Place: N933
Title: Introduction to the classical methods for computing the unstable homotopy groups
Abstract: In this talk, we review classical Toda-style methods for computing unstable homotopy groups of spheres, 2-cell complexes, and SO(n), and present recent progress based on these techniques.
Place: N933
Title: A “Periodicity” phenomenon of the attaching map of the 2-cell complexes
Abstract: In this talk, I would like to introduce our recent work on the “Periodicity” phenomenon of the attaching map of the suspended two-cell CW complex. I will first introduce Selick-Wu’s A^{min}-theory, a theory on the homotopy functor decomposition and becoming a bridge between homotopy theory and the representation theory. Employing this theory, we deduce that the attaching map of the CW complex above manifests a“periodicity”phenomenon. The result essentially furnishes, up to the present, one of the most effective methods for computing the unstable homotopy groups of 2-cell complexes in the J_{4}-range.
Place: N820
Title: The equivalence between two real Seiberg-Witten-Floer homologies
Abstract: Recently, 3&4 manifolds with finite group actions has become a popular topic. Real manifolds form the simplest class among them. Following the strategy of Manolescu, Kronheimer-Mrowka, respectively, Konno-Miyazawa-Taniguchi and Li introduced two versions of real Seiberg-Witten-Floer homologies and developed many interesting applications. In this work, we show the two homology theories are equivalence whenever they are both defined following the strategy developed by Lidman and Manolescu. As application, we identify Froyshov-type invariants from two theories and proved some Smith-type inequalities. In this talk, we first review the two approaches to (real) Seiberg-Witten-Floer homologies, then sketch a proof of our main theorem. If time admits, we talk about the applications and some basic examples.
Place: N820
Title: Lightbulb theorem, embedded surfaces and isotopy of symplectic structures
Abstract: Gabai’s lightbulb theorem classifies embedded spheres in 4-manifolds with a geometric dual sphere. It is a breakthrough in 4-dimensional topology. In this talk, I discuss a joint work with Weiwei Wu, Yi Xie and Boyu Zhang, which classifies the isotopy classes of embeddings of a surface F into the product manifold F cross S2 with a geometric dual. This answers a question of Gabai regarding the generalized lightbulb theorem. Second, we show that the space of symplectic forms on an irrational ruled surface in a fixed cohomology class has infinitely many connected components. This gives the first such example among closed 4–manifolds and answers Problem 2(a) in McDuff–Salamon’s problem list. The proofs are based on a generalization of the Dax invariant to embedded closed surfaces. In the proof, we also establish several properties of the smooth mapping class group of a surface cross S2 is infinitely generated.
Place: Zoom
Title: Endotrivial modules, Picard groups and chromatic Jokers
Abstract: Associated to a finite dimensional cocommutative (graded) Hopf algebra there is a symmetric monoidal \emph{stable module category} and a Picard group whose elements are \emph{endotrivial modules}. These objects have been the focus of a lot of activity in the following cases: finite group algebras over fields of positive characteristic, finite subHopf algebras of the Steenrod algebra $\mathcal{A}$.
I will discuss these and then explain what the algebraic Joker module over $\mathcal{A}(1)$ is and how it generalises to modules over the $\mathcal{A}(n)$ subHopf algebras. For small values of $n$ these can be realised as cohomology of spectra.
Then I will explain how double Joker spectra when viewed in chromatic level 2 give rise to interesting endotrivial modules over $\mathbb{F}_4Q_4$, the group ring of the quaternion group of order 8.
Place: N820
Title: Universal L^2-torsion detects fibered 3-manifolds
Abstract: It is well-known that the Alexander polynomial of a fibered knot must be monic. But in general the converse is not true. In this talk, we introduce the universal L^2-torsion of a 3-manifold, an invariant defined in analogy with the classical Reidemeister torsion, but using tools from L^2-theory. We show that this invariant detects fibered 3-manifolds. Our proof is based on a study of the leading term map on Linnell’s skew field.
Place: N820
Title: On the mapping class group of 2-sphere bundles over the complex projective plane
Abstract: Mapping class groups of smooth manifolds are important objects in geometric topology. In this talk, I will introduce some of my recent progress on the computation of the mapping class groups of the total spaces of a class of 2-sphere bundles over the complex projective plane. In particular, these include some smooth divisors in the product of two complex projective planes as such the Milnor hypersurface. Our strategy is a combination of homotopy theory and surgery theory.
Place: N820
Title: On 7-manifolds with b_2=2: diffeomorphism classification and nonconnected moduli spaces of positive Ricci curvature metrics
Abstract: Understanding how topology influences existence and variation of positive Ricci curvature metrics remains a central theme in manifold topology and geometry. Many potential sources of nontrivial moduli behavior, especially those lacking the geometric structures required by existing general results, remain poorly understood.In this talk we present a partial classification of simply connected 7-manifolds with b_2=2 arising as circle bundles over (CP^1×CP^2)# CP^3, using modified surgery theory to develop and compute refined invariants. The classification provides new manifolds whose spaces and moduli spaces of positive Ricci curvature metrics have infinitely many path components, extending such phenomena beyond previously accessible settings.
Place: Zoom
Title: Equivariant localizing motives for finite groups
Abstract: In this talk, I will give a proposal for a definition of genuine equivariant localising motives for finite groups. This notion will be based on that of idempotent complete equivariantly stable categories. Using isotropy separation arguments on equivariant cubes and the recent insights of Ramzi-Sosnilo-Winges, we will see how to use this version of motives to enhance the algebraic K-theory functor with the structure of multiplicative norms. Time permitting, we will also discuss other applications such as showing that all genuine G-spectra are the K-theory of a G-stable category. This reports on joint work-in-progress with Maxime Ramzi.
Place: N820
Title: Universal simplicial complexes and the mod p Buchstaber invariant
Abstract: Universal simplicial complexes are important objects in toric topology, closely related to the classification of quasitoric manifolds and to small covers and Buchstaber’s invariant computation. In this talk, we present some recent results about the Tor algebra of the Stanley-Reisner ring of the universal simplicial complexes over the field of characteristic p, where p is a prime number and some new applications in toric topology.
Place: N820
Title: Cosmetic surgery and filtered instanton homology
Abstract: In joint work with Lidman and Daemi, we show that if K is a non-trivial knot in S^3, no two integer homology sphere surgeries on K can be oriented homeomorphic. It follows from this and known results that if a knot K in S^3 admits a cosmetic surgery, then the cosmetic slopes must be {2,-2} and K must have trivial Alexander polynomial.
Place: zoom
Title: Detection methods with synthetic spectra
Abstract: The use of synthetic (or motivic or filtered or…) spectra for studying spectral sequences from a homotopical perspective has been a major theme in many recent computations in the stable homotopy groups of spheres. In this talk, I would like to discuss some basic tools and techniques for using synthetic spectra, paired with a detection spectrum X, like real K-theory or topological modular forms, together with some operations on X, to produce infinite periodic families in the stable homotopy groups of spheres. In particular, we want to focus on the utility of the synthetic Hurewicz image of X to detect these families in the classical stable homotopy groups of spheres. We will begin with the height one example of real K-theory together with its natural Adams operations in some detail, before moving onto various generalisations at height 2. If there is time, we will comment on some more recent advances. This is all joint work with Christian Carrick.
Place: MCM410
Title: Spectral Flow, Eta Invariant and Llarull’s Rigidity Theorem in Odd Dimensions
Abstract: In this talk, I will present the application eta invariant and spectral flow on the proof of the odd-dimensional part of Llarull’s Theorem and two of its extensions. Generally speaking, Atiyah-Singer index theory is one of the major tools in the study of Riemannian metrics of positive scalar curvature. In odd dimensions, the spectral flow of a family of twisted Dirac operators on a compact spin manifold can be used to provide a direct proof of Llarull’s rigidity theorem and the so called “spin-area convex extremality theorem”. Furthermore, combining with the deformed Dirac operator introduced by Bismut and Cheeger, this method can be used to prove noncompact extension of Llarull’s theorem, which provides a final answer to a question by Gromov. This talk is based on joint works with Guangxiang Su, Xiangsheng Wang and Weiping Zhang.
Place: Zoom
Title: Slice spectral sequences through synthetic spectra
Abstract: We define a $t$-structure on the category of filtered $G$-spectra such that for a Borel $G$-spectrum $X$ the slice filtration of $X$ is the connective cover of the homotopy fixed-point filtration of $X$. Using this, we show that the slice spectral sequence for the norm $N_{C_2}^GMU_{\mathbb{\R}}$ of Real bordism theory refines canonically to a $\mathbb{E}\infty$-algebra in $MU$-synthetic spectra, when $G$ is a cyclic $2$-group. Concretely, this gives a map of multiplicative spectral sequences from the classical Adams–Novikov spectral sequence of $\mathbb{S}$ to the slice spectral sequence for $N{C_2}^GMU_{\mathbb{\R}}$ that respects the higher $\mathbb{E}_\infty$ structure, such as Toda brackets and power operations. We speculate further on a relationship to the equivariant ANSS based at tom Dieck’s homotopical complex bordism $MU_G$.
Place: MCM 110
Title: A infinity multi-modules from quilts and applications
Abstract: Invariants defined by counting pseudo-holomorphic maps from Riemann surfaces with boundary are uniquitous in symplectic geometry, low dimensional topology and mathematical physics. Some examples include various types of Floer homologies, Fukaya categories and Gromov-Witten theory. In this talk I will give a short survey about a generalization of this developed by Wehrheim-Woodward called pseudo-holomorphic quilts. Then I will present some applications of quilted invariants including a proof of a conjecture of Seidel regarding the Floer homology of composed Dehn twists.
Place: N820
Title: Compactification of homology cells and the complex projective space
Abstract: 很多例子表明正曲率（各种意义下）流形和具圆周作用的流形在一些情形具有相同或类似的拓扑推论。在这个报告中我们将回顾若干这方面的经典结果，并给出一个新的例子：在很多维数同调胞腔的凯莱紧化都是标准形式，或者具有圆周作用的辛紧化在同调意义下都是标准形式。
Place: N820
Title: A Bilinear Form for Spinc Manifolds
Abstract: Let M be a closed oriented spinc manifold of dimension (8n+2) with fundamental class [M], and let ρ2 : H4n(M; Z) → H4n(M; Z/2) denote the mod 2 reduction homomorphism. For any torsion class t ∈ H4n(M; Z), we establish the identity ⟨ρ2(t) · Sq2 ρ2(t), [M]⟩ = ⟨ρ2(t) · Sq2 v4n(M), [M]⟩, where Sq2 is the Steenrod square, v4n(M) is the 4n-th Wu class of M, x · y denotes the cup product of x and y, and ⟨· , ·⟩ denotes the Kronecker product. This result generalizes the work of Landweber and Stong from spin to spinc manifolds. As an application, let β Z/2: H4n+2(M; Z/2) → H4n+3(M; Z) be the Bockstein homomorphism associated to the short exact sequence of coefficients Z×2−−→ Z → Z/2. We deduce that βZ/2(Sq2v4n(M)) = 0, and consequently, Sq3v4n(M) = 0, for any closed oriented spinc manifold M with dim M ≤ 8n+1.
Place: Zoom: 6817169181 Password: 211748
Title: Classifying modules of equivariant Eilenberg–MacLane spectra
Abstract: Classically, since Z/p is a field, any module over the Eilenberg—MacLane spectrum HZ/p splits as a wedge of suspensions of HZ/p itself. Equivariantly, cohomology and the module theory of G-equivariant Eilenberg–MacLane spectra are much more complicated.For any p-group and the constant Mackey functor Z/p, there are infinitely many indecomposable HZ/p-modules. Previous work together with Dugger and Hazel classified all indecomposable HZ/2-modules for the group G=C2. The isomorphism classes of indecomposables fit into just three families. By contrast, joint work with Jacob Grevstad shows for G=Cp with p an odd prime, the classification of indecomposable HZ/p-modules is wild.
Place: N820
Title: Quantized nonabelianization
Abstract: Gaiotto-Moore-Neitzke introduced the notion of spectral network in the study of supersymmetric gauge theory. It provides a way to define a nonabelianization map (UV-IR map) from gl(1) local systems on a spectral curve to gl(N) local systems on the base surface. Neitzke-Yan considered a quantization of this map between skein algebras. In this work, we consider a different quantization in the sense of braid skein algebras using both Floer and Morse approach. This is joint work with Ko Honda and Yin Tian.
Place: N820
Title: Turaev-Viro invariant from the modular double of UqSL(2;R)
Abstract: In this talk we introduce recent collaboration work with Shuang Ming, Xin Sun, Baojun Wu, and Tian Yang. We define a family of Turaev-Viro type invariants of hyperbolic 3-manifolds with totally geodesic boundary from the 6j-symbols of the modular double of Uqsl(2;R), and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds.
Place: N933
Title: Homotopy theoretic properties of gyrations
Abstract: If M is a manifold, a gyration is a surgery on M\times S^k for some positive integer k. These appear prominently in work of Gitler-Lopez de Medrano on intersections of quadrics and recent work of Duan on the classification of certain types of manifolds. We will discuss homotopy theoretic properties of gyrations, such as loop space decompositions and classifications of homotopy types depending on how the surgery may be twisted by a nontrivial diffeomorphism. This is based on joint projects with Ruizhi Huang and Sebastian Chenery.