| Aug 27 (Mon) | Aug 28 (Tue) | Aug 29 (Wed) | Aug 30 (Thu) |
| Registration 8:30 - 9:30 |
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| Murillo 9:30 - 10:30 |
Strom 9:30 - 10:30 |
Grant 9:30 - 10:30 |
Sakai 9:30 - 10:30 |
| Coffee 10:30 - 11:00 |
Coffee 10:30 - 11:00 |
Coffee 10:30 - 11:00 |
Coffee 10:30 - 11:00 |
| Menichi 11:00 - 12:00 |
Tsutaya 11:00 - 12:00 |
Hasui 11:00 - 11:30 |
Abbaspour 11:00 - 12:00 |
| Lunch 12:00 - 13:30 |
Lunch 12:00 - 13:30 |
Asao 11:35 - 12:05 |
Lunch 12:00 - 13:30 |
| Stanley 13:30 - 14:30 |
Grbić 13:30 - 14:30 |
Free | Tamaki 13:30 - 14:30 |
| Matsushita 14:45 - 15:15 |
Lahtinen 14:45 - 15:15 |
Wakatsuki 14:45 - 15:15 |
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| Coffee 15:15 - 15:45 |
Coffee 15:15 - 15:45 |
Coffee 15:15 - 15:45 |
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| Cutler 15:45 - 16:15 |
Theriault 15:45 - 16:15 |
So 15:45 - 16:15 |
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| Huang 16:20 - 16:50 |
Iwase 16:20 - 17:20 |
Tanaka 16:20 - 16:50 |
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| Dinner 18:30 - ?? |
| Invited Lecture | Contributed Lecture | Survey |
| Invited Lectures | |
| Hossein Abbaspour | $A_\infty$ Structure of Morse Complex |
| (Université de Nantes) | We lay out the details of Fukaya's $A_\infty$-structure of the Morse complexe of a manifold possibily with boundary. As an application we show that when manifold is closed, this $A_\infty$-structure is homotopically independent of the made choices. (Joint work with François Laudenbach) |
| Mark Grant | Topological complexity of symplectic manifolds |
| (University of Aberdeen) | Topological complexity is a numerical homotopy invariant defined by Farber as part of his topological approach to robot motion planning problems. While it is closely related to the more classical Lusternik-Schnirelmann (LS) category, it is in general harder to compute.
I will present recent joint work with Stephan Mescher, in which we prove that the topological complexity of any closed symplectically atoroidal manifold is equal to twice its dimension. The condition `symplectically atoroidal' means that the symplectic class vanishes on images of tori. This result is the analogue for topological complexity of a theorem of Oprea and Rudyak from 1999, which stated that the LS category of any closed symplectically aspherical manifold equals its dimension. Whereas those authors employed the notion of category weight, we use the notion of TC weight due to Farber and myself. Our proof involves a careful analysis of the Mayer-Vietoris sequence for fibred joins of the free path fibration, and differential forms on the free loop space. |
| Jelena Grbić | $LS$-category of moment-angle manifolds and Massey products |
| (University of Southampton) | We give various bounds for the Lusternik-Schnirelmann category of moment-angle complexes $\mathcal{Z}_K$ and show how this relates to vanishing of Massey products in $H^*(\mathcal{Z}_K)$. In particular, we characterise the Lusternik-Schnirelmann category of moment-angle manifolds $\mathcal{Z}_K$ over triangulated $d$-spheres $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. This characterisation is given in terms of the combinatorics of $K$, the cup product length of $H^*(\mathcal{Z}_K)$, as well as a certain Massey products. Some of the applications include calculations of the Lusternik-Schnirelmann category and the description of conditions for vanishing of Massey products for moment-angle complexes over fullerenes and $k$-neighbourly complexes. |
| Norio Iwase | Monoidal Topological Complexity |
| (Kyushu University) | We know the monoidal topological complexity is a strong version of topological complexity, while the upper bound known by now for the monoidal topological complexity is only twice the dimension. In this talk, we discuss about a more effective upper bounds for it.
Joint with M.Sakai and M.Tsutaya. |
| Luc Menichi | The BV algebra in String Topology of classifying spaces |
| (Université d'Angers) | Let $M$ be a closed oriented manifold.
Chas and Sullivan have defined a product on the shifted homology
of the free loop space $\mathbb{H}_*(LM)$, making this algebra into a Batalin-Vilkovisky algebra.
This Batalin-Vilkovisky algebra is hard to compute.
Dually, let $G$ be a compact connected Lie group. Chataur and myself have defined a product on the shifted cohomology of the free loop space of the classifying space $\mathbb{H}^*(LBG)$, making this algebra into a Batalin-Vilkovisky algebra. Using the computation of the cup product on cohomology of the free loop space of the classifying space $H^*(LBG)$ due to Kuribayashi, for almost any compact connected Lie group $G$ and any field $\mathbb{F}_p$, we compute the Batalin-Vilkovisky algebra $\mathbb{H}^*(LBG;\mathbb{F}_p)$. This is joint work with K. Kuribayashi. |
| Aniceto Murillo | An extension of Quillen approach to rational homotopy theory |
| (Universidad de Málaga) | Having as motivation the understanding of the rational homotopy type of mapping spaces, we extend with full generality the classical Quillen approach to rational homotopy of simply connected spaces. This is based in a new model category structure for (complete) differential graded Lie algebras and the core of this structure lies in the construction of the "Eckmann-Hilton dual" of the classical differential forms on the standard simplices. In fact, the non-existence of this object in the Lie setting has puzzled (rational) homotopy theorists since the beginning of the subject. |
| Keiichi Sakai | The space of short ropes and the classifying space of the space of long knots |
| (Shinshu University) | We will see that the classifying space of the topological monoid of long knots is weakly equivalent to the space of certain 1-manifolds, and that the latter space is weakly equivalent to the space of J. Mostovoy's short ropes. This affirmatively solves the conjecture of Mostovoy. Some generalizations will be discussed.
This is joint work with Syunji Moriya (Osaka Prefecture University). |
| Donald Stanley | LS category from Ganea Conjecture to TC |
| (University of Regina) | We discuss some of the results that came out of the new perspective offered by the counterexample to Ganea's Conjecture, and some related open problems that remain. |
| Jeff Strom | The relation $\mathrm{map}_*(X,Y)\sim *$ |
| (Western Michigan University) | The relation $\mathrm{map}_*(X,Y)\sim *$ provides a powerful lens for studying homotopy-theoretical problems. Its usecan be discerned in retrospect in classical homotopy theory dating back to the 1950s (Eilenberg-Mac Lane spaces, Postnikov systems), more clearly in the theory of localization that began in the late 1960s and early 1970s and in sharp focus in the period since Haynes Miller proved the Sullivan conjecture in theearly 1980s.
If we study this relation in its own right, we are led to a Galois-type correspondence between the (strongly) closed classes of spaces on the domain side (introduced by Bousfield and studied by Farjoun, Chachólski and many others) and, on the target side, classes of spaces called resolving kernels. Strongly closed classes $\mathcal{C}$ have a strange closure property: if $F\to E\to B$ is a fiber sequence, and $F, B\in \mathcal{C}$, then $E\in \mathcal{C}$ as well --- $\mathcal{C}$ is closed under extensions by fibrations. The dual result for resolving classes is not true in general. I'll give an overview of this general theory and solve the puzzle of the non-dualizable closure property byintroducing a dual pair of closure properties (implicitly present in the literature) that hold in the same way and for the same formal reasons in their dual environments. Then we'll feed some topology into the formal setup and derive new closure properties, specific to the application of these ideas to the category of spaces. In particular, we'll see that the closure of strongly closed classes under extensions by fibrations results from the formal closure property together with a theorem of Ganea. The Hilton-Milnor theorem and Mather's second cube theorem work together with the formal closure property for resolving kernels to establish closure properties having to do with wedges and extensions by cofibrations. We'll sketch a proof of the Sullivan conjecture based on these latter closure properties. We'll also see that if $X$ is simply-connected and of finite type, and if $\mathrm{map}_*(X,S^{n}\wedge S^m) \sim *$ for just one choice of $n$ and $m$ (both positive, and not both $1$), then $\mathrm{map}_*(X, K)\sim *$ for every finite-dimensional CW complex $K$. |
| Dai Tamaki | Stratifications on Classifying Spaces of Acyclic Categories |
| (Shinshu University) | This is a report of two projects; one with Hiro Tanaka [TT], and the other with Vidit Nanda and Kohei Tanaka [NTT].
We introduced a stratification on the classifying space $BC$ of a small acyclic (topological) category $C$ in such a way that strata are labelled by objects of $C$. We show that, under a reasonable condition, this stratification is cylindrically normal in the sense of [T] and that the original category $C$ can be recovered as the face category of $BC$. This stratification allows us to show that the "exit-path category" of a cylindrically normal stellar stratified space is a quasicategory [TT]. This is the first step towards the understanding of the relation between two "categories" attached to stratified spaces. As another application, a functorial "cell decomposition" of the canonical $2$-categorical model for discrete Morse theory introduced in [NTT] is obtained. Reference [NTT] Vidit Nanda, Dai Tamaki and Kohei Tanaka, Discrete Morse theory and classifying spaces, http://arxiv.org/abs/1612.08429. [T] Dai Tamaki, Cellular stratified spaces, Combinatorial and Toric Homotopy: Introductory Lectures, vol. 35, Lecure Notes Series, Institute for Mathematical Sciences, National University of Singapore, pp. 305--453, 2018. (http://arxiv.org/abs/1609.04500) [TT] Dai Tamaki and Hiro Lee Tanaka, Stellar stratifications on classifying spaces, http://arxiv.org/abs/1804.11274. |
| Mitsunobu Tsutaya | $A_n$-maps and mapping spaces |
| (Kyushu University) | $A_n$-maps are morphisms between $A_n$-spaces introduced by Sugawara, Stasheff, Boardman-Vogt and Iwase. Sugawara, Stasheff and Iwase characterised the condition when a map between $A_n$-spaces admits an $A_n$-map structure in terms of projective spaces. In this talk, we see that a refinement of this result is realised as a weak homotopy equivalence between the space of $A_n$-maps $A_n(G,H)$ and the space of based maps $Map_*(B_nG,BH)$ from the $n$-th projective space $B_nG$ to the classifying space $BH$. We also see some applications of this results to extension of an evaluation fibration and homotopy commutativity. |
| Contributed Lectures | |
| Yasuhiko Asao | The loop homology algebra of discrete torsion |
| (University of Tokyo) | Let $M$ be a closed oriented manifold with a finite group action by $G$. We denote its Borel construction by $M_{G}$. As an extension of string topology due to Chas-Sullivan, Lupercio-Uribe-Xicoténcatl constructed a graded commutative associative product (loop product) on $H_{*}(LM_{G})$, which plays a significant role in the “orbifold string topology”. They also showed that the constructed loop product is an orbifold invariant. In this talk, we describe the orbifold loop product by determining its "twisting" out of the ordinary loop product in term of the group cohomology of $G$, when the action is homotopically trivial.Through this description, the orbifold loop homology algebra can be seen as R. Kauffmann's “algebra of discrete torsion”, which is a group quotient object of Frobenius algebra. As a corollary, we see that the orbifold loop product is a non-trivial orbifold invariant. |
| Tyrone Cutler | The Homotopy Types of Certain Spinor Gauge Groups Over $S^4$ |
| (Universität Bielefeld) | The gauge group $\mathcal{G}$ of a principal $G$-bundle $P\rightarrow X$ is the group of all $G$-equivariant bundle automorphisms covering the identity on $X$. If $G$ is a compact, connected Lie group then $\mathcal{G}$ will in general be an infinite-dimensional Lie group. However much of its topological structure will be dictated by the topological features of the base space $X$ and by the group-theoretic properties of $G$. In particular, when $X=S^n$, the homotopy-commutativity of $G$ plays a direct role in determining the number of distinct homotopy types amongst all the possible $G$-gauge groups - a number which will in fact be finite for such $G$ and $X$ according to a result of Crabb and Sutherland, despite the fact that the number of bundle-isomorphism classes may be infinite.
In this talk I will discuss recent work relating to the homotopy types of $Spin(6)\cong SU(4)$- and $Spin(7)$-gauge groups over $S^4$. The first part of this work is joint with Stephen Theriault. We completely determine the number of $p$-local homotopy types amongst these gauge groups for $p$ an odd prime, and give strict bounds on the number of $2$-local homotopy types. Many of the interesting features of a typical calculation of this sort will be discussed and highlighted. |
| Sho Hasui | On the quasitoric bundles |
| (University of Tsukuba) | A quasitoric manifold is a $2n$-dimensional smooth manifold with a good action of $T^n=(S^1)^n$ for which the orbit space is a convex polytope. If the orbit space is a product $P_1\times P_2$, then many of the quasitoric manifolds are fiber bundles whose the bottom spaces and the fibers are quasitoric manifolds over $P_i$. In this talk, we'll give a geometric characterization of this kind of fiber bundles and see that $BT^n$ works as the classifying space of them in a certain sense. Moreover, I'll introduce some applications of this fact to the topological classification of bundle-type quasitoric manifolds. |
| Ruizhi Huang | Higher associativity and classification problem of homotopy associative $H$-spaces |
| (Chinese Academy of Sciences) | $A_n$-spaces known as higher associativity were found by J. Stasheff, as the generalisations of $H$-spaces. And there are exactly generalised Hopf constructions whose cofibres are the $n$-projective spaces of the $A_n$-spaces, which somehow reflect the $A_n$-structures themselves analogous to Milnor's classifying spaces for topological groups.
In a joint work with Jie Wu, we proved a necessary condition for the existence of the $A_p$-structures on mod $p$ spaces, and also derived a simple proof for the Hubbuck-Mimura's finiteness of the number of mod $p$ $A_p$-spaces of given rank. As a direct application, we computed a list of possible types of rank 3 mod 3 homotopy associative $H$-spaces. |
| Anssi Lahtinen | String topology of finite groups of Lie |
| (University of Copenhagen) | I will discuss a surprising connection between finite groups of Lie type and string topology of classifying spaces of compact connected Lie groups recently discovered by Jesper Grodal and myself: the cohomology of a finite group of Lie type is a module over the cohomology of the free loop space of the classifying space of the corresponding compact Lie group when the latter cohomology groups are equipped with a string topological multiplication. This module structure provides in particular a new perspective towards the Tezuka conjecture asserting that under certain conditions, the cohomologies of the two objects are isomorphic. |
| Takahiro Matsushita | Relative phantom maps |
| (University of the Ryukyus) | This is a joint work with Kouyemon Iriye and Daisuke Kishimoto.
The de Bruijn-Erdős theorem is a theorem of the graph coloring problem in combinatorics, which states that the chromatic number of an infinite graph coincides with the maximum of the chromatic numbers of the finite subgraphs. The box complex is a well-known functor from the category of graphs to the category of $\mathbb{Z}/2$-spaces, and this connects the chromatic number with $\mathbb{Z}/2$-equivariant topology. From this viewpoint, the de Bruijn-Erdős theorem naturally leads to the following definition of the relative phantom maps: A relative phantom map from a space $X$ to a map $\varphi \colon B \to Y$ is a map $f \colon X \to Y$ such that the restriction $f|_{X^n} \colon X^n \to Y$ to every finite dimensional skeleton has a lift with respect to $\varphi$, up to homotopy. This is a natural generalization of usual phantom maps. A relative phantom map $f \colon X \to Y$ from $X$ to $\varphi \colon B \to Y$ is trivial if $f$ itself has a lift. Moreover, a usual phantom map is clearly a relative phantom map. If $X$ is a suspension, we can consider a finite sum of these two types of relative phantom maps, and we call these phantom maps relatively trivial. In this talk, we consider some conditions that every relative phantom map from $X$ to $\varphi \colon B \to Y$ is trivial, or relatively trivial. |
| Tse Leung So | Classifying homotopy types of gauge groups over 4-manifolds |
| (University of Southampton) | A gauge group arises in physics and is a collection of local symmetries of a physical system. From the point of view of homotopy theory, a gauge group is a special type of mapping space. Atiyah, Bott and Gottlieb showed that it is homotopy equivalent to the loop space of some path-component of mapping space $Map(M, BG)$, where $M$ is a topological space and $BG$ is the classifying space of a topological group $G$. When $M$ is a 4-manifold, the topology of gauge groups has a deep connection to the study of topology and geometry of 4-manifolds. Many topologists have been studying the homotopy types of gauge groups over 4-manifolds for different cases. In this talk, I will talk about the decomposition and classification of gauge groups over 4-manifolds and my recent work on gauge groups over non-spin 4-manifolds. |
| Kohei Tanaka | Topological complexity and L-S category for finite spaces |
| (Shinshu University) | The topological complexity of a space is a numerical homotopy invariant closely related to the motion planning problem. Motion planning rules assign a path between each two points in a space, and the topological complexity tells how many local motion planning rules we should design. It is well known that topological complexity is estimated by Lusternik-Schnirelmann (L-S) category. In this talk, I will present a combinatorial way to compute the topological complexity and the L-S category of a finite space and the associated complex. The central methods used in the calculation are simplicial approximation and barycentric subdivision for simplicial complexes. |
| Shun Wakatsuki | Coproducts in brane topology on rational Gorenstein spaces |
| (University of Tokyo) | We extend the loop product and the loop coproduct to the mapping space from the $k$-dimensional sphere, or more generally from any $k$-manifold, to a $k$-connected space with finite dimensional rational homotopy group, $k \geq 1$. The key to extend the loop coproduct is the fact that the embedding $M \to M^{S^{k-1}}$ is of "finite codimension" in a sense of Gorenstein spaces. Moreover, we give some properties and (non-trivial) examples of them. |
| Survey | |
| Stephen Theriault | Norio Iwase: Some mathematical highlights |
| (University of Southampton) | This talk surveys some of Norio's best known work, focusing on his results in Lusternik-Schnirelmann category, co-H-spaces, and A-infinity spaces. |