부산대학교  기하학, 위상수학  세미나
(PNU Geometry and Topology seminar)

Title: Homology Manifolds and $g_2$

Abstract: The $g$-vector of a simplicial complex contains significant information about the combinatorial and topological structure of the complex. Several classification results concerning the structure of normal pseudomanifolds and homology manifolds have been established in relation to the value of $g_2$, the third component of the $g$-vector. It is known that when $g_2=0$, all normal pseudomanifolds of dimensions at least three are stacked spheres. Walkup proved that a homology $3$-manifold with $g_2\leq 9$ is a triangulated sphere. In this talk, we demonstrate that for $d\geq 3$, the homology $d$-manifolds with $g_2\leq 3$ are triangulated spheres. In particular, if $d=4$, then this bound can be extended until $g_2\leq 5$, and it is the best possible bound in dimension $4$ for homology manifolds being triangulated spheres.

Title: Hirzebruch genera and rigidity equations

Abstract: A Hirzebruch genus is a ring homomorphism from the complex or oriented bordism ring to some ring R. Any such homomorphism can be extended in the standard way to a homomorphism from the corresponding bordism ring of manifolds with a k-dimensional torus action to the power series ring R[[x_1, ..., x_k]]. A genus is called rigid on a T^k-manifold M if the value of its extension on the bordism class of M is a constant series. The rigidity of genera has been intensively studied in the works of V. Buchstaber, T. Panov and N. Ray, and in many cases this property is equivalent to the multiplicativity of a genus with respect to fiber bundles with fiber M. There is a localization formula expressing the value of the equivariant extension of a genus in terms of the fixed points data of M. Using this formula, the property of rigidity is written as a functional equation for the exponential of a genus. I will give a brief introduction and discuss some of the results obtained using this approach.

Title: Right-angled Artin groups in hierarchically hyperbolic groups

Abstract: Hierarchically hyperbolic groups form a broad class of groups characterized by axioms inspired by mapping class groups. In this talk, we will first show that any finite collection of elements in a mapping class group generates a right-angled Artin subgroup once taken to sufficiently large powers. We will then discuss how this phenomenon extends, under suitable assumptions, to the setting of hierarchically hyperbolic groups.

Title. Geometry Through the Eyes of a Computational Geometer

Abstract. This talk looks at geometry through the eyes of a computational geometer, focusing on two themes close to my own research: distance queries in curved spaces and exact space partitions in three dimensions. I will discuss nearest and farthest neighbor search, hyperbolic geometry, Voronoi diagrams, and symbolic exact computation. I will conclude by presenting several open problems at the interface of geometry, topology, and algorithms, where close collaboration between computational geometers and mathematicians would be especially valuable.

Title. Equivalence of boundary measures on metric graphs and extensions to hyperbolic buildings

Abstract. On the boundary of the universal covering tree of a finite metric graph, three canonical measures arise: the visibility measure, the Patterson–Sullivan measure, and the harmonic measure. For simplicial graphs (unit edge lengths), Lyons (1994) characterized when any two of these measures coincide. We generalize this to metric graphs with arbitrary real edge lengths, replacing coincidence by equivalence (mutual absolute continuity). In the real-length setting Lyons' combinatorial Markov-matrix construction is no longer available, and equivalence — a weaker and more natural relation than coincidence, measuring how closely two measures capture the same boundary geometry — emerges as the appropriate framework. The key device is the notion of an integrated potential, an analogue of the geometric potential on manifolds, which realizes each boundary measure as the Patterson density and Gibbs measure of the geodesic flow associated with a suitable potential. In this framework, equivalence between two boundary measures reduces to a cohomology relation between the corresponding integrated potentials, and the three pairwise equivalence conditions become explicit algebraic identities in the edge lengths and vertex degrees. In the second part, we report on ongoing work extending these results to right-angled Fuchsian 2-dimensional hyperbolic buildings — CAT(-1) buildings whose apartments are right-angled hyperbolic hexagons with a prescribed thickness along each side.

Title: Dynamics on hyperbolic planes and trees: critical exponents, horocycle equidistribution, and spectrum

Abstract. The hyperbolic plane and regular trees are, at first sight, very different geometric objects: one is a continuous negatively curved space, while the other is a discrete combinatorial space. Nevertheless, discrete group actions on these spaces exhibit strikingly parallel dynamical, geometric, and spectral phenomena. In this talk, we discuss these analogies from the viewpoint of the critical exponent. For a discrete group acting on a hyperbolic space or on a tree, the critical exponent is defined through the convergence of the associated Poincaré series. It measures the exponential growth rate of an orbit, but its significance goes far beyond orbit counting. In the hyperbolic plane, it is closely related to Patterson–Sullivan measures, the dimension of the limit set, the entropy of the geodesic flow, the spectrum of the Laplacian, Eisenstein series, and the rate of horocycle equidistribution. On the tree side, analogous roles are played by boundary conformal measures, geodesic shifts, adjacency and Hecke operators, transfer operators, and Ihara-type zeta functions. We will compare geodesic flow and horocycle flow on hyperbolic surfaces with their counterparts on trees, where continuous flows are replaced by symbolic geodesic dynamics and horospherical group actions. Particular attention will be given to how quantitative equidistribution results are governed by spectral gaps, resonances, and scattering data. The goal of the talk is not to give a complete account of these theories, but rather to highlight a common structure: the critical exponent serves as a bridge connecting orbit growth, boundary geometry, spectral theory, zeta functions, and homogeneous dynamics.