Seminars are held Tuesday 3pm-4pm (unless otherwise noted)
Spring 2025
4/1: (No seminar)
4/8: Dev Sinha (UO)
4/15: Alison Tatsuoka (Princeton)
4/22: No seminar this week.
4/29: No seminar this week.
5/6: No seminar this week.
5/13: Daren Chen (Caltech)
5/20: Qianhe Qin (Stanford)
5/27: Kai Nakamura (Stanford)
6/3: Gary Guth (Stanford)
Abstracts
(6/3) Gary Guth (Stanford)
Title: Real Heegaard Floer homology
Abstract: There has been a burst of interest in gauge theoretic invariants of 3- and 4-manifolds equipped with an involution, developed in various contexts by Tian-Wang, Nakamura, Konno-Miyazawa-Taniguchi, and Li. Notably, Miyazawa proved the existence of an infinite family of exotic RP^2-knots using real Seiberg-Witten theory. In joint work with Ciprian Manolescu, we construct an invariant of based 3-manifolds with an involution, called real Heegaard Floer homology. This is the analogue of Li’s real monopole Floer homology. We prove that real Heegaard Floer homology is indeed a topological invariant of the underlying pointed real 3-manifold. Further, we study the Euler characteristic of our theory, which is the Heegaard Floer analogue of Miyazawa’s invariant for twist-spun 2-knots. This quantity is algorithmically computable and, indeed, appears to agree with Miyazawa’s invariant.
(5/27) Kai Nakamura (Stanford)
Title: Torus surgeries on knot traces
Abstract: This talk has a simple thesis statement: torus surgeries are a powerful tool to study 4-manifolds, we apply this technology to knot traces. The key insight is that annulus twisting a knot's Dehn surgery can be realized 4-dimensionally as a torus surgery on the knot's trace. We will then sketch a few applications of this idea.
(5/20) Qianhe Qin (Stanford)
Naturality in Heegaard Floer theory
Juhász, Thurston, and Zemke showed that Heegaard Floer homology is natural over Z/2. In this talk, we take a step toward the sign issues over Z. We will first review the notion of strong Heegaard invariants, then consider how to modify this to better suit the development of naturality over Z. We will show that the modified version induces strong Heegaard invariants by analyzing loops of handleslides for sutured diagrams.
(5/13) Daren Chen (Caltech)
Title: Two-component L-space links, satellites, and the tau-invariant
Abstract: A result by Ozsvath and Szabo states that the knot Floer complex of an L-space knot is a staircase. In this talk, we will discuss a similar result for two-component L-space links: the link Floer complex of such links can be thought of as an array of staircases. We will describe an algorithm to extract this array directly from the H-function of the link. (If this sounds familiar to you, that's right--Ian gave a talk last fall on the formality of link Floer homology of some classes of L-space links, and we are focusing on the case of 2-component L-space links.) As an application, we will discuss how to use this and the link surgery formula to compute the knot Floer complex and the tau-invariant of satellite knots under certain classes of satellite operations. This is joint work with Ian Zemke and Hugo Zhou.
(4/15) Alison Tatsuoka (Princeton)
Title: Splitting Spheres for S^2’s in S^4
Abstract: If K_1 \sqcup K_2 is a split link in S^4, a splitting sphere for K is an S^3 in S^4 such that K_1 lies in one connected component of S^4\S^3, and K_2 lies in the other. We show that there exist infinitely many pairwise non-isotopic splitting spheres for two unlinked, unknotted S^2’s in S^4. Along the way, we introduce barbell diffeomorphisms of 4-manifolds, as constructed by Budney-Gabai in their paper “Knotted 3-balls in S^4”.
(4/8) Dev Sinha (UO)
Title: New link invariants from algebraic topology. Or: What Milnor and Moore missed.
Abstract. J.W. Milnor and J.C. Moore were long-time colleagues, almost certainly first meeting in 1952 when Moore started as faculty at Princeton and Milnor was a graduate student. After Milnor got his PhD and was retained as faculty, they coauthored a famous paper on Hopf algebras. But they missed another opportunity to work together. Milnor’s senior and PhD theses were both on link theory. At the time Moore was leading development and application of the bar construction, in particular to cohomology of loop spaces. In this talk I share progress with co-authors Greg Friedman, Nir Gadish, Robin Koytcheff and Ben Walter, building on work with Aydin Ozbek, using analysis in the bar construction to generalize Milnor invariants. We produce new invariants for links in S^3 and define invariants for links in arbitrary three-manifolds. By using geometric cochains (developed with Greg Friedman and Anibal Medina), calculations are made by filling in curves by surfaces and intersecting those (to obtain new curves, which can then get filled in…).
Winter 2025
1/7: Colloquium
1/14: Colloquium
1/21: Colloquium
1/28: Halley Fritze (UO)
2/4: Chung-Ping Lai (Oregon State)
2/11: Seppo Niemi-Colvin (Indiana)
2/18: Qiuyu Ren (Berkeley)
2/25: Gheehyun Nahm (Princeton)
3/4: Austin Bosgraaf (Oregon State)
3/11: Neda Bagherifard (UO)
Abstracts
(2/25) Gheehyun Nahm (Princeton)
Title: An unoriented skein exact triangle in unoriented link Floer homology
Abstract: We show that band maps in unoriented link Floer homology give rise to an unoriented skein exact triangle, and describe potential connections with Khovanov homology.
(2/18) Qiuyu Ren (UC Berkeley)
Title: Lasagna s-invariant detects exotic $4$-manifolds
Abstract: We introduce a generalization of Rasmussen's $s$-invariant, called the lasagna $s$-invariant, which assigns either an integer or $-\infty$ to each second homology class of a smooth $4$-manifold. The construction is based on the construction of skein lasagna modules by Morrison-Walker-Wedrich. We present a few properties enjoyed by lasagna $s$-invariants, and we show that they detect the exotic pair of knot traces $X_{-1}(-5_2)$ and $X_{-1}(P(3,-3,-8))$, an example first discovered by Akbulut. This gives the first gauge/Floer-theory-free proof to the existence of compact orientable exotic $4$-manifolds. This is joint work with Michael Willis.
(2/11) Seppo Niemi-Colvin (Indiana University)
Title: A kitchen sink surgery formula for knot lattice homotopy
Abstract: I compute the knot Floer complex for the regular fiber of \Sigma(2,3,7), and I show that its Seifert genus and genus in a self homology cobordism agree. A step used for this result was providing upgrades to the surgery formula for knot lattice homotopy. Ozsv\'ath, Stipsicz, and Szab\'o showed that knot lattice homology satisfies a surgery formula similar to the one relating knot Floer homology and Heegaard Floer homology. I provide an iterable version of this formula that given the doubly filtered knot lattice space with involutive maps for a generalized algebraic knot produces the corresponding data for the dual knot post-surgery.
(2/4) Chung-Ping Lai (OSU):
Title: Complex of Groups and Homology
Abstract: In this talk, we apply the complex of groups framework developed by Haefliger and Corson to compute the homology of a simplicial complex acted upon regularly by a finite cyclic group. This framework compresses the original complex into its quotient simplicial complex and additional local data. By combining this local data with suitable boundary matrices of the quotient, we have constructed a matrix over a group ring for each dimension. We call them G-boundary matrices and we will show that the homology of the original simplicial complex can be obtained by decomposing the G-boundary matrices. This algebraic algorithm complements the geometric compression algorithm by Carbone, Nanda, and Naqvi.
(1/28) Halley Fritze (UO):
Title: Topological exploration through higher dimensional mapper graphs.
Abstract: The mapper graph M of a dataset X is a soft clustering under chosen parameters and allows us to visualize the connectivity of X as the graph M. To understand and visualize the higher-dimensional properties of X, we extend mapper to a simplicial complex construction called 2-mapper. Using techniques from persistent homology theory, we can show that 2-mapper is stable under certain choices of the cover parameter. We apply this construction to climate data to approximate and visualize pseudo-periodic trajectories.
Seminars are held Tuesday 3pm-4pm (unless otherwise noted)
Fall 2024
11/19: Ian Zemke (UO)
11/26: Siavash Jafarizadeh (UO)
12/3: Hanming Liu (UO)
12/10: Neda Bagherifard (UO)
Abstracts
Date: 12/3
Speaker: Hanming Liu
Title: Legendrian contact homology and Morse flow trees
Abstract: I will present Ekholm’s work on computing Legendrian contact homology by counting Morse flow trees. Legendrian contact homology defined by Eliashberg, Givental, and Hofer is an invariant of Legendrian submanifolds of (nice enough) contact manifolds, defined by counting holomorphic disks in the symplectization of the contact manifold. Ekholm showed that in the case where the contact manifold is a 1-jet space J^1(M) of some manifold M, we can count Morse flow trees on M instead of counting holomorphic disks in the symplectization of J^1(M), thus reducing the computation to a problem in finite dimensional Morse theory.
Date: 11/26
Speaker: Siavash Jafarizadeh (UO)
Title: Equivariant Khovanov homology for periodic knots.
Abstract: Periodic knots are those knots in 3-space that are invariant under a rotation around z-axis. Khovanov homology has been studied to periodic knots and it is referred to as Equivariant Khovanov homology. Some of the interesting properties of Khovanov homology of links are the functoriality of this invariant, and its behavior under special class of cobordisms, known as concordances. In this talk, we will mention analogous results for the Equivariant Khovanov homology of periodic links.
Date: 11/19
Speaker: Ian Zemke (UO)
Title: L-space links and formality
Abstract: An L-space knot is a knot whose Dehn surgeries have as simple Heegaard Floer homology as possible. Ozsvath and Szabo proved that the knot Floer complexes of L-space knots are staircase complexes, making them very easy to compute combinatorially. There is a natural extension of this definition for link, and such a link is called an L-space link. This is an important family of links, since it includes all algebraic links. We will describe an analog of their result for two important families of L-space links: plumbed L-space links, and 2-component L-space links. The former includes all algebraic links, and the latter includes families of links which are relevant for many common satellite operations. In this talk, we will describe the key algebraic properties we use to describe the link Floer complexes of these families, and also how to compute them algorithmically.