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Place: N802
Title: An Introduction of the Classical Toda Bracket Theory
Abstract: Toda bracket is an art of constructing homotopy liftings and homotopy extensions of maps, it plays a fundamental role on dealing with composition relations of homotopy classes, there are both classical and category-theoretic descriptions of Toda bracket, and it’s the one of the most important tools to determine homotopy groups. In this talk, we will introduce some properties of this important tool based on the classical view, and introduce some developments and applications of Toda bracket theory we got in recent years.
Place: N802
Title: The quotient spaces of moment-angle complexes
Abstract: Buchstaber-Panov introduced the notion of the moment-angle complex Z. It is defined by a union of certain product spaces of discs and circles with a natural action of a torus T. Topologically, a moment-angle complex allows us to understand a simplicial toric variety as its quotient Z/H, where H is a closed subgroup of T. The computation of the cohomology groups and cup products for such quotient spaces involves using techniques from combinatorics, algebras, and homotopy theory with applications in other fields. This talk summarises known results and problems in the research field of toric topology and reports on recent progress.
Place: N933
Title: Equivariant algebraic K-theory and L-functions of Galois representations
Abstract: The profound connection between the algebraic K-theory and zeta functions was first hinted in two classical results in algebraic number theory: Dirichlet’s unit theorem and the class number formula. Those results were later generalized to Borel’s theorem on ranks of algebraic K-groups of number fields and the celebrated Quillen-Lichtenbaum Conjecture (QLC), proved by Voevodsky and Rost.
In this talk, I will explain how to generalize the QLC to L-functions associated to Galois representations of finite, function, and number fields. On the K-theory side, we twist equivariant algebraic K-theory with equivariant Moore spectra associated to Galois representations. Those equivariant algebraic K-groups with coefficients in Galois representations are then computed by an equivariant Atiyah-Hirzebruch spectral sequence. This is joint work in progress with Elden Elmanto.
Place: N802
Title: The Formality of Sphere Bundles
Abstract: A manifold is called formal when it shares the same rational homotopy type as its cohomology ring. We first study the formality of a sphere bundle over a formal manifold. In this case the formality can be totally determined by the Bianchi-Massey tensor, a 4-tensor on a subspace of the cohomology ring, initially introduced by Crowley and Nordstrom. As a special case, we observe that if a manifold and its unit tangent bundle are both formal, then the manifold has either a zero Euler characteristic or a rational cohomology ring generated by one element. Lastly, we delve into the scenario of a general base manifold, presenting an obstruction of formality.
Place: N802
Title: On configuration space integrals
Abstract: In this talk, I will first recall the configuration space integral for classical knots, a powerful invariant first proposed by Kontsevich and further studied by many people including Bott, Taubes and Thurston. Then I will talk about Watanabe’s work on the 4-dimensional Smale conjecture on the diffeomorphism group of disks. In the end, I will briefly discuss a joint work with Yi Xie regarding relation between configuration space integral and formal smooth structures. This gives an evidence towards a more general conjecture about the relation between configuration space integrals, little disk operad and embedding calculus.
Place: N802
Title: Unital operads, monoids and monads
Abstract: Operads have played an important role in both topology and algebra. It is well known that operads may be viewed as monoids in symmetric sequences. In topology, it is often sensible to work with unital operads and their (reduced) monads. I will discuss a variant of symmetric sequences monoids in which give unital operads. This is joint work with Peter May and Ruoqi Zhang.
Place: N802
Title: Modularity of WRT invariants via resurgence theory
Abstract: We will talk about the (quantum) modularity of WRT invariants or GPPV invariants in general of Seifert fibered homology spheres. It turns out that they are related to the Stokes phenomena of the Ohtsuki asymptotic series through resurgence theory. The talk is based on joint work with Li HAN, Yong LI and David SAUZIN.
Place: N802
Title: Complex geometry of moment-angle manifolds
Abstract: Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds with a holomorphic torus action. A complex structure on a moment-angle manifold Z is defined by a marked complete simplicial fan. When the fan is rational, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In this case, the invariants of the complex structure of Z, such Dolbeault cohomology and the Hodge numbers, can be analysed using the Borel spectral sequence of the holomorphic bundle.
In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F which is equivariant with respect to the algebraic torus action. Examples of moment-angle manifolds include the Hopf manifolds, Calabi-Eckmann manifolds, and their deformations. The holomorphic foliated manifolds (Z,F) are models for irrational toric varieties.
We describe the basic de Rhama and Dolbeault cohomology algebras of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology algebra of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg-Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence and bring it to bear on the study of smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.
We also provide a DGA model for the ordinary Dolbeault cohomology algebra of Z. The Hodge decomposition for the basic Dolbeault cohomology is proved by reducing to the transversely Kaehler (equivalently, polytopal) case using a foliated analogue of toric blow-up.
The talk is based on joint works with Hiroaki Ishida and Roman Krutowski.
Place: N802
Title: Hempel pairs and Turaev Viro Invariants
Abstract: Hempel pairs are pairs of periodic surface bundles with isomorphic profinite completions of the fundamental group. In this talk, I will discuss whether Hempel pairs have identical Turaev Viro invariants.
Place: N802
Title: A link invariant from higher-dimensional Heegaard Floer homology
Abstract: We define a higher-dimensional analogue of symplectic Khovanov homology. Consider the standard Lefschetz fibration of a 2n-dimensional Milnor fiber of the A_{k-1} singularity. We represent a link by a k-strand braid, which is expressed as an element of the symplectic mapping class group. We then apply the higher-dimensional Heegaard Floer homology machinery to this element. We prove its invariance under arc slides and Markov stabilizations, which shows that it is a link invariant.
Place: N802
Title: Configuration space integral, graph cohomology and the diffeomorphism groups of 4-manifolds
Abstract: In 2018, Watanabe used a version of configuration space integral to detect non-trivial smooth families of disk bundles and disproved the 4-dimensional Smale conjecture. More precisely, he related the homotopy groups of Diff(D^4) to the so called graph cohomology. In this talk we will show that the configuration space integral only depends the formal smooth structure, i.e. a lift of the tangent microbundle to a vector bundle. As an application, we can generalize Watanabe’s result and obtain information on the diffeomorphism group of a general compact oriented 4-manifold in terms of the graph cohomology. This is joint work with Jianfeng Lin.
Place: N802
Title: From the real cobordism theory to the Kervaire invariant one problem
Abstract: The real cobordism theory plays an important role in Hill—Hopkins—Ravenel’s solution to the Kervaire invariant one problem and leads to new computations in chromatic homotopy theory at the prime 2. I will report on my joint work in progress with Po Hu, Igor Kriz, Petr Somberg, and Foling Zou on the construction of odd prime analogues of real cobordism theory.
Place: N802
Title: The inverse topology of $K(n)$-local homotopy theory
Abstract: Homotopy theory can be studied through its $p$-localization for all primes $p$. Chromatic homotopy theory allows one to further decompose $p$-local homotopy theory into $K(n)$-local homotopy theory for all heights $n$. In this talk, we introduce an inverse topology on $K(n)$-local homotopy theory. This allow us to identify the entire E1 and E2-pages of the descent spectral sequence for Picard groups of profinite Galois extensions of the $K(n)$-local sphere. This is joint work with Ningchuan Zhang.
Place: N802
Title: Topological complexity of enumerative problems
Abstract: How complex is the solution of a polynomial equation? One way to measure such complexity is provided by topological complexity in the sense of Smale, who proved that any algorithm that finds roots of a polynomial must be sufficiently complicated. However, finding roots of a polynomial is just one basic example of enumerative problems in algebraic geometry. In this talk, we will consider the problem of finding flex points on smooth cubic plane curves. We prove a lower bound for the topological complexity of this problem, and show that our bound is close to be best possible. This talk is based on joint work with Zheyan Wan.
Place: N802
Title: Topological classification of Bazaikin spaces
Abstract: Manifolds with positive sectional curvature have been a central object dates back to the beginning of Riemannian geometry. Up to homeomorphism, there are only finitely many examples of simply connected positively curved manifolds in all dimensions except in dimension 7 and 13, namely, Aloff-Wallach spaces and Eschenburg spaces in dimension 7, and the Bazaikin spaces in dimension 13. The topological classification modelled on the 7-dimensional examples has been carried out by Kreck-Stolz which leads to a complete solution for the Aloff-Wallach spaces. The main goal of this talk is the topological classification of 13-dimensional manifolds modelled on the Bazaikin spaces.