The scientific activities of the group are focused on different aspects of Algebraic and Arithmetic Geometry, Combinatorics of Evolutionary Structures, Complex and Differential Geometry, and Geometric Topology.<\/p>\n\n\n\n
The group runs the following Seminars Series:<\/p>\n\n\n\n
Usually, the group organizes workshops, conferences, and summer schools. A list of the upcoming ones is available on this page<\/a>.<\/p>\n\n\n\n Below is the list of the specific topics studied in this research area, each of them with a detailed description and the lists of members and collaborators.<\/p>\n\n\n\n Given an algebraic variety $X$ defined over a field $k$, one can associate to it various cohomology groups: coherent, \u00e9tale, $p$-adic, or even motivic. These groups reflect the geometry, and in case $k$ is of arithmetic interest, the arithmetic of the variety $X$. Using cohomological methods we study, among other things:<\/p>\n\n\n\n There is a deep connection between fundamental groups and rational solutions of polynomial equations. This was first clearly identified by A. Grothendieck in his “anabelian” program, though its roots go back at least to E. Galois. <\/p>\n\n\n\n In order to study it, we blend several techniques, old and new: classical algebraic geometry, p-adic geometry, algebraic topology, cohomology, and stacks.<\/p>\n\n\n\n Our approach is characterized by the application of the stacky formalism to questions in arithmetic geometry. With this approach, we also study minimal fields of definition of varieties, non-abelian cohomology, Brauer groups, singularities.<\/p>\n\n\n\n Logarithmic geometry is an enhanced version of algebraic geometry, where spaces are equipped with an additional structure sheaf, which encodes information of a combinatorial nature (e.g. toric varieties). This recent theory has been fruitfully applied to questions regarding special kinds of degenerations of varieties or other more complicated objects, and compactifications of moduli spaces, for example in the context of mirror symmetry. There are also very interesting interactions with the field of tropical (and non-Archimedean) geometry. Our activity focuses for example on<\/p>\n\n\n\n The moduli space of surfaces of general type is well known to have an intricate structure. Its \u201cgeography\u201d has been extensively studied. Furthermore, a modular compactification of it is the moduliResearch Topics<\/h4>\n\n\n\n
Algebraic and arithmetic geometry<\/h5>
Members<\/h6>
\n<\/i>giulio.bresciani1@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>andrea.dilorenzo@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>marco.franciosi@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>roberto.fringuelli@dm.unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>greg.pearlstein@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>francesco.sala@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>tamas.szamuely@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>mattia.talpo@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\nCohomology of varieties and arithmetic questions<\/h5>
\n
Members<\/h6>
\n<\/i>tamas.szamuely@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\nCollaborators<\/h6>
\n\n<\/div>\n<\/div>\n<\/div>\n
\n\n<\/div>\n<\/div>\n<\/div>\n
\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\nFundamental groups, arithmetic geometry and stacks<\/h5>
Members<\/h6>
\n<\/i>giulio.bresciani1@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\nCollaborators<\/h6>
\n<\/i>angelo.vistoli@sns.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\nLogarithmic and tropical algebraic geometry<\/h5>
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Members<\/h6>
\n<\/i>andrea.dilorenzo@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>mattia.talpo@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\nCollaborators<\/h6>
\n\n<\/div>\n<\/div>\n<\/div>\n
\n\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>siebert@math.utexas.edu<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n\n<\/div>\n<\/div>\n<\/div>\n
\n<\/i>ulirsch@math.uni-paderborn.de<\/a>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\nModuli spaces of surfaces<\/h5>
space of stable surfaces, i.e. semi-log-canonical surfaces with ample canonical divisor. The activity of the group is focused on the analysis of the compactified moduli space $\\mathcal{M}(a,b)$ (where $a=K^2$ and $b$ is the holomorphic Euler characteristic), with particular attention to the case of surfaces with low numerical invariants. Such analysis is given by studying log-canonical pairs via a classical approach and analyzing the singularities, via Deformation Theory, and a detailed study of the canonical ring.
Related goals are to extend the knowledge of the moduli space by analyzing $\\mathbb{Q}$-Gorenstein surfaces and to study the Hodge theoretic approach, by associating to a variety its cohomology and analyzing the induced variation of Hodge structures.<\/p>\n\n\n\n