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Columbia Geometric Topology SeminarFall 2024 

Organizers: Ross Akhmechet, Deeparaj Bhat, Siddhi Krishna, Francesco Lin
The GT seminar typically meets on Fridays at 2:00pm Eastern time in Room 407, Mathematics Department, Columbia University.
Other area seminars. Our email list. Archive of previous semesters
Date  Time (Eastern)  Speaker  Title 

September 13 
2pm 
Ben Lowe (UChicago) 
Rigidity and Finiteness of Totally Geodesic Hypersurfaces in Negative Curvature 
September 20 
2pm  Qiuyu Ren (UC Berkeley) 
Lasagna sinvariant detects exotic 4manifolds 
September 27 
2pm 
John Baldwin (BC) 
Instanton Floer homology from Heegaard diagrams 
October 4 
2pm  No seminar 

October 11 
2pm  No seminar 

October 18 
2pm  Rohil Prasad (UC Berkeley) 
Lowaction holomorphic curves and invariant sets 
October 25 (Algebraic Topology seminar of interest!) 
11am in Room 507  Rachael Boyd (Glasgow) 
Diffeomorphisms of reducible 3manifolds 
October 25 (double header!) 
2pm in Room 407  Mark Powell (Glasgow) 
Corks for diffeomorphisms 
October 25 (double header!) 
4pm in Room 520  Chris Leininger (Rice)  Atoroidal surface bundles 
November 1 
2pm 
No seminar 

November 8 (double header!) 
2pm 
Anubhav Mukherjee (Princeton) 
Complete Riemannian 4manifolds with uniformly positive scalar curvature metric 
November 8 (double header!) 
4:10pm in Room 417 
Peter Feller (ETH Zurich) 
Visual primeness of knots and Murasugi sums of open books 
November 15 
2pm 
Xinle Dai (Harvard)  
November 22 
2pm 
Casandra Monroe (UT Austin)  
November 29 
2pm 
no seminar! 
Thanksgiving break 
December 6 (double header!) 
2pm  Giulio Tiozzo (UToronto) 

December 6 (double header!) 
2pm  Matt Hedden (Michigan State) 

Name: Ben Lowe
Title: Rigidity and Finiteness of Totally Geodesic Hypersurfaces in Negative Curvature
Abstract: There is a broad body of work devoted to proving theorems of the following form: spaces with infinitely many special subspaces are either nonexistent or rare. Such finiteness statements are important in algebraic geometry, number theory, and the theory of moduli space and locally symmetric spaces. I will talk about joint work with Simion Filip and David Fisher proving a finiteness statement of this kind in a differential geometry setting. Our main theorem is that a closed negatively curved analytic Riemannian manifold with infinitely many totally geodesic hypersurfaces must be isometric to an arithmetic hyperbolic manifold.
Name: Qiuyu Ren
Title: Lasagna sinvariant detects exotic 4manifolds
Abstract: We introduce a lasagna version of Rasmussen's sinvariant coming from the study of Khovanov/Lee skein lasagna modules, which assigns either an integer or \infty to each second homology class of a given smooth 4manifold. After presenting some properties of the lasagna sinvariant, we show that it detects the exotic pair of knot traces X_{1}(5_2) and X_{1}(P(3,3,8)). This gives the first gauge/Floertheoryfree proof of the existence of exotic compact orientable 4manifolds. Time permitting, we mention some other applications of lasagna sinvariants. This is joint work with Michael Willis.
Name: John Baldwin
Title: Instanton Floer homology from Heegaard diagrams
Abstract: Heegaard Floer homology and monopole Floer homology are known to be isomorphic thanks to the monumental work of Taubes et al. But is there a simpler, more axiomatic explanation? And how is instanton Floer homology related to these other theories? I'll talk about work in progress with Zhenkun Li, Steven Sivek, and Fan Ye motivated by these questions. In particular, I'll sketch the construction of a chain complex that computes sutured instanton homology, which is isomorphic as a vector space to the Heegaard Floer chain complex of the sutured manifold. We are currently trying to prove that the differentials on the two sides agree.
Name: Rohil Prasad
Title: Lowaction holomorphic curves and invariant sets
Abstract: Holomorphic curves are a very useful tool for studying the topology and dynamics of symplectic manifolds. I will start with an overview of how holomorphic curves can detect periodic orbits of symplectic diffeomorphisms, taking the viewpoint pioneered by Hofer in 1993. Then, I will discuss a new method using “lowaction” holomorphic curves to detect closed invariant subsets that might be more general than periodic orbits. This has a few applications. I will mention one of them: a generalization to higher genus surfaces of a theorem by Le Calvez and Yoccoz. The talk is based on joint work with Dan CristofaroGardiner, and will not assume any prior knowledge of holomorphic curves.
Name: Mark Powell
Title: Corks for diffeomorphisms
Abstract: I will present a cork theorem for diffeomorphisms of simply connected 4manifolds, showing that one can sometimes localise a diffeomorphism to a contractible submanifold. I will sketch the proof and describe some applications. This is joint work with Slava Krushkal, Anubhav Mukherjee, and Terrin Warren.
Name: Chris Leininger
Title: Atoroidal surface bundles
Abstract: I will discuss joint work with Autumn Kent in which we construct the first known examples of compact atoroidal surface bundles over surfaces for which the base and fiber genus are both at least 2. This is a consequence of our construction of a typepreserving embedding of the fundamental group of the figure eight knot complement into the mapping class group of a thricepunctured torus.
Name: Anubhav Mukherjee
Title: Complete Riemannian 4manifolds with uniformly positive scalar curvature metric
Abstract: In three dimensions, geometry plays a crucial role in classifying the topology of manifolds. Inspired by this, we set out to explore the intricate world of smooth 4manifolds through the lens of geometry. Specifically, we aim to understand under what conditions a contractible 4manifold admits a uniform positive scalar curvature metric. In collaboration with Otis Chodos and Davi Maximo, we demonstrated that in certain cases, the existence of such a metric can provide insight into the topology of 4manifolds. Moreover, by utilizing Floer theory, we identified obstructions to the existence of such metrics in 4manifolds.
Name: Peter Feller
Title: Visual primeness of knots and Murasugi sums of open books
Abstract: Knots—circles embedded in R^3—are most commonly represented by knot diagrams. The latter are the result of projecting a knot to a generic 2dimensional plane retaining crossing information. Many simple 3dimensional properties of knots are, in general, difficult to detect from a knot diagram. For example the prime decomposition (much like its namesake for integers) is hard to discern. A conjecture of Cromwell asserts that, if a knot diagram satisfies a certain minimality condition, the prime decomposition is readily visible in the knot diagram. We discuss known partial results (for alternating and positive diagrams) and present an approach that allows us to generalize all prior results. As a concrete application, we establish Cromwell's conjecture for those diagrams that arise from braids. Methodologically, we consider knots via minimal surfaces that span them (which are fiber surfaces of open books in case the knot is fibered). A key result is a criterion that assures, under certain veeringconditions, that the Murasugi sums of two prime open books is prime. As a fun aside we note that iterative figure 8 knot plumbing preserves primeness of an open book, while iterative trefoil plumbing does not. Based on joint work in progress with Lukas Lewark and Miguel Orbegozo Rodriguez.