几何拓扑

Let $S$ be a closed surface of genus $g \geq 2$. We study the cocompact domain of discontinuity $\Omega_{\rho}$ in the Einstein universe $\mathrm{Ein}^{p-1,p}$ defined by Guichard-Wienhard and Kapovich-Leeb-Porti for a class of $p$-Anosov…

We present several structural results on closed, nonorientable, smooth $4$--manifolds, extending analogous results and machinery for the orientable case. We prove the existence of simplified broken Lefschetz fibrations and simplified…

We show that the $(4,5)$- and $(5,6)$-torus knots admit ghost characters. Consequently, these knots provide counterexamples to Ng's conjecture, which proposes an isomorphism between the complexification of degree $0$ abelian knot contact…

We study the structure underlying Ng's conjecture, which relates the degree $0$ abelian knot contact homology of a knot $K$ to the coordinate ring of the $SL_2(\mathbf{C})$-character variety $X(\Sigma_2 K)$ of the $2$-fold branched cover of…

We classify Riemannian foliations of manifolds homeomorphic to CROSSes.

We introduce the non-self OU sequence and the OU number for link diagrams. Using these, we give a lower bound for the number of necessary Reidemeister moves of type III between two diagrams of the same link.

In this paper, we consider the action of the Goeritz group $\mathcal G_p$ for the genus two Heegaard splitting of the lens space $L(p,1)$ with $p\ge 2$ on the homology of the Heegaard surface. We describe the action in terms of matrices in…

In this paper, we study $6$-dimensional GKM manifolds with $4$ fixed points. We classify all possible GKM graphs, and for each type of graph we construct a manifold, proving the existence. We show that six types occur. (P1) complex…

We introduce a new diffeomorphism invariant of smooth compact oriented 4-manifolds $X$ with a framed oriented 1-link $L$ in the boundary, where $L$ may be the empty set, and call it {\it Khovanov-Lipshitz-Sarkar skein lasagna homotopy type}…

This paper proves the Giroux Correspondence in dimension three using Heegaard splittings of contact manifolds. In two of the authors earlier paper they proved the Giroux Correspondence for tight contact 3-manifolds via convex Heegaard…

In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various…

We introduce a new numerical knot invariant, termed the \textit{segment number}, which is derived from partitioned knot diagrams subject to specific over/under-crossing constraints. We prove that a knot is non-trivial if and only if its…

Ratner's theorem implies topological rigidity of immersed totally geodesic subspaces of noncompact type in finite-volume locally symmetric spaces. In higher rank and infinite volume, however, counter-examples to this rigidity have remained…

We study fibrations of the projective model for the symmetric space associated with $\text{SL}(2n,\mathbb{R})$ by codimension $2$ projective subspaces, or pencils of quadrics. In particular we show that if such a smooth fibration is…

The modular group PSL(2;Z) acts on the hyperbolic plane HP with quotient the modular surface M, whose unit tangent bundle U is a 3-manifold homeomorphic to the complement of the trefoil knot in the 3-sphere. The hyperbolic conjugacy classes…

We study the space of measured laminations ML on a closed surface from the valuative point of view. We introduce and study a notion of Newton polytope for an algebraic function on the character variety. We prove for instance that trace…

We prove that semisimple 4-dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth 4-manifolds and homotopy equivalent simply connected…

We study several properties of $\ZZ_2^n$-equivariant triangulations of $\RR P^n$. We show that a $\ZZ_2^n$-equivariant triangulation of $\RR P^n$ induces a triangulated subdivision of the orbit space $\bigtriangleup^n$. We show that any…

A surface $\Sigma$ in a 4-manifold $M$ is called flexible if any mapping class of the surface arises as the restriction of a diffeomorphism $(M,\Sigma) \to (M,\Sigma)$. We construct flexible surfaces in $\mathbb{C}P^2$ and $S^2 \times S^2$…

Han and Liu gave a geometric characterization of uniform perfectness for the Morse boundary of a proper geodesic metric space: the Morse boundary is uniformly perfect if and only if the space is Morse geodesically rich, equivalently…