In this talk, I will present a complete coarse classification of strongly exceptional Legendrian realizations of the connected sum of two Hopf links in contact 3-spheres. This is joint work with Sinem Onaran.
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). This is collaborative work with Hokuto Konno, Jianfeng Lin, and Juan Munoz-Echaniz.
A Dehn surgery slope p/q is said to be characterizing for a knot K if the homeomorphism type of the p/q-Dehn surgery along K determines the knot up to isotopy. I discuss advances towards a conjecture of McCoy that states that for any knot, all but at most finitely many non-integral slopes are characterizing.
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. We prove this infinitesimal rigidity by calculating the twisted cohomology of the level-ℓ mapping class group with coefficients in the Prym representation, and more generally in the r-tensor powers of the Prym representation. Our results also show that when r ≥ 2, this twisted cohomology does not satisfy cohomological stability, i.e. it depends on the genus g.
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. In a joint work with Hyunki Min and Agniva Roy, we build on the works of Lisi--Van Horn-Morris--Wendl in using spinal open books to further delve into the classification problem of symplectic fillings of higher genus open books. In particular, we provide the local model of the mysterious "exotic fibers" in a generalized version of Lefschetz fibrations, which captures a natural type of singularity at infinity. We will give some applications to classifying symplectic fillings via this new phenomenon.
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. As an application, we characterize the Stipsicz-Vertesi map in terms of A-infinity action on CFA. If time permits, we will further discuss applications to contact surgery.
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. In joint work with Alex Nolte, we obtain a new description of the SL(3,R)-Hitchin component in terms of concave foliated flag structures on T^1S.
Geometric mechanics is a tool for mathematically modeling the locomotion of animals or robots. In this talk I will focus on modeling the locomotion of a very simple robot. This modeling involves constructing a principal SE(2)-bundle with a connection. Within this bundle, the base space is parametrized by variables that are under the control of the robot (the so-called control variables). A loop in the base space gives rise to some holonomy in the fiber, which is an element of the group SE(2). We interpret this holonomy as the locomotion that is realized when the robot executes the path in the base space (control) variables.
Now, we can put a metric on the base space and ask the following natural question: What is the shortest path in the base space that gives rise to a fixed amount of locomotion? This is an extension of the isoperimetric problem to a principal bundle with a connection.
In this talk I will describe how to compute holonomy of the simple robot model, described above. Then I will solve the isoperimetric problem to find the shortest path with a fixed holonomy.
No prior knowledge of geometric mechanics will be assumed for this talk.
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. For some small Seifert fibered spaces, we provide explicit and new examples of such fillings, and for most others we provide strong restrictions. In the talk, we will review these concepts and provide further context; give some details of the techniques involved and finally mention a few applications. This will report on recent joint work with J. Etnyre and B. Özbağcı.
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.
In this talk, I show how to categorify the m-color polynomial of a surface with a CW complex. This polynomial is based upon Roger Penrose’s seminal 1971 paper on abstract tensor systems and can be thought of as the ``Jones polynomial’’ for CW complexes. The homology theory that results from this categorification is called the bigraded m-color homology and is based upon a topological quantum field theory (that will be suppressed from this talk due to time). The construction of this homology shares some similar features to the construction of Khovanov homology—it has a hypercube of states, multiplication and comultiplication maps, etc. Most importantly, the homology is the $E_1$-page of a spectral sequence whose $E_\infty$-page has a basis that can be identified with proper m-face colorings, that is, each successive page of the sequence provides better approximations of m-face colorings than the last. Since it can be shown that the $E_1$-page is never zero, it is safe to say that a non-computer-based proof of the four color theorem resides in studying this spectral sequence! (This is joint work with Ben McCarty.)
