# Topology & Singularities

Topology is the mathematical study of those properties that are preserved through continuous deformations of objects. Topology began with the study of curves, surfaces, and other objects in the plane and three-space. It can be used to abstract the inherent connectivity of objects while ignoring their detailed form.

The basic language of topology is known as point-set topology. Algebraic topology is the study of algebraic objects attached to topological spaces. The algebraic invariants reflect some of the topological structure of the spaces. The algebraic tools include homology groups, cohomology rings, homotopy groups, derived functors, and spectral sequences. Differential topology is the field dealing with differentiable functions on differentiable manifolds, vector fields, and foliations. It arises naturally from the study of differential equations, and is closely related to differential geometry. These fields have many applications in physics, notably in the theory of relativity. Geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. It has come over time to be almost synonymous with low-dimensional topology, concerning in particular objects of two, three, or four dimensions.

David Massey studies the local topology of singular spaces, especially complex analytic singular spaces. Many of his results revolve around finding effectively calculable algebraic data that describe or control the topology of the singularities in the space.

Egon Schulte studies discrete structures in geometry and combinatorics, such as polytopes, maps on surfaces, tessellations on manifolds, complexes, and graphs. The classification of regular abstract polytopes by global or local topological type is a prominent part of his *Abstract Regular Polytopes* research monograph with Peter McMullen.

Alex Suciu‘s research interests are in topology, and how it relates to algebra, geometry, and combinatorics. He studies cohomology jumping loci, hyperplane arrangements, and polyhedral products, as well as various problems concerning the topology and geometry of knots, links, and manifolds, and the homology and lower central series of discrete groups.

The award received by Terence Gaffney from the Brazilian government as a visiting researcher carries with it support for a graduate student for each year. Coupling this with the rise in international stature of our singularities group, we have three visiting students who are working with us this year, with another post-doc, Nivaldo de Góes Grulha Júnior, beginning in February. Here are short biographies of our students to introduce them to the department.

**Guillermo Peñafort Sanchis** is a Spanish PhD student. He obtained his master’s degree from Universitat de València and has recently submmited his PhD Thesis, supervised by Juan José Nuño Ballesteros (Valencia) and Washington Luiz Marar (Universidade de São Paulo). His work is about multiple-point schemes of smooth maps, and his main interests are Algebraic Geometry and Singularity Theory. He visits regularly Brazil (USP and IMPA) and England (Warwick University), and he will stay in Boston for the next three months, working with Prof. Terence Gaffney.”

**Michelle L. S. Molino** is a Brazilian from Espírito Santo state. She went to the Federal University of Espírito Santo, where she got a Bachelor’s degree in Mathematics and later a Master’s degree, studying Singularities while being advised by Prof. Dr. Valmecir Bayer. After that, she taught Calculus and Linear Algebra for engineers at a local college. Two years later she entered the doctoral program at Fluminense Federal University, focusing on Algebraic Geometry. She remained interested in studying singularities, and at the beginning of her second year, after taking a short course on singularities with Prof. Dr. Terence Gaffney, was selected by her adviser Prof. Dr. Abramo Hefez, to receive a Special Visiting Researcher scholarship, given by the Brazilian government, for study at Northeastern University.

**Thiago Filipe da Silva** is a Brazilian from Espirito Santo state. He went to Federal University of Espirito Santo, where he did his undergraduate degree in Mathematics, and later a Master`s degree, studying Algebraic Geometry while being advised by Prof. Dr. Jose Gilvan de Oliveira. Afterwards, he became a professor at Federal University of Espirito Santo. Three years later he entered the doctoral program at University of São Paulo, focusing on Singularity Theory, advised by Prof. Nivaldo de Góes Grulha Junior. During his second year, he received a Special Visiting Researcher scholarship given by the Brazilian government for study at Northeastern University where he will work with Prof. Terence Gaffney.”