| Takuro Abe | Hyperplane arrangements and Hessenberg varieties
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Abstract:
Hessenberg varieties were introduced by De Mari, Procesi and Shayman as a generalization of
flag varieties. Recently, for the regular nilpotent and regular semisimple cases, their topologies are intensively studied, and related to combinatorial and (geometric) representational aspects. However, the algebraic structure of their cohomology groups have been unknown except for the case of type A.
Recalling the fact that their cohomology rings are isomorphic to the coinvariant algebras, and
Kyoji Saito's original proof of the freeness of Weyl arrangements by using basic invariants,
we give a presentation of the cohomology group of a regular nilpotent
Hessenberg variety by using a logarithmic derivation module of
certain hyperplane arrangements (ideal arrangements) coming
from the Hessenberg variety. Also, several properties of
cohomology groups like complete intersection, hard Lefschetz
properties and Hodge-Riemann relations are shown.
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| Karim Adiprasito |
T < 4 E |
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| Pauline Bailet |
A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements |
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Let S be a smooth proper complex variety of dimension equal to or
greater than 2, and let D= D_1 + ... + D_n a divisor on S (all D_i irreducible).
Consider a rank one local system L on U=S\D, with
monodromy t_i in C* around D_i. We give a general
vanishing result for the first twisted cohomology group H^1(U,L),
generalizing a result due to Cohen-Dimca-Orlik. Then we give some
applications in the context of hyperplane arrangement, namely local
system cohomology of line arrangement complements. In particular, we
will apply our result to determine the monodromy action on the Milnor
fiber of two hyperplane arrangements: the Ceva arrangement and the
exceptional reflection arrangement of type G_{31}. (Joint work with A. Dimca and M. Yoshinaga)
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Christin Bibby |
A generalization of Dowling lattices
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Abstract: We are interested in certain arrangements of subvarieties on which a complex reflection group acts. We give a combinatorial description of its poset of layers (connected components of intersections) as a generalization of Dowling lattices. While these posets are not in general lattices, they still share some nice properties with Dowling and partition lattices. This combinatorial structure is an aid in trying to understand the cohomology of the complement as a representation of the complex reflection group. |
| Nero Budur |
Absolute sets and the Decomposition Theorem
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Abstract: Cohomology jump loci of rank one local systems for
complements of hyperplane arrangements are of a very special nature.
Later, it was discovered that the special properties hold for all
smooth quasi-projective varieties. In this talk we report on a new
finding: replace "cohomology" by any other naturally-defined functor,
and the corresponding jump loci for rank one local systems satisfy the
same special properties. For higher ranks, we end up with a conjecture
of André-Oort type for special loci of local systems. The conjecture
is true in rank one, and if true in general, it would provide a simple
proof in all generality of the DecompositionTheorem of
Beilinson-Bernstein-Deligne-Gabber. Joint work with Botong Wang. |
| Daniel Cohen |
Topological complexity of the Klein bottle
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Abstract:
Topological complexity is a numerical homotopy-type invariant introduced by Farber about 15 years ago, motivated by the motion planning problem from robotics. For a space X, this invariant TC(X), the sectional category of the fibration sending a path in X to its two endpoints, provides a measure of the complexity of navigation in X. Computing TC(X) is sometimes easy, sometimes hard. I will attempt to illustrate this, primarily with surfaces, including the Klein bottle, which is recent joint work with Lucile Vandembroucq (Univ. Minho). |
| Graham Denham |
What are the critical points of the master function of a nonrealizable
matroid?
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Abstract:
It is well-known that complex hyperplane arrangements can be
conveniently resolved to normal crossing divisors with the help of the
permutohedral toric variety. The cohomology algebras of the resulting
wonderful compactifications are not only matroid invariants, but
Adiprasito, Huh and Katz (2015) found that Hodge-theoretic constraints
imposed on them by complex geometry persist for arbitrary matroids.
The maximal likelihood variety of a complex arrangement captures the
set of critical points of all rational functions with poles and zeros
on the arrangement. Its bidegree (as a biprojective variety) agrees
with the h-vector of the underlying matroid's broken circuit complex.
I will describe work with Federico Ardila and June Huh in which we
construct a combinatorial analogue of the maximal likelihood variety
for arbitrary matroids, based on a construction that doubles the
permutohedron. Although the concept in the title is fictitious, the
analogy leads to a proof that the h-vector of the broken circuit
complex is a log-concave sequence. |
| Clément Dupont |
Bi-arrangements related to zeta values and polylogarithms
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Abstract: I will describe some geometric configurations
(bi-arrangements of hyperplane and toric type) whose associated
cohomology groups encapture arithmetic datum about values of
the Riemann zeta function at integers, and classical
polylogarithms. In particular, we will see how topological and
combinatorial methods allow one to compute the coefficients of
some linear forms in zeta values that are related to problems
in diophantine approximation. This is partly joint with Javier
Fresàn (ETHZ). |
| Michael Falk |
Milnor fibers and characteristic varieties of 3-arrangements
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| Let A be a union of linear hyperplanes in
C^3 with defining equation Q(x,y,z)=0. The Milnor fiber F
of A is the affine variety defined by Q(x,y,z)=1. Let S be
the projective closure of F in P^3, defined by
Q(x,y,z)=w^n, n=deg(Q). We define a resolution of S
determined by the incidence graph of the irreducible
components of A, and show that the rational homology of F
is determined by the local cohomology of the boundary
divisor F \cap {w=0}. As a consequence we obtain a combinatorial algorithm to compute the first betti number of F. The same argument applies when Q is not reduced, which implies that rational points on the first characteristic variety of A can be detected combinatorially. Applying a 2013 result of Artal-Bartolo, Cogolludo-Agustín, and Matei, one concludes that the first characteristic variety of A, the jumping locus for cohomology of rank-one local systems over the complement of A, is determined combinatorially. By general results, these statements apply to arrangements of arbitrary rank. |
| Eva-Maria Feichtner |
A Leray model for Orlik-Solomon algebras |
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Although hyperplane arrangement complements are rationally formal,
we note that they have non-minimal rational (CDGA) models which are
topologically and combinatorially significant, in view of recent
work of Bibby and Dupont, as well as foundational results of De Concini
and Procesi. We construct a family of CDGAs which interpolates
between the Orlik-Solomon algebra and the cohomology algebras of
arrangement compactifications.
Our construction is combinatorial and extends to all matroids,
regardless of their (complex) realizability. The approach makes
use of the notion of combinatorial blowups [F & Kozlov] as well as
ideas in previous work with Yuzvinsky.
This is joint work with Christin Bibby and Graham Denham.
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| David Garber | On chamber counting formula for conic-line arrangements. |
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In this talk, we start by presenting Zaslavsky's well-known
deletion-restriction chamber counting formula, and then we present its
generalization for the case of conic-line arrangements. We conclude
the talk by remarks on possible connections to other
restriction-deletion formulas. This is joint work with Michael Friedman.
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| June Huh |
Negative correlation and Hodge-Riemann relations
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Abstract: All finite graphs satisfy the two properties mentioned in the title. I will explain what I mean by this, and speculate on generalizations and interconnections. This talk will be non-technical: Nothing will be assumed beyond basic linear algebra. |
| Toshitake Kohno |
Higher category extensions of holonomy maps
for hyperplane arrangements
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Abstract:
We explain a method to construct higher category extensions of
holonomy representations of homotopy path groupoids
based on Chen's formal homology connections.
Applying this general method, we describe an explicit form of
higher holonomy for homotopy path groupoids in the case
of the complement of hyperplane arrangements.
In particular, by means of a 2-functor from the path
2-groupoid of the configuration space, we construct
representations of the 2-category of braid cobordisms. |
| Anatoly Libgober |
Representations of Manin-Schechtman groups.
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Abstract. I will outline a constriction of representations of
Manin-Schechtman higher braid groups over the field of rational
functions in several variables and over the ring of Laurent
polynomials in one variable. This construction is inspired by one of constrictions of Burau
representation and generlizes it. |
| Ivan Marin |
Lattice extensions of Hecke algebras
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Abstract : We shall first recall the construction of Hecke algebras associated
to a (finite) Coxeter group, and then more generally to
a complex reflection group. We then introduce extensions of them,
which are related to natural actions of the reflection group
on lattices, and describe their structure. |
| Luis Paris |
Artin groups, symmetries, and linear representations
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Abstract:
The talk is based on a join work with Olivier Geneste. One of the most popular questions on braid groups has been for a long time whether these groups are linear. This question was solved in the late 90s by Bigelow and Krammer. Krammer's construction was then extended to all simply laced Artin groups of spherical type by Cohen--Wales and Digne, and, afterwards, to all simply laced Artin groups without triangles by myself. Now, we would like to extend the construction to the other Artin groups, or, at least, to some Artin groups that are not simply laced. An answer partially lies in some works by Digne and Castella that, in particular, provide such a construction for the Artin groups of type $B_n$, $F_4$, and $G_2$, by means of symmetries of Coxeter graphs. We will explain this story in more detail, show how Digne's ideas can be extended to other Artin groups, and what are the limits of such a construction. |
| Giovanni Paolini |
Shellability of posets of components associated to toric arrangements
defined by root systems
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Abstract:
The poset of intersections of a linear hyperplane arrangement is a geometric lattice, and in particular it is shellable. It is natural to ask if the poset of (connected components of) intersections of a toric arrangement is also shellable. This is indeed the case for arrangements associated to root systems, and I will give an idea of how to construct such a shelling (joint work with Emanuele Delucchi and Noriane Girard). |
| Mario Salvetti |
Cohomology of Configuration Spaces and Artin groups: applications |
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Several (topologically and combinatorially based) methods for the cohomology of Artin groups were previously developed. As an application to a geometric situation which is of a very wide interest, we completely calculate the integral (co)homology of the so called hyperelliptic locus, namely the space $E_n$ of genus $g$ curves ramified over $n=2g+1$ points.
The main part of such (co)homology is described by the (co)homology of the braid group with coefficients in a symplectic representation, namely the braid group acts on the first homology group of a genus $g$ surface in a standard way.
Our computations confirm some previous experimental results, showing that such groups have only $2-$torsion. We also find the Poincar\'e series for the (co)homology, in particular the series for the stable groups (joint work with F. Callegaro).
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| Henry K. Schenck |
Multidimensional Persistent Homology
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Abstract: A fundamental tool in topological data analysis is persistent homology, which allows
detection and analysis of underlying structure in large datasets. Persistent homology (PH)
assigns a module over a principal ideal domain to a filtered simplicial complex. While the theory
of persistent homology for filtrations associated to a single parameter is well-understood, the
situation for multifiltrations is more delicate; Carlsson-Zomorodian introduced multidimensional
persistent homology (MPH) for multifiltered complexes via multigraded modules over a polynomial
ring. We use tools of commutative and homological algebra to analyze MPH, proving that the MPH
modules are supported on coordinate subspace arrangements, and that restricting an MPH module to
the diagonal subspace V(x_i-x_j | i \neq j) yields a PH module whose rank is equal to the rank
of the original MPH module. This gives one answer to a question asked by Carlsson-Zomorodian.
This is joint work with Nina Otter, Heather Harrington, Ulrike Tillman (Oxford).
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| Simona Settepanella |
Intersection lattice of Discriminantal arrangement and hypersurfaces
in Grassmannian.
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Abstract: In 1989 Manin and Schechtman considered a family of arrangements of hyperplanes
generalizing classical braid arrangements which they called the Discriminantal arrangements. Such an arrangement consists of parallel translates of collection of n hyperplanes in general position in C^k which fail to form a generic arrangement in C^k.
In 1994 Falk showed that the combinatorial type of Discriminantal arrangement depends on the
collection of n hyperplanes in general position in C^k. In 1997 Bayer and Brandt divided generic arrangements in C^k in "very generic" and "non very generic" depending of the intersection lattice of associated Discriminatal arrangement.
In 1999 Athanasiadis provided a full description of intersection lattice for Discriminantal arrangement in the very generic case.
More recently, in 2016, Libgober and Settepanella gave a description of rank 2 intersection lattice of Discriminantal arrangement in non very generic case, providing a sufficient geometric condition for a generic arrangement in C^k to be non very generic. In this talk we will recall their result and we will show that non very generic arrangements in C^3 satisfying their condition correspond to points in a degree 2 hypersurface in the complex Grassmannian Gr(3,n). This is a joint work with S. Sawada and S. Yamagata |
| Alexander Varchenko |
Potentials of a family of arrangements of hyperplanes
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There are three sources of commutative algebras. The first is quantum integrable models. One has a vector space, the space of states, and an algebra of commuting linear operators, quantum Hamiltonians. The second is the cohomology algebras of algebraic varieties, if the odd cohomology is zero. The third source is the algebras of functions on the critical sets of functions. In interesting cases the algebras of different sources become isomorphic and that indicates some interesting dualities. It is important to study these algebras and their interrelations.
I will discuss the algebra of functions on the critical set of a master function associated with an arrangement of hyperplanes and show how all information about that algebra can be packed into two potential functions.
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| Max Wakefield |
Some algebra and combinatorics of matroid Kazhdan-Lusztig
polynomials
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Abstract: The Kazhdan-Lusztig polynomial of a matroid is a
combinatorial invariant which computes the intersection
cohomology of the reciprocal plane if the matroid is
representable over some field. There are many unknown
properties about this polynomial. For example, it is
conjectured that the coefficients are non-negative for all
matroids. In this talk we will briefly review how these
polynomials are similar to the classical Kazhdan-Lusztig
polynomials. Then we will discuss a combinatorial formula for
their coefficients in terms of flags in the matroids lattice of
flats. This formula is given by sums of ``top-heavy'' pairs of
flag Whitney numbers of the second kind and we will note the
recent result of Huh and Wang on the top-heavy conjecture.
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