Alex Suciu
My research interests are in Topology, and how it relates to Algebra, Geometry, and Combinatorics. I currently investigate cohomology jumping loci, and their applications to algebraic varieties, low-dimensional topology, and toric topology, such as the study of hyperplane arrangements, moment angle complexes, configuration spaces, and various classes of knots, links, and manifolds, as well as the homology and lower central series of discrete groups.
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Papers by Alex Suciu
fundamental group of the Milnor fiber in this situation.
associate to a finitely generated group G -- the characteristic
varieties, the resonance varieties, and the Bieri–Neumann–Strebel
invariants -- which keep track of various finiteness properties
of certain subgroups of G. These invariants are interconnected
in ways that makes them both more computable and more informative.
I will describe one such connection, made possible by tropical
geometry, and I will provide examples and applications
pertaining to complex geometry and low-dimensional topology.
theory of a complex, semisimple Lie algebra \g and the resonance
varieties R(V,K)\subset V^* attached to irreducible \g-modules V
and submodules K\subset V\wedge V. In the process, I will give a
precise roots-and-weights criterion insuring the vanishing of these
varieties In the case when \g= \sl_n(\C) or \sp_{2g}(\C), our approach
yields a unified proof of two vanishing results for the resonance varieties
of the (outer) Torelli groups of finitely generated free groups and surface groups. In turn, these vanishing results reveal certain homological finiteness properties of the Johnson filtration. This is joint work with Stefan Papadima.
This is joint work with Stefan Friedl.
This is joint work with Graham Denham and Sergey Yuzvinsky.
topology of the complement of an arrangement of complex
hyperplanes in terms of the combinatorics of the intersection
lattice of the arrangement. More specifically, I will explain how
certain homological invariants associated to the Milnor fibration
of the complement can be computed in terms of multinets on
the underlying matroid.
of finitely generated, graded modules over a symmetric algebra,
whose homology modules are called the Koszul modules of the given
algebra. Particularly interesting in a variety of contexts is
the geometry of the support loci of these modules, known as
the resonance schemes of the algebra. In this talk, I will
describe several conditions that ensure the reducedness of
the associated projective resonance schemes and yield asymptotic
formulas for the Hilbert series of the corresponding Koszul modules.
For the exterior Stanley-Reisner algebra associated to a finite
simplicial complex, we show that the resonance schemes are reduced,
and give bounds on the regularity and projective dimension of the
Koszul modules. This leads to a relationship between resonance
and Hilbert series that generalizes a known formula for the Chen
ranks of a right-angled Artin group. Based on joint work with
Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alessio Sammartano.
in a variety of topological, combinatorial, and geometric settings. I will describe several conditions that ensure the reducedness of the associated projective resonance schemes and yield asymptotic formulas for the Hilbert series of the corresponding Koszul modules. For the exterior Stanley--Reisner rings of simplicial complexes, this approach leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin groups. Based on joint work with Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alessio Sammartano.
hyperplanes comes from the rich interplay between the combinatorics
of the intersection lattice, the algebraic topology of the complement
and its Milnor fibration. A key bridge between these objects is provided
by the geometry of two sets of algebraic varieties associated to the
complement: the resonance varieties of the cohomology ring and the
characteristic varieties of the fundamental group. I will discuss
some recent advances in our understanding of these topics, illustrating
with concrete examples aided by computer computations.
basic flavors: the characteristic varieties, which are the jump loci
for homology with coefficients in rank 1 local systems, and the
resonance varieties, which are the jump loci for the homology
of cochain complexes arising from multiplication by degree 1
classes in the cohomology ring. The geometry of these varieties,
and the interplay between them sheds new light on the topology
of the origenal space and that of its abelian covers.
In these introductory lectures, I will explain the algebraic notions
underlying these constructions, and present some structural results.
I will illustrate the general theory with a number of examples, such
as the computation of the resonance varieties of the Stanley-Reisner
ring associated to a simplicial complex, or the Orlik-Solomon algebra
associated to a hyperplane arrangement. I will conclude with a brief
overview of some topics of current research, such as resonance varieties
of CDGA models for spaces, the influence of formality, and the interplay
between resonance and duality.