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Building on a classical solution to the pentagon equation, constructed earlier by the author and E.V. Martyushev and related to the flat geometry invariant under the group SL(2), we construct an algebraic complex corresponding to a…

For fixed large genus, we construct families of complete immersed minimal surfaces in R3 with four ends and dihedral symmetries. The families exist for all large genus and at an appropriate scale degenerate to the plane.

We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic…

We study noncompact, complete, finite volume, Riemannian 4-manifolds $M$ with sectional curvature $-1<K<0$. We prove that $\pi_1 M$ cannot be a 3-manifold group. A classical theorem of Gromov says that $M$ is homeomorphic to the interior of…

The triangulation complexity of a compact 3-manifold is the minimal number of tetrahedra in any triangulation of the 3-manifold. We compute the triangulation complexity of all elliptic 3-manifolds and all sol 3-manifolds, to within a…

In this work, we establish a relationship between the sum of irreducible character degrees and the number of twisted involutions associated with the automorphisms of a finite group. We develop algorithmic frameworks for evaluating these…

This paper concerns complete noncompact manifolds with nonnegative Ricci curvature. Roughly, we say that M has the loops to infinity property if given any noncontractible closed curve, C, and given any compact set, K, there exists a closed…

For a closed orientable connected 3-manifold $M$, its complexity $\boldsymbol{T}(M)$ is defined to be the minimal number of tetrahedra in its triangulations. Under the assumption that $M$ is prime (but not necessarily atoroidal), we…

We prove that for every closed, connected, orientable, irreducible 3-manifold, there exists an alternating group A_n which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group…

We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and…

We classify all closed non-orientable P2-irreducible 3-manifolds with complexity up to 7, fixing two mistakes in our previous complexity-up-to-6 classification. We show that there is no such manifold with complexity less than 6, five with…

We show that for a smooth, closed 2-connected manifold $M$ of dimension $d \geq 6$, the topological mapping class group $\pi_0 \mathrm{Homeo}(M)$ is residually finite, in contrast to the situation for the smooth mapping class group $\pi_0…

We construct infinitely many examples of pairs of isospectral but non-isometric $1$-cusped hyperbolic $3$-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an…

In this paper, we first construct $k$-dimensional compact complex manifolds from automorphisms of $\mathbb{C}^k$ which admit a fixed attracting point at infinity. Then, we charactize the fundamental group as well as the universal covering…

We classify all closed non-orientable P2-irreducible 3-manifolds having complexity up to 6 and we describe some having complexity 7. We show in particular that there is no such manifold with complexity less than 6, and that those having…

We prove that any compact, orientable 3-manifold with empty or toral boundary is profinitely almost rigid among all compact, orientable 3-manifolds. In other words, the profinite completion of its fundamental group determines its…

The double scaling limit of a new class of the multi-matrix models proposed in \cite{MMM91}, which possess the $W$-symmetry at the discrete level, is investigated in details. These models are demonstrated to fall into the same universality…

We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G$_2$-structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G is…

Firstly, we characterise the embeddability of simply connected locally 3-connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski's characterisation of graph planarity, by nine excluded minors. This answers…

We prove that given two compact oriented $3$-manifolds $N$ and $M,$ with $M$ satisfying only a mild hypothesis, there is a hyperbolic $3$-manifold $N'$ arbitrarily ``closely related'' to $N,$ and such that $N'$ does not embed in $M.$ For…