A basic question for any knot invariant asks which knots the invariant detects. For example, it is famously open whether the Jones polynomial detects the unknot. I'll focus in this talk on the detection question for knot invariants coming from Floer theory and the Khovanov--Rozansky link homology theories. I'll survey the progress made over the past twenty years, and will describe some of the topological ideas that go into my recent work with Sivek on these questions.
The uniformization theorem states that every Riemann surface is a quotient of some subset of the complex projective line by a group of Mobius transformations. However, a number of closely related questions regarding the structure of uniformization maps remain open. For example, it is unclear how one might associate a uniformizing map to a given Riemann surface. In this talk we will discuss an approach to this question due to Gunning by attaching a projective line bundle to a Riemann surface and studying its analytic properties.
In this talk, we introduce contact invariants in bordered sutured Floer homology. Given a contact 3-manifold with convex boundary, we apply a result of Zarev to derive contact invariants in the bordered sutured modules BSA and BSD. We show that these invariants satisfy a pairing theorem, which is a bordered extension of the Honda-Kazez-Matic gluing map for sutured Floer homology. We also show that there is a correspondence between certain A-infinity operations in bordered modules and bypass attachment maps in sutured Floer homology.
Pseudoholomorphic curves are pivotal in establishing uniqueness and finiteness results in the classification of symplectic manifolds. In a series of works, Wendl used punctured pseudoholomorphic foliations to classify symplectic fillings of contact three-manifolds supported by planar open books, turning it into a problem about monodromy factorizations.
Mathematically, what is a 5 feet divided by 2 seconds? Is it 2.5 ft/sec? What is a foot per second? We go through several examples of basic mathematical terms you learned in elementary, middle, and high school and understand them at a deeper, graduate student level. You may be surprised to learn that things you thought you knew were actually put on very weak mathematical foundations. The goal is to learn what those foundations are so that you can bring these basic ideas into your classroom in a non-pedantic-but-mathematically sound way.
In this talk, I will present recent advancements in the study of smooth mapping class groups of 4-manifolds. Our work focuses on diffeomorphisms arising from Dehn twists along embedded 3-manifolds and their interaction with Seiberg-Witten theory. These investigations have led to intriguing applications across several areas, including symplectic geometry (related to Torelli symplectomorphisms), algebraic geometry (concerning the monodromy of singularities), and low-dimensional topology (involving exotic diffeomorphisms).
The four color theorem states that each bridgeless trivalent planar graph has a proper 4-face coloring. It can be generalized to certain types of CW complexes of any closed surface for any number of colors, i.e., one looks for a coloring of the 2-cells (faces) of the complex with m colors so that whenever two 2-cells are adjacent to a 1-cell (edge), they are labeled different colors.
It is an important and rather difficult problem in low dimensional topology to determine which rational homology 3-spheres bound smooth rational homology 4-balls. This is largely open even in the case of Brieskorn spheres—a special class of Seifert fibered spaces. In this talk, we will focus on symplectic version of this question, and (almost) determine when a small Seifert fibered space admits a symplectic rational homology ball filling.
The Prym representations of the mapping class group are an important family of representations that come from abelian covers of a surface. They are defined on the level-ℓ mapping class group, which is a fundamental finite-index subgroup of the mapping class group. One consequence of our work is that the Prym representations are infinitesimally rigid, i.e. they can not be deformed.
In 1992 Hitchin discovered distinguished components of the PSL(d,R) character variety for closed surface groups pi_1S and asked for an interpretation of those components in terms of geometric structures. Soon after, Choi-Goldman identified the SL(3,R)-Hitchin component with the space of convex projective structures on S. In 2008, Guichard-Wienhard identified the PSL(4,R)-Hitchin component with foliated projective structures on the unit tangent bundle T^1S. The case d \ge 5 remains open, and compels one to move beyond projective geometry to flag geometry.
