Papers

We study asymptotic cones of Lie groups, presenting a link between the geometry of the weights and the highly non-simply-connected nature of asymptotic cones. We show that a Lie group that admits a group of SOL-type as a quotient is such that the fundamental group of its asymptotic cone contains the fundamental group of the Hawaiian earring space. We also obtain a strong non-vanishing result for the first homology group of the asymptotic cone. The assumption on the Lie group, introduced by Abels and called the "SOL obstruction", is equivalent to an explicit geometric condition on the weights. Our work builds upon the results of Burillo, who proves the statement on asymptotic cones in the case of SOL, and on those of Cornulier and Tessera who show that the SOL obstruction implies exponential growth of the Dehn function.

Hyperbolic length functions are among the most fundamental ones on Teichmüller space, and they are quasi-convex along Teichmüller geodesics. In this paper, we investigate the same question for energy functions of harmonic maps in two natural settings, which may be viewed as nonlinear and higher-dimensional analogs of the length functions. For a fixed domain and a varying hyperbolic target, we prove the energy is quasi-convex along Teichmüller geodesics under a filling hypothesis. Furthermore, we generalize Masur's result on asymptotic growth of the length function along the Teichmüller geodesic determined by a Jenkins-Strebel differential to the energy functions. We also prove the quasi-convexity for covering maps between closed hyperbolic surfaces with fixed target and varying domains. We derive first and second variation formulas of energy functions along Teichmüller geodesics and explain why the natural global statement is quasi-convexity rather than genuine convexity.

We solve the Nielsen realisation problem for high-dimensional aspherical manifolds in a new class of special cases. Mainly, we focus on actions of cyclic groups of prime order. A novelty is that it gives topological solutions to the problem with certain high-dimensional fixed point sets, whereas previous solutions of this type were restricted to discrete fixed point sets. This is enabled by recent joint work with Kirstein on isovariant Poincaré duality spaces, and Farrell-Lück-Steimle's obstruction theory to finding approximate fibrations within a homotopy class.

We construct a smooth, equivariantly formal, simply connected torus $8$-manifold over a contractible $4$-manifold with the boundary a Brieskorn homology $3$-sphere having a nontrivial fundamental group. We prove extension to a torus graph for GKM graphs of GKM$_{4}$ manifolds in complexity $2$, and $3$ under a finite fundamental group assumption (that is not satisfied, in general).

We investigate the filtration structure of the skein algebra induced by the (quantum) trace map. Utilizing the structure, we prove that relative $\operatorname{SL}(3, \mathbb{C})$ character varieties of punctured surfaces admit log Calabi-Yau compactifications and prove the (weak) geometric P=W conjecture.

R-equivalence is an equivalence relation on the set of colorings of an oriented knot diagram by a quandle. In this paper, we show that a certain subgroup of the inner automorphism group of a quandle acts on the R-equivalence class of a given coloring by the quandle. We also determine the R-equivalence classes of colorings of a diagram of a $2$-bridge knot by a dihedral quandle completely, under a certain condition.

We study the geography of taut foliations on compact $3$-manifolds with toroidal boundary components. The central object of our work is the set of boundary multislopes realized by foliations transverse to a fixed flow on such a manifold. We prove that these sets exhibit remarkable structural properties (rationality, rigidity, and convexity) which motivate the name ziggurats. Our main tool, of independent interest, is a two-way correspondence between foliations and contact structures on $3$-manifolds with boundary: we generalize the Eliashberg-Thurston theorem, which produces pairs of positive and negative contact structures from foliations, and a construction of the first author, which builds foliations from such contact pairs. Using both directions of this correspondence, we bring contact-geometric methods to bear on the architecture of ziggurats.

This survey explores the rich interplay between (taut) foliations and (tight) contact structures in dimension three, highlighting recent work of the author. We outline a new method for constructing taut foliations from suitable pairs of contact structures, and discuss some applications and future research directions. In particular, we give a brief account of work in progress with Jonathan Zung on ziggurats for taut foliations transverse to pseudo-Anosov flows, and we propose a (mostly speculative) contact perspective on the $L$-space conjecture.

We define a 'full' version of Khovanov-Rozansky homology: a chain complex CH(K) over F[U,V] whose chain homotopy type is a knot invariant and which has the property that setting U = V = 0 recovers the reduced Khovanov-Rozansky homology. We explore the algebraic structure of CH(K) and show that a variation of the structure theorem for knot Floer homology applies. This allows us to define counterparts to knot Floer concordance invariants $\tau$, $\epsilon$ and $\phi_j$ in the Khovanov-Rozansky setting.

Let $M$ be a finite-volume hyperbolic $d$-manifold with $d\ge3$. We prove that the number of primitive nonsimple closed geodesics has exponential growth rate strictly smaller than $d-1$. Consequently, asymptotically almost every primitive closed geodesic in $M$ is simple. In contrast, we show that on an arithmetic hyperbolic manifold of type~I, the unit vectors tangent to nonsimple closed geodesics are dense in the unit tangent bundle.

Given any closed hyperbolic n-manifold M ($n\ge 3$), we show that for a fixed $\kappa>\frac{3}{n-2}$, a random closed geodesic of length close to r has the collar of width $r^{-\kappa}$ when r is large enough. In particular, most closed geodesics of length close to r are simple. This theorem positively answers Problem 3.16 from Kirby's list which asks whether every closed hyperbolic 3-manifold contains infinitely many simple closed geodesics.

We introduce a variant of reduced Khovanov-Rozansky homology defined in terms of the variables alpha_1, ..., alpha_{n-1} that correspond to the regions between the strands. We rigorously reestablish several folklore results in terms of our framework, including the equivalence between Khovanov's and Rasmussen's approaches. Furthermore, we show that Rasmussen's chain complex is a free resolution of the Rouquier chain complex.

In the spirit of Conway and Coxeter, we classify all $\mathbf{Y}$-frieze patterns of type $A_n$. In particular, we settle a conjecture made by de Saint Germain that all such $\mathbf{Y}$-frieze patterns arise from Conway-Coxeter frieze patterns. Moreover, our approach naturally leads to the enumeration of these $\mathbf{Y}$-frieze patterns in terms of Fuss-Catalan numbers.

Let $J_r^{SU(n)}(K;q)$ denote the reduced $SU(n)$ quantum invariant of a zero-framed knot $K$, colored by the $r$th symmetric power of the defining representation and normalized to be $1$ for the unknot. For every fixed $n\ge2$ we prove the Chen–Liu–Zhu cyclotomic expansion conjecture: there are unique coefficients $H_k^{(n)}(K;q)\in\mathbb{Z}[q^{\pm1}]$ such that \[ J_r^{SU(n)}(K;q)=\sum_{k=0}^{r} \left(\prod_{i=0}^{k-1}\{r-i\}\{r+n+i\}\right) H_k^{(n)}(K;q), \] where $\{m\}=q^m-q^{-m}$. The finite dual interpolation formula of Beliakova–Gorsky gives an integral one-sided factorial expansion. After identifying their reduced scalar with the Habiro–Lê convention, we restrict the completed center to one-row colors. Completed Harish–Chandra reflection then yields inversion symmetry in the variable $z$, and integral descent through $X=z+z^{-1}$ converts the one-sided expansion into the two-sided Newton basis. Cyclotomic-local interpolation and a UFD denominator-removal argument prove Laurent integrality of the Newton coefficients. We also determine a natural coefficient ring for a rank-uniform expansion. For every zero-framed knot there are unique Laurent differential coefficients $G_k(K;A,q)\in\mathbb{Z}[A^{\pm1},q^{\pm1}]$. The associated Newton coefficients are Laurent polynomials in $A$ over $\mathbb{Q}(q)$ whose values at every geometric node $A=q^n$, $n\ge2$, lie in $\mathbb{Z}[q^{\pm1}]$. They define a two-variable Newton inverse-limit element whose positive-rank specializations recover all symmetric-color HOMFLY–PT polynomials. The completion is taken in the Newton kernels rather than coefficientwise at roots of unity. After a positive rank and a color have been fixed, the series is finite and may be evaluated at a root of unity.

We study cabling operations on braids, and characterize the matrices that can be realized as crossing matrices of reducible pure braids. We also use cabling to construct enhanced conjugacy invariants of braids.

We first give a complete classification of bi-affine representations of mapping class groups of surfaces with finitely many boundary components or punctures. We also show that every linear representation of the mapping class group of a genus-$g$ surface with two boundary components of dimension at most $3g-3$ is bi-affine. We then classify low-dimensional symplectic representations of the mapping class group associated to triple covers. Let $[\beta]\in H_1(S_g;\mathbb{Z}/3\mathbb{Z})^*$, and let $\widetilde{S}\to S_g$ be the corresponding triple cover with deck transformation $\sigma$. For $h\le g$, every non-abelian homomorphism from either $\mathrm{Mod}(S_g,[\beta])$, the stabilizer of $[\beta]$ in $\mathrm{Mod}(S_g)$, or $\mathrm{Mod}(\widetilde{S},\sigma)$, the centralizer of $\sigma$ in $\mathrm{Mod}(\widetilde{S})$, to $\mathrm{Sp}_{2h}(\mathbb{Z})$ is, up to conjugation, the standard symplectic representation on $H_1(S_g;\mathbb{Z})$. As an application, we obtain a rigidity theorem for holomorphic maps from the moduli space $R_g^{(3)}$ of genus-$g$ curves equipped with a $3$-sheeted (unbranched) normal covering to the moduli space $\mathcal{A}_h$ of $h$-dimensional principally polarized abelian varieties. We prove that, for $g\ge 6$ and $h\le g$, the unique nonconstant holomorphic map from $R_g^{(3)}$, equipped with either of its two natural complex-orbifold structures, to $\mathcal{A}_h$ is the period map sending a cover $Y\to X$ to the Jacobian of the base curve $X$.

We show that the first derivative of the Burau matrix at $t=1$ coincides with the Laplacian of the crossing matrix for pure braids. We also discuss conjugacy invariants derived from these matrices.

Fox $p$-colorings of knots and links admit a linear-algebraic description in terms of braid representations. For each braid $\beta \in B_n$ we associate a representation $M : B_n \to \mathrm{GL}(n,\mathbb{F}_p)$ such that Fox $p$-colorings of the closed braid $\widehat{\beta}$ correspond to fixed points of $M(\beta)$. As a consequence, $$ \mathrm{Col}_p(\widehat{\beta}) = p^{\dim \ker(M(\beta)-I)}, $$ yielding an explicit formula for the number of Fox $p$-colorings of arbitrary knots and links. We apply the method to torus links and spiral links, obtaining explicit descriptions of their Fox 3-coloring spaces.

We define the notion of weighted polyhedron which can be thought of as the geometric realization of a weighted simplicial complex introduced by Dawson. Moreover, we will define a weighted version of singular homology theory for a weighted polyhedron and prove that it is isomorphic to the weighted simplicial homology of the weighted polyhedron. This implies that weighted simplicial homology is an invariant under isomorphisms and more generally under certain type of homotopy equivalences of weighted polyhedra. Moreover, we will generalize the cup product and cap product to weighted singular cohomology. In addition, we will interpret some known theories of orbifolds in terms of our weighted singular homology and cohomology.

In 2023, Richard Schwartz introduced a new dynamical system which is a marriage of two types of familiar billiards, tiling billiards and symplectic billiards. In this paper we investigate this dynamical system played on tilings of the plane which arise from non-Euclidean geometries on the torus. We review the affine analogue of the flat conformal structures on the torus through the work of Oliver Baues and William Goldman, and define an open subset of this deformation space corresponding to markings of complete affine tilings of the plane. We make this definition precise, and provide algebraic conditions on the symmetries of the tiling to define it. We then analyze the dynamics of symplectic tiling billiards played on these types of tilings and investigate the long-term dynamics of the system to prove a stability result concerning divergent trajectories. The divergence is defined in terms of geometric invariants arising from the tiling symmetry group. We argue that in some sense this divergence is a consequence of the tiles of a non-Euclidean tiling becoming 'thin' as one moves far away in the tiling. To do so we introduce a notion of thinness that is well adapted to the non-Euclidean affine tilings.

We define an invariant of alternating links—a homogeneous, four-variable Laurent polynomial—that encodes the symmetrized Alexander polynomial, the signature, and other topological data. Along the way, we extend a spanning tree formulation of the Alexander polynomial due to Murasugi and Stoimenow from special alternating links to all alternating links. This project is motivated by Fox's trapezoidal conjecture; accordingly, we prove certain sequences associated to our invariant are trapezoidal for all alternating links. We also conjecture our polynomial has $M$-convex support, and that it satisfies symmetry and log-concavity properties. We prove a partial symmetry result.

We prove Mühlherr's Twist Conjecture: any two angle-compatible Coxeter generating sets of a Coxeter group differ by a finite sequence of elementary twists and a conjugation. Combined with earlier work of Howlett-Mühlherr and Marquis-Mühlherr, this completes the resolution of the Isomorphism Problem for Coxeter groups. A further consequence is that ${\rm Aut}(W)$ is finitely generated for every Coxeter group $W$, and there is an algorithm producing a finite set of generators for ${\rm Aut}(W)$ starting from any Coxeter matrix. Of the vast literature on the Twist Conjecture, we utilise only two results in an essential way: strong rigidity of $2$–spherical Coxeter systems, due to Caprace and Mühlherr, and the framework of markings and hierarchies developed by Caprace and Przytycki for the twist-rigid case. We also exploit in a fundamental way some soft ideas from JSJ theory and an observation of Mihalik-Tschantz on splittings of Coxeter groups. No form of AI was used in the writing of this manuscript, nor in the research that it presents.

Bonk and Kleiner proved that if $G$ is a Gromov hyperbolic group whose boundary $\partial_{\infty}G$ is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, and the Ahlfors regular conformal dimension of $Z$ is attained and equal to $Q$, then $G$ acts discretely, cocompactly, and isometrically on $\mathbb{H}^3$. In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if $(G,\mathcal{H})$ is a relatively hyperbolic group whose Bowditch boundary is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, with the Ahlfors regular conformal dimension of $Z$ attained and equal to $Q$, then $G$ acts discretely and isometrically on $\mathbb{H}^3$, and every subgroup in $\mathcal{H}$ is virtually $\mathbb Z^2$.

Let $S$ be a closed oriented surface of genus $g\geq 2$. We consider an $n$-pleated representation $\rho: \pi_1(S) \to \mathrm{PSL}_n(\mathbb{C})$ obtained by bending a Hitchin representation $\rho_0:\pi_1(S) \to \mathrm{PSL}_n(\mathbb{R})$ along a maximal geodesic lamination. The space of such $n$-pleated representations was recently introduced by Maloni-Martone-Mazzoli-Zhang who provided a parametrization via shear-bend cocycles. Our main result is that $\rho_0$ dominates $\rho$ in the Hilbert length spectrum and the translation-length spectrum, with a strict domination for $\textit{most}$ curves, that we call $\textit{statistical}$ domination. Using this, we prove some entropy rigidity results: namely, the Hilbert entropy of any quasi-Hitchin representation in the bending fiber is strictly greater than that of $\rho_0$, the same for the translation length entropy when $\rho_0$ is $n$-Fuchsian, and in the latter case a new proof that for hyperconvex representations the Hausdorff dimension of the full limit set increases. The proof involves analyzing the weighted planar networks for finite approximants of the monodromy matrix, and establishing a strict matrix domination for generic monodromy using the equidistribution of closed geodesics in the unit tangent bundle of $S$.

We introduce an example of biquandle and define an invariant of long virtual knots through it. As an application, we give a necessary condition for two long virtual knots commuting.

We prove that there exist hyperbolic integer homology spheres with arbitrarily large Dehn surgery number. Previously, no integer homology sphere was known to have a surgery number larger than $2$. Our approach uses Froyshov's invariant $q_3$ of integer homology spheres, which is defined in terms of mod 2 instanton homology. We show that if $W: Y \to Y'$ is a cobordism between integer homology spheres with no $2$-torsion in its first homology, then $-b^+(W) \le q_3(Y') - q_3(Y) \le b^-(W)$. We also extend both $q_3$ and the inequality to rational homology spheres.

We prove that two invariants of smooth 4-manifolds defined in terms of trisections are uncomputable: the trisection genus and the Kirby-Thompson L-invariant. That is, there does not exist an algorithm that takes as input a triangulated closed orientable 4-manifold and outputs one of these quantities. The result on the L-invariant resolves the second part of Problem 4.116 in the K3 problem list. In the same spirit, we show that the PL multisection genus of PL manifolds is uncomputable in dimensions at least four.

The category TFS of thin flat surfaces, introduced by Khovanov, Qi, and Rozansky, is a strict symmetric monoidal category whose morphisms are compact oriented surfaces with corners arising as neighbourhoods of immersed graphs in a strip. We prove two main results. First, a normal form theorem: every connected viewable tf-cobordism is equal, in TFS, to a canonical composite determined by its topological type. Second, a sufficiency theorem: we establish a list of relations in the TFS category which are sufficient, meaning that any two composites of generators representing the same tf-cobordism are related by a finite sequence of the listed relations. Together, these results give a complete generators-and-relations presentation of TFS.

We prove that mapping class groups of surfaces and of locally finite connected graphs support a unique Polish group structure.

The Andreadakis-Johnson filtration and its associated construction, known as the Johnson homomorphism, are useful tools in group theory. They provide a step-by-step approach to studying the automorphisms of a given group. After explaining the basics we survey both classical and recent results around the Andreadakis-Johnson filtration, with emphasis on the mapping class group of a once-bordered surface.