# Postgraduate studies at MIC

## Staff Research Fields and Research Options

Postgraduate supervision is available in the following specialist areas:

**Mathematics:**
Commutative Algebra, Algebraic Geometry, Complex Analysis, Homological Algebra

**Computer Studies:**
Applying Computer Algebra software to problems in Algebra or Algebraic
Geometry

Algebraic Geometry is one of the oldest of the classical mathematical disciplines. Traditionally, it deals with the study of algebraic varieties, that is, the solution sets of (systems of) polynomial equations. There are deep connections with various other areas of mathematics and theoretical physics, such as complex analysis, number theory, homological algebra, topology, differential geometry or string theory. This interaction makes algebraic geometry a highly dynamic subject and relevant for many "real world" problems.

The development of algebraic geometry during the past centuries was
influenced by different schools, each of them using a different language. In
the second half of the previous century, the foundations of algebraic geometry
were completely reformulated. Commutative algebra was
established as a solid basis for geometric considerations; the language
of *schemes*, *vector bundles* and *sheaf cohomology* was
introduced. This has led to a unified language, many new results and a deeper
understanding of classical ones; it has allowed simplification of proofs as
well as substantial generalisation of results.

In algebraic geometry, the computer has become an increasingly important tool for complicated calculations. Computer algebra is concerned with the development of efficient algorithms and their implementation. It emerged in the 1970's and gained popularity via computer algebra systems such as Macsyma, Reduce, Maple, Mathematica, MuPAD, etc. The special purpose computer algebra systems SINGULAR, and Macaulay have been designed for computations in commutative algebra, algebraic geometry and singularity theory. They are freely available and can be applied to a variety of problems in modern algebraic geometry.

More detailed information is available from
Dr. Bernd Kreussler

e-mail: | Bernd.Kreussler@mic.ul.ie |

phone: | 061 – 204578 |

office: | N29 |