scholarly journals Spider evaluation and representations of web groups

2019 ◽  
Vol 28 (04) ◽  
pp. 1950021
Author(s):  
Charles Frohman
Keyword(s):  
Irreducible Component ◽  
Rooted Tree ◽  
Fundamental Groups ◽  
Text Representation ◽  
Poincaré Polynomial ◽  
Representation Variety ◽  
Representation Varieties ◽  
The Web

The topology of [Formula: see text]-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to [Formula: see text] are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate to each geodesic [Formula: see text] from the root of the tree to the tip of a leaf an irreducible component [Formula: see text] of the representation variety of the web, and a graded subalgebra [Formula: see text] of [Formula: see text]. The spider evaluation of geodesic [Formula: see text] is the symmetrized Poincaré polynomial of [Formula: see text]. The spider evaluation of the web is the sum of the symmetrized Poincaré polynomials of the graded subalgebras associated to all maximal geodesics from the root of the tree to the leaves.

scholarly journals TOPOLOGY OF THE REPRESENTATION VARIETIES WITH BOREL MOLD FOR UNSTABLE CASES

2011 ◽  
Vol 91 (1) ◽  
pp. 55-87 ◽  
Author(s):  
KAZUNORI NAKAMOTO ◽  
TAKESHI TORII
Keyword(s):  
Configuration Space ◽  
Cohomology Group ◽  
Cohomology Ring ◽  
Picard Group ◽  
Homotopy Equivalent ◽  
Representation Variety ◽  
Stable Case ◽  
Representation Varieties

AbstractIn this paper we show that, in the stable case, when m≥2n−1, the cohomology ring H*(Repn(m)B) of the representation variety with Borel mold Repn(m)B and $H^{\ast }(F_{n}(\mathbb {C}^m)) \otimes H^{\ast }(\mathrm {Flag}(\mathbb { C}^n)) \otimes \Lambda (s_{1}, \ldots , s_{n-1})$ are isomorphic as algebras. Here the degree of si is 2m−3 when 1≤i<n. In the unstable cases, when m≤2n−2, we also calculate the cohomology group H*(Repn(m)B) when n=3,4 . In the most exotic case, when m=2 , Rep n (2)B is homotopy equivalent to Fn (ℂ2)×PGL n (ℂ) , where Fn (ℂ2) is the configuration space of n distinct points in ℂ2. We regard Rep n (2)B as a scheme over ℤ, and show that the Picard group Pic (Rep n (2)B) of Rep n (2)B is isomorphic to ℤ/nℤ. We give an explicit generator of the Picard group.

scholarly journals RANK TWO TOPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI OF QUASI-PROJECTIVE MANIFOLDS

2018 ◽  
Vol 19 (2) ◽  
pp. 451-485 ◽  
Author(s):  
Stefan Papadima ◽  
Alexander I. Suciu
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Linear Algebraic Group ◽  
The Other ◽  
Fundamental Groups ◽  
Flat Connections ◽  
Other Hand ◽  
Projective Manifolds ◽  
Representation Varieties

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\text{SL}_{2}(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\text{SL}_{n}(\mathbb{C})$ for some $n\geqslant 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.

scholarly journals Representation varieties of the fundamental groups of compact orientable surfaces

1996 ◽  
Vol 93 (1) ◽  
pp. 29-71 ◽  
Author(s):  
A. S. Rapinchuk ◽  
V. V. Benyash-Krivetz ◽  
V. I. Chernousov
Keyword(s):  
Fundamental Groups ◽  
Representation Varieties ◽  
Orientable Surfaces

scholarly journals Learning Sequential Correlation for User Generated Textual Content Popularity Prediction

2018 ◽  
Author(s):  
Wen Wang ◽  
Wei Zhang ◽  
Jun Wang ◽  
Junchi Yan ◽  
Hongyuan Zha
Keyword(s):  
Real World ◽  
Information Overload ◽  
Representation Learning ◽  
Text Representation ◽  
Sequential Model ◽  
Content Popularity ◽  
Popularity Prediction ◽  
Real World Datasets ◽  
Textual Content ◽  
The Web

Popularity prediction of user generated textual content is critical for prioritizing information in the web, which alleviates heavy information overload for ordinary readers. Most previous studies model each content instance separately for prediction and thus overlook the sequential correlations between instances of a specific user. In this paper, we go deeper into this problem based on the two observations for each user, i.e., sequential content correlation and sequential popularity correlation. We propose a novel deep sequential model called User Memory-augmented recurrent Attention Network (UMAN). This model encodes the two correlations by updating external user memories which is further leveraged for target text representation learning and popularity prediction. The experimental results on several real-world datasets validate the benefits of considering these correlations and demonstrate UMAN achieves best performance among several strong competitors.

scholarly journals The Topology of Compact Lie Group Actions Through the Lens of Finite Models

2018 ◽  
Vol 2019 (20) ◽  
pp. 6390-6436 ◽  
Author(s):  
Stefan Papadima ◽  
Alexander I Suciu
Keyword(s):  
Lie Group ◽  
Orbit Space ◽  
Algebraic Model ◽  
Characteristic Classes ◽  
Fundamental Groups ◽  
Algebraic Models ◽  
Sasakian Manifolds ◽  
Finite Dimensional ◽  
Surface Singularities ◽  
Representation Varieties

AbstractGiven a compact, connected Lie group K, we use principal K-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let M be a compact, connected, smooth manifold, which supports an almost free K-action. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe an algebraic model for M with commensurate finiteness and partial formality properties. The existence of such a model has various implications on the structure of the cohomology jump loci of M and of the representation varieties of π1(M). As an application, we show that compact Sasakian manifolds of dimension 2n + 1 are (n − 1)-formal, and that their fundamental groups are filtered-formal. Further applications to the study of weighted-homogeneous isolated surface singularities are also given.

Gifts From the Web: Mentoring

2008 ◽  
Vol 11 (2) ◽  
pp. 83-85
Author(s):  
Howard Wilson
Keyword(s):  
The Web
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