Combinatorial and Toric Homotopy (1 - 31 Aug 2015)

on the occasion of Professor Frederick Cohen's 70th Birthday
Jointly organized with Department of Mathematics, NUS

~ Abstracts ~

Exact sequences for braid and mapping class groups
Jon Berrick, National University of Singapore

This talk describes work done/being done at NUS on exact sequences that relate braid groups and mapping class groups of surfaces, beginning with work by Wong, Wu and Berrick jointly with Fred Cohen, and including work with two former postdoctoral fellows of Wu and Berrick.

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Free loop cohomology of complete flag manifolds
Matthew Burfitt, University of Southampton, UK

We will discuss the calculation of the cohomology of the free loops space of complete flag manifolds. To demonstrate this we will focus on the case of the complete flag of the special unitary group, as our primary example.

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Koszul complexes and Toric manifolds
Haibao Duan, Chinese Academy of Sciences, China

We discuss two types of Koszul complexes arising from free actions of torus groups on manifolds.

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Cohomology rings of moment-angle complexes
Feifei Fan, Nankai University, China

Associated to every finite simplicial complex $K$, there is a moment-angle complex $\mathcal {Z}_{K}$; $\mathcal {Z}_{K}$ is a compact manifold if and only if $K$ is a generalized homology sphere. In this talk, I will introduce some results on the cohomology rings of moment-angle complexes. First, I will give the cohomological transformation formulae of $\mathcal {Z}_{K}$ induced by some combinatorial operations on the base space $K$, such as the connected sum operation on Gorenstein* complexes and the stellar subdivisions on simplicial spheres. Second, I will show that the indecomposable property of $\mathcal {Z}_{K}$ (i.e. $\mathcal {Z}_{K}$ is a \emph{prime manifold}) when $K$ is a flag $2$-sphere by proving the indecomposable property of their cohomology rings. Our results can be used to solve the cohomological rigidity problem for some moment-angle manifolds.

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Topology of large random spaces
Michael Farber, Queen Mary University of London, UK

In the first part of the talk I will survey several most popular models producing large random spaces and manifolds. The second part of the talk will focus on a new probabilistic model involving several probability parameters describing the statistical properties in random simplicial complexes of various dimensions. The multi-parameter random simplicial complexes interpolate between the Linial-Meshulam random complexes and the clique complexes of random graphs. The Homological Domination Principle states that the Betti number in one specific dimension (the Critical Dimension), which depends on the probability multi-parameter, significantly dominates all other Betti numbers. Attempting to understand the general picture of properties of random simplicial complexes with a fixed critical dimension leads to some interesting conjectures, which I will discuss in my talk.

I will also describe some results about the probabilistic treatment of the Whitehead Conjecture concerning aspherical 2-dimensional complexes.

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Topology of large random spaces
Michael Farber, Queen Mary University of London, UK

Large random spaces (simplicial complexes and manifolds) are interesting mathematical objects which can be used to model large systems in various practical applications. Large random spaces can be also used in pure mathematics for creating examples with curious combinations of topological properties.

In my lectures I will describe several mathematical models producing random simplicial complexes and some recent results about topological properties of these spaces.I will also describe some results about the probabilistic treatment of the Whitehead Conjecture concerning aspherical 2-dimensional complexes.

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The role of the Moore space in the unstable homotopy groups of spheres
Brayton Gray, The University of Illinois at Chicago, USA

I will trace the beginning of the appearance of Moore space as a key differential in unstable homotopy, and explain how this led to the secondary suspension and finally the secondary EHP sequences. These functioned via universal mapping properties which have their own interesting history.

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Linearization of algebraic structures via functor calculus
Manfred Hartl, University of Valenciennes and Hainaut-Cambresis, France

As the result of a long optimization process in categorical algebra, the notion of semi-abelian category allows for developing highly non-trivial algebraic theory in a very general framework which encompasses almost all algebraic structures usually studied, and even certain types of objects having additional topological or analytic structures, such as compact topological groups and C$^*$-algebras. In particular, a new approach to commutator theory, internal object actions including the important special case of representations (Beck modules), to crossed modules and cohomology is being developed in this framework. This even leads to the foundation of categorical Lie theory generalizing both classical Lie theory (for groups) and recent non-associative Lie theory (for loops) to a broad variety of other non-linear algebraic structures. The key new tool consists of an (algebraic) functor calculus in the framework of semi-abelian categories.

In this talk I will focus on basic functor calculus, categorical commutator and Lie theory which so far culminates in associating a linear operad (i.e. type of algebras) to any suitable type $\mathbb{T}$ of non-linear algebraic structure (actually, algebraic theory in the sense of Lawvere); the algebras over this operad are supposed to play the same role for objects with structure $\mathbb{T}$ as Lie algebras play for groups and Sabinin algebras turned out to play for loops. This is proven for certain aspects and is conjectural for others (especially some concerning object bialgebras generalizing group and loop algebras); these are the subject of current joint investigations with J. Mostovoy, J.-M. Pérez-Izquierdo and I. Shestakov who developed the corresponding theory in the case of loops.

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Higher limits, homology theories and fr-language
Sergei Ivanov, The Chebyshev Laboratory of St. Petersburg State University, Russia

The aim of these lectures is to present a new approach to some 'homology theories' of algebraic structures (groups, algebras, abelian groups) like homology of groups, Hochschild and Cyclic homology of algebras and others. The origin of this approach was made by Quillen in his work "Cyclic cohomology and algebra extensions" and developed by R.Mikhailov, I.B.S.Passi and others. This approach is based on consideration of the category of presentations of an algebraic structure. The global aim of this approach is to make a link between homological algebra and combinatorial group (algebra) theory in the style of Hopf's formula. To provide required technique we discuss such topics as, classifying space of a category and its homology, higher limits of representations of categories, monoadditive representations and their suspensions.

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On nontriviality of homotopy groups of spheres
Sergei Ivanov, The Chebyshev Laboratory of St. Petersburg State University, Russia

The talk will be devoted to the proof of the fact that n-th homotopy group of 2-sphere is not trivial for n>1. Moreover, we will discuss the non-stable Adams spectral sequence and some conjectures about its structure that allow to prove this result simplier.

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Local structure in atomic systems
Emanuel Lazar, University of Pennsylvania, USA

Many physical systems can be abstracted as large sets of point-like particles. Understanding how such particles are arranged is thus a very natural problem, though aside from perfect crystals and ideal gases, describing this structure in an insightful yet tractable manner can be difficult. We consider a configuration space of local arrangements of neighboring points, and consider a stratification of this space via Voronoi topology. Theoretical results help explain limitations of purely metric approaches to this problem, and computational examples illustrate the unique effectiveness of topological ones. Applications in computational materials science are considered.

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Neural systems from an algebraic topology point of view
Ran Levi, The University of Aberdeen, UK

The brain is without a doubt the most complicated complex system science ever studied. However, at a basic level, the brain, or any part of it, is a network of neurons which can be described as a directed graph. It is also natural to think about connections among various brain regions in graphical terms. Electrical activity in the brain can similarly be viewed as highlighting certain subgraphs of an ambient graph. This approach has been used by theoretical neuroscientist for a while, employing mostly the tools of classical graph theory.

The Blue Brain model is an intricate and biologically accurate computer simulation of the neocortical column - a formation of roughly 31,000 simulated neurons. The data used in our study arises from forty two columns in six clusters of seven columns each, generated by the Blue Brain algorithm. The first five clusters are based on biological data extracted from five individual rats, while the sixth cluster is based of the data averaged across the five individuals. In each case the algorithm is ran seven times to create the columns. Like in a biological brain the resulting models are similar, but not identical, as the algorithm is in part stochastic in nature. The simulation allows scentists to ask questions of the model which are intractable by wet lab techniques. In particular, it is very easy to get the entire connectivity matrices of these columns, as well as activate them and obtain information about emergent chemical and electrical properties.

Combinatorial and algebraic topology are naturally suited to associating various invariants and metrics to directed graphs. In this talk I will report on an ongoing collaboration with the Blue Brain team. In particular I will show how rather naive techniques of algebraic topology are used to extract useful information from the system. These techniques can also be used in the study of neurological fMRI data and other large networks. This project is the practical, experimental part of a larger initiative which includes a highly theoretical component.

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On the Lie algebra of braid groups and Lambda algebra
Jingyan Li, Shijiazhuang Tiedao University, China

In this talk, we give a connection between the intersections of some Lie ideals from the Lie algebra of pure braid group over the sphere and the Lambda algebra.

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Invariants of 3-manifolds from intersecting kernels of Heegaard splittings
Fengling Li, Dalian University of Technology, China

In this talk, we will derive some invariants of 3-manifolds from the intersecting kernels of their Heegaard splittings. This is a joint work with Prof. Fengchun Lei and Prof. Jie Wu.

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The fixed point data and equivariant Chern numbers
Jun Ma, Fudan University, China

First, we follow the line of tom Dieck to consider the unitary $T^k$-equivariant cobordism and prove equivariant Chern numbers determine the bordism class of a stable almost complex $T^k$-manifold. For given set of trivial $T^k$ action connected compact unitary manifolds and G~complex vector bundle of them, we find necessary and sufficient conditions for the existence of a stable almost complex $T^k$ manifold with the given fixed point data by using equivariant Chern numbers.

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Characteristic classes of surface bundles and configuration spaces
Miguel Xicoténcatl Merino, Cinvestav, Mexico

If $S_g$ is an orientable surface of genus $g$, characteristic classes of $S_g$-bundles are given by the cohomology of $B \text{Diff}^+(S_g)$, the classifying space of the group of orientation preserving diffeomorphisms of $S_g$. For $g \geq 2$, it is well known that $H^*( B \text{Diff}^+ (S_g)) = H^*(\Gamma_g)$, where $\Gamma_g$ is the mapping class group of $S_g$. The case $g=1$ is special and the cohomology of $B\text{Diff}^+(T)$ is given in terms of modular forms. Moreover, the case with marked points was treated by F. Cohen who expressed $H^*(\Gamma_1^k)$ in terms of modular forms and the cohomology of configuration spaces and certain mapping spaces. In the non-orientable setting we carry out similar computations for the case of the projective plane and the Klein bottle and show everything boils down to computing the cohomology of configuration spaces.

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Estimating thresholding levels for random fields via Euler characteristics
Anthea Monod, Duke University, USA

We introduce Lipschitz-Killing curvature (LKC) regression, a new method to produce (1-\alpha) thresholds for signal detection in random fields that does not require knowledge of the spatial correlation structure. The idea is to fit the observed empirical Euler characteristics to the Gaussian kinematic formula via generalized least squares, which quickly and easily provides statistical estimates of the LKCs -- complex topological quantities that are otherwise extremely challenging to compute, both theoretically and numerically. With these estimates, we can then make use of a powerful parametric approximation of Euler characteristics for Gaussian random fields to generate accurate (1-\alpha) thresholds and p-values. Furthermore, LKC regression achieves large gains in speed without loss of accuracy over its main competitor, warping, which we demonstrate in a variety of simulations and applications. This is joint work with Robert Adler (Technion), Kevin Bartz (Renaissance Technologies), and Samuel Kou (Harvard), and supported in part by the US-Israel BSF, ISF, NIH/NIGMS, and NSF.

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On the cohomology of partial quotients of moment-angle manifolds
Taras Panov, Moscow State University, Russia

We describe the cohomology of the quotient Z_K/H of a moment-angle complex Z_K by a freely acting subtorus H in T^m by establishing a ring isomorphism of H*(Z_K/H,R) with an appropriate Tor-algebra of the face ring R[K], with coefficients in an arbitrary commutative ring R with unit. The quotients Z_K/H include moment-angle manifolds themselves, projective toric manifolds (the result was known for both these cases), and also 'projective' moment-angle manifolds. The latter admit non-Kaehler complex-analytic structures as LVM-manifolds. We prove the collapse of the corresponding Eilenberg-Moore spectral sequence using the extended functoriality of Tor with respect to `strongly homotopy multiplicative' maps in the category DASH, following Gugenheim-May and Munkholm.

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Regular maps on spheres and projective spaces
Shiquan Ren, National University of Singapore

A map from a topological space to a Euclidean space is said to be k-regular if the image of any distinct k-points are linearly independent. For k-regular maps on Euclidean spaces, lower bounds of the dimension of the ambient space were extensively studied. In this talk, we study the lower bounds of the dimension of the ambient space for 2,3-regular maps on spheres as well as on real and complex projective spaces. Moreover, generalizing the notion of 2-regular maps, we study complex 2-regular maps on spheres and complex projective spaces.

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Persistent homology analysis of porous and granular materials
Vanessa Robins, Australian National University, Australia

Physical properties of porous and granular materials critically depend on the topological and geometric details of the material micro-structure. For example, the way water flows through sandstone depends on the connectivity of its pores, the balance of forces in a grain silo on the contacts between individual grains, and the impact resistance of metal foam in a car door on the arrangement of its cells. These materials are therefore a natural application area for persistent homology.

My colleagues at ANU have developed a state-of-the-art x-ray micro-CT facility for imaging porous materials from bone to sandstone to metal foam. Our work with these images has required the development of topologically valid and efficient algorithms for studying and quantifying their intricate structure. This talk will describe our discrete Morse theory based image analysis algorithms for skeletonization and partitioning, and give a detailed analysis of what the persistent homology diagrams tell us about the micro-structure of various porous and granular materials. Most recently, we have demonstrated an intriguing connection between percolation thresholds and maximally-persistent pairs.

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An approximate nerve theorem
Primoz Skraba, Jozef Stefan Institute, Slovenia

The nerve theorem [Borsuk 48] states that the homotopy type of a sufficiently nice topological space is captured by the nerve of a good cover of that space. Its application is crucial in computational and applied topology as it allows us to replace continuous spaces with combinatorial representation (e.g. simplicial complexes) with which we can then compute homology, cohomology, persistence, etc. of the underlying space.

In the case of persistence, we rarely compute the persistence diagram of a filtration exactly, but rather an approximation of it. In this talk we consider a notion of an epsilon-good cover and its application to computing persistence. Rather than require a good cover, one where all the elements of the cover and their finite intersections are contractible, we first replace the notion of contractible with homologically trivial and then define the notion of an epsilon good cover - one where its all elements and finite intersections are homologically trivial modulo epsilon-persistent classes (i.e. each element can have a small amount of topological noise).

We show an approximation result for the persistence diagram of a filtration of the nerve and the underlying space which depends on epsilon and the maximal dimension of the nerve and show that this bound is tight.

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Algebraic models for configuration spaces
Don Stanley, University of Regina, Canada

The rational homotopy type of a topological space is contained in its model which is a commutative differential graded algebra. We give a model for the ordered configuration space of three points in a closed manifold.

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Discrete morse theory and classifying spaces of 2-categories
Dai Tamaki, Shinshu University, Japan

Given a "good" discrete Morse function $f$ on a finite regular cell complex $K$, we construct an acyclic category $C(f)$, called the flow category of $f$, whose objects are critical cells. Morphisms between critical cells are certain sequences of cells, called flow paths, which serve as an analogue of gradient flows in the case of smooth Morse theory. It turns out that the set of morphisms $C(f)(c,c')$ between two critical cells has a structure of poset, inducing a structure of $2$-category on $C(f)$. We show that there is a zigzag of homotopy equivalences between the classifying space $BC(f)$ of the flow category as a $2$-category and $K$. This result can be regarded as a discrete analogue of a refinement of smooth Morse theory announced by R.~Cohen, J.D.S.~Jones, and G.B.~Segal in early 1990's. This is a joint work with Vidit Nanda and Kohei Tanaka.

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The dual polyhedral product, cocategory and nilpotence
Stephen Theriault, University of Southampton, UK

The notion of a dual polyhedral product is introduced as a generalization of Hovey's definition of Lusternik-Schnirelmann cocategory. Properties established from homotopy decompositions that relate the based loops on a polyhedral product to the based loops on its dual are used to show that if X is a simply-connected space then the weak cocategory of X equals the homotopy nilpotency class of the based loops on X.

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Toric homotopy theory
Stephen Theriault, University of Southampton, UK

These lectures discuss some of the homotopy theory surrounding Davis-Januszkiewicz spaces, moment-angle complexes and their generalizations to polyhedral products. These spaces are defined by gluing together products formed from pairs of spaces (X,A), where the gluing is determined by the faces of a simplicial complex K. The emphasis is on determining homotopy types - of the spaces themselves, their suspensions and their based loop spaces - and showing how these homotopy types depend on a beautiful interplay between the topology of the pairs (X,A) and the combinatorics of the simplicial complex K.

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Persistent homology methods for digraphs and quasi metric spaces with corresponding stability results
Katharine Turner, École Polytechnique Fédérale de Lausanne, Switzerland

One of the most common objects in Applied Topology is the persistent homology of Rips complexes constructed from finite metric spaces. This uses a filtration of clique complexes where each edge is added at the distance between the two corresponding points. Due to the asymmetry of the distance function in quasi metric spaces we can build many analogous objects to Rips complexes. Some involve creating filtrations of clique complexes and performing persistent homology as usual. Other methods also arise by considering filtrations of posets (using poset topology) and of digraphs (to consider strongly connected components). Each of these methods have related stability results bounding the interleaving distance between the persistence modules in terms of the Gromov Hausdorff distances of the quasi metric spaces. If time permits, I will also state other stability results involving filtrations by sub-level sets of Lipschitz like functions over quasi metric spaces.

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Braids and Lie algebras
Vladimir Vershinin, University of Montpellier II, France

In the talk we give a survey of the works that study Lie algebras associated with the groups close to the pure braid groups. We start with the joint work with Fred Cohen, J.Pakianathan and Jie Wu where the Lie algerbra of the upper triangular McCool group was calculated. Then we describe the Lie algebras of surface pure braid groups and the pure mapping class groups of the punctured sphere. Finally we decribe the joint work with Jing Yan Li and Jie Wu on the Lie algebra analogue of Brunnian braids.

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Braids and some other groups arising in geometry and topology
Vladimir Vershinin, University of Montpellier II, France

There will be discussed in detail such notions as configuration spaces, braid groups, their various generalisations and relations with other fields of Mathematics. Among different groups that we shall define and study we shall pay special attention to the Richard Thompson groups and their relations with braid groups.

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Univalent morphisms

There is a class of fibrations in the category of simplicial sets that satisfy the property that, intuitively speaking, every fiber appears in such a fibration at most one time. I will tell about these fibrations and some of their remarkable properties and then outline a proof of the theorem that asserts that the fibration that is universal for Kan fibrations whose fibers belong to a given universe of sets is univalent. This theorem is one of the key points in the construction of the univalent model of Martin-Lof inference rules in simplicial sets which is in turn the core element of the program to build univalent foundations of mathematics.

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Toric elements in some bordism groups
Wei Wang, Shanghai Ocean University, China

The main purpose of this talk is to use some toric objects (small covers, quasitoric manifolds, moment angle manifolds) to represent some elements of some bordism groups (Spin bordism groups, framed bordism groups, etc.). This talk will divide into two parts. In the first part, we will collect some basic facts about the structures of various bordism groups. In the second part, we will discuss the possibility to represent the elements of some bordism groups by toric objects. This work is in progress.

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Homotopy decompositions of gauge groups over type $1$ Klein surfaces
Michael West, University of Southampton, UK

Let $P$ be a principal $G$-bundle over a space $X$. The gauge group of $P$ is defined to be its (topological) group of automorphisms. A useful way to calculate some invariants of gauge groups is to decompose them, up to homotopy, as a product of other spaces usually involving $G$ and $X$. In this talk we will discuss decompositions of gauge groups of principal $G$-bundles associated to Real vector bundles in the sense of Atiyah.

In addition, here are some brief remarks on my research. Klein surfaces can be thought of as a Riemann surface endowed with an orientation reversing involution i.e. a $\mathbb{Z}_2$-action. My research concerns Real principal U(n)-bundles over Klein surfaces. The 'Real' is a reference to Atiyah's paper 'K-Theory and Reality' where one considers a lift of the involution to the total space. An automorphism of such a bundle must be equivariant with respect to both $U(n)$ and the lifted involution and it must be a lift of the identity. There has been significant work to calculate some of the invariants of gauge groups (the group of automorphisms) of such bundles. For instance, $\pi_1$ and $\pi(2)$ of the classifying space of the gauge groups were presented in 'The moduli space of stable vector bundles over a real algebraic curve' by Biswas, Huismann and Hurtubise. My results provide homotopy decompositions of the gauge groups and therefore significantly build on this work, providing a large number of higher homotopy groups.

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Self-dual binary codes from small covers and simple polytopes
Li Yu, Nanjing University, China

We explore the connection between simple polytopes and self-dual binary codes via the theory of small covers. We first show that a small cover $M^n$ over a simple $n$-polytope $P^n$ produces a self-dual binary code through the method of Puppe--Kreck if and only if $P^n$ is $n$-colorable and $n$ is odd. These self-dual binary codes can be described by the combinatorics of $P^n$. Moreover, we construct a family of binary linear codes $\mathfrak{B}_k(P^n)$, $0\leq k\leq n$, for a general simple $n$-polytope $P^n$ and discuss when $\mathfrak{B}_k(P^n)$ is self-dual. A spinoff of our investigation gives us some new ways to judge whether a simple $n$-polytope $P^n$ is $n$-colorable in terms of the associated binary codes $\mathfrak{B}_k(P^n)$. In addition, we discuss some applications to doubly-even binary codes.

This is a joint work with Bo Chen and Zhi Lu. This work relates the theory of self-dual binary codes with toric topology and combinatorics of simple polytopes.

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