dan

Cristian Anghel

Marius Vladoiu

Alin Stefan

Nicolae Buruiana

Ovidiu Pasarescu

Gabriel Chiriacescu

Anca Daniela Macinic

Dumitru Stamate

Mircea Cimpoeas

Mihai Epure

Mihai Fulger

Alina Ostafe

Lavinia Ostafe

Viviana Ene – responsabil proiect Univ. Ovidius - Constanta

Mirela Stefanescu

Denis Ibadula

Alexandru Bobe

Marian George Ciuca

Irina Cristea

Mariana Dumitru

Anda Olteanu

'Lefschetz properties on complete intersection algebras of embedding dimension three', authors Dorin Popescu and Marius Vladoiu;

'On the Hilbert series of transversal polymatroids', author Alin Stefan;

'Finite simplicial multicomplexes', author Mircea Cimpoeas.

'The Groebner region of an ideal', author Alexandru Bobe;

'Groebner fan. The state polytope', author Alexandru Bobe;

'Monomial ideals: some examples and computations with Singular', author Viviana Ene.

'Classes of integral domains characterized by their homological behaviour' authors T. Dumitrescu and C. Ionescu.

'Finitely generated-fragmented domains' authors T. Dumitrescu and J. Coykendall.

'Fitting conditions for symmetric algebras of modules of finite projective dimension', authors C. Ionescu, G. Restuccia and R. Utano.

'On the Structure of Maximal Cohen-Macaulay Modules over the Ring k[[x,y]]/(x^n), authors: Viviana Ene and Dorin Popescu.

'On the Structure of Nilpotent Endomorphisms and Applications', author: Viviana Ene.

'Beilinson type spectral sequences on scrolls', authors M. Aprodu and V. Brinzanescu;

'The Classification of the Non-Degenerated Plane Cubics over Qp From the Point of View of the Associated Igusa Local Zeta Function', author Denis Ibadula;

'Algebraic invariants for Bestvina-Brady groups', authors S. Papadima and A. Suciu;

'The arboreal structure of the metric space X:=GL_2(Qp)/Qp*GL_2(Zp)', author Denis Ibadula;

'On the Plane Cubics over Qp and the Associated Igusa Zeta Function', author Denis Ibadula;

'Nonarchimedean Analysis and Applications', ed. Acad. Romane, authors M. Vajaitu and A. Zaharescu.