Both algebra and geometry are mathematical disciplines with a very long history. Algebra can be said to have its origins in the solving of equations which goes back (at least) to Babylonian mathematics while geometry has been with us since the very beginning of mathematics.
In the last two centuries or so both algebra and geometry have expanded enormously, splitting up into many different subfields. Large areas of these subjects have been very closely connected with the "other side" during this development. The inception of homological algebra for instance was heavily inspired by algebraic geometry and topology but also has had a profound influence on these disciplines.
The research in algebra and geometry at SMC is focused on areas that are closely connected with both fields. Interestingly enough, these areas are also subjects that have received a lot of inspiration from some modern areas of theoretical physics such as string theory.
My research is in (arithmetic) algebraic geometry and concerns moduli spaces of curves and of abelian varieties. I am in particular interested in the cohomology of these spaces and the connected automorphic forms. Together with Carel Faber and Gerard van der Geer we have been finding information about the cohomology for low genera/dimension by using the method of counting points over finite fields.
My research interests center around algebraic theories of differential equations, and their use in algebraic geometry, e. g. algebraic D-module theory (how to interpret topology in terms of DE) or asymptotic behaviour of zero-sets of solutions to parameter dependent systems of DE.
Sandra Di Rocco
My recent research is about projective toric varieties and computational Algebraic Geometry.
Algebraic projective varieties are geometrical shapes defined by polynomial equations. There are various ways in which one can study an algebraic variety. One way is to relate it to projective spaces by trying to understand its various embeddings. Quantifying a given embedding is a challenging and fundamental problem in projective geometry. This problem can be looked at from different points of view. Important algebraic invariants are defined using tangential and osculating properties of the embedded variety. These properties are typically analyzed by investigating corresponding properties of associated line bundles. More generally one can look at the the entire derived category of complexes of coherent sheaves or from a different point of view at the equations that cut out the projective variety and the syzygies among them.
I am currently working on some open problems in projective toric geometry aiming at studying embedded toric varieties from the different directions described above. The one-to-one correspondence between projective toric varieties and convex polytopes provides not only a useful computational device but also an opportunity to import results between the two areas.
My research is in algebraic geometry; the moduli spaces of curves and their intersection theory form the main subject. On the one hand, I am interested in the relations between certain standard classes, the tautological classes, on the moduli space of smooth curves and its Deligne-Mumford compactification. A typical question: does every relation on the former space extend to the latter? A large part of my work in this area is in collaboration with Pandharipande. On the other hand, I am interested in describing all the cohomology of the moduli spaces of curves and abelian varieties. In joint work with Bergström and van der Geer, the method of counting curves over finite fields has led to a good theory in low genus/dimension, large parts of which have now been established. Along the way, we have obtained a lot of information on Siegel modular forms, and more recently, on Teichmüller modular forms as well.
My area of research is commutative algebra. I have mainly been concerned with the theory of noetherian rings and modules and made some research on chain conditions, lengths, generating sets of ideals etc.
I am working on various aspects of supersymmetry and their applications. Some of them are unexpected, e.g., EVERY differential equation possesses a supersymmetry (this phenomenon is manifest in terms of Cartan's exterior differential systems). I intend to describe the Lie superalgebras of classical equations of mathematical physics. I intend to describe highest weight representations of distinguished simple Lie superalgebras of string theories (extending current results of B.Feigin et al). The answer might be interpreted in terms of critical phenomena of such materials as graphene. The new examples of simple finite dimensional Lie (super)algebras over fields of characteristic 2 are being obtained. Most of the results are obtained in collaboration with mathematicians from Equa Simulation AB, Stockholm and Independent U. of Moscow.
I am an algebraic geometer broadly interested in questions related to moduli problems. In loose terms, a moduli space is a geometric object that parameterizes other geometric objects (curves, surfaces, cycles, etc). The last few years I have among other things studied foundational questions for algebraic stacks and algebraic spaces --- the modern setting for moduli problems --- and this is an on-going interest. My present research activities also include log geometry, root stacks, non-archimedean geometry, derived algebraic geometry, wild ramification and resolution of singularities.
My research interests are in algebraic geometry and commutative algebra, more specifically Hilbert schemes, Quot schemes and related moduli problems.
Moduli spaces are algebraic objects solving particular moduli problems. The Hilbert scheme parameterizing subvarieties in a given ambient variety is an example of such a moduli space. Important for many branches in algebraic geometry is that many moduli spaces exist quite generally. My research is centered around moduli spaces, their existence and their explicit construction.
Recently, together with several colleagues, I work on the construction of Quot schemes on specific varieties, on specific properties of the Quot space in general, and on the moduli of projective curves with a specific resolution.