The abstract conjectures a formula for virtual elliptic genera of sheaves on surfaces.
problem Computing virtual elliptic genera of sheaves on surfaces.
method Expressed conjectures in terms of Igusa cusp form, quasi-Jacobi forms, and universal functions.
result Conjectures a formula involving virtual cobordism classes and universal functions.
The paper compares two torsion invariants in complex vector bundles.
problem Comparing two torsion invariants in complex vector bundles.
method Constructing Bismut-Lott analytic torsion classes and showing they coincide with Igusa-Klein torsions.
result Bismut-Lott analytic torsion classes coincide with Igusa-Klein torsions for trivial flat line bundles.
According to Kiyoshi Igusa a generalized Morse function on an n-dimensional manifold M is a smooth function with only Morse and birth-death singularities and a framed function is a generalized Morse function with an additional structure: a framing of the negative eigenspace at each critical point of the function f. In …
Faltings heights help in Kahler geometry stability.
problem Stability of polarized manifolds and csck metrics.
method Faltings heights, Igusa local zeta functions, stability conjectures.
result Stability of (X,L) implies existence of csck metric.
We study the structure of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion. It was known before that the invariant is expressed as the Petersson norm of an automorphic form on the moduli space. When the rank of the invariant sublattice of the K3-lattice with respect to …
Study calculates Cheeger constants and small eigenvalues of Maass cusp forms.
problem Understanding small eigenvalues of Maass cusp forms on hyperbolic surfaces.
method Computed Cheeger constants of hyperbolic surfaces and searched for small eigenvalues of Maass cusp forms numerically.
result Found evidence and computed small eigenvalues of Maass cusp forms.
We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.
Analytic surgery and gluing formula for torsion forms in fiber bundles.
problem Behavior of torsion forms under analytic surgery in fiber bundles.
method Analytic surgery and gluing formula for Bismut-Lott torsion and eta forms.
result Gluing formula for Bismut-Lott analytic torsion and eta forms under surgery limit.
Equivalence proven between two torsion invariants for flat vector bundles.
problem Equivalence of Igusa-Klein and Bismut-Lott torsion invariants for flat vector bundles.
method Reduction to trivial flat line bundles using Artin's induction theorem.
result Igusa-Klein and Bismut-Lott torsion invariants are equivalent for flat vector bundles with finite holonomy.
Study of Eisenstein series linked to hyperbolic cusps.
problem Understanding Eisenstein series associated with hyperbolic cusps.
method Analyzing cohomology classes and intertwining operators.
result Different cusps correspond to linearly independent cohomology classes.
We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…
We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension 2 singularity…
Study bounds on harmonic forms in hyperbolic 3-manifolds using Thurston norm and minimal surfaces.
problem Bounding the L2-norm of harmonic forms in hyperbolic 3-manifolds. method Using Thurston norm and interaction with minimal surfaces.
result Generalizes inequalities of Brock-Dunfield and studies sharpness in closed and cusped cases.
This thesis studies cusp forms on complex hyperbolic spaces.
problem Analyzing cusp forms on complex hyperbolic spaces.
method Defined Schwartz space, cusp forms, and used representation theory.
result Direct sum decomposition of cusp forms in L2 space. Infinite volume moduli spaces of hyperbolic surfaces are redefined with exponential forms.
problem Infinite volume of moduli spaces of hyperbolic surfaces with cusps.
method Introduce exponential volume form exp(-W)Vol(K,L) where W is a function of hyperbolic areas.
result Exponential volume forms make moduli spaces finite and relevant to open string theory.
New method calculates winding of geodesics on surfaces.
problem Understanding the distribution of geodesics on surfaces.
method Introducing a new construction of winding numbers for geodesics on cusped hyperbolic orbifolds.
result Winding numbers can be expressed by Rademacher symbols for various arithmetic families of surfaces.
Continuous process closes cusps in complex algebraic surfaces.
problem Developing cusps in Kähler-Einstein metrics on algebraic surfaces.
method Continuous cusp closing process via gluing construction.
result Cusps form in Kähler-Einstein metrics near isolated singularities.
Computes invariant for smooth h-cobordisms families, proving duality and vanishing theorems.
problem Computing invariants for smooth h-cobordisms families.
method Using Dwyer, Weiss, and Williams work, fiberwise generalized Morse function, fiberwise Poincaré--Hopf theory.
result Duality theorem for smooth structure class, vanishing theorem for Rigidity Conjecture.
We prove a quantitative statement of the quantum ergodicity for Hecke--Maass cusp forms on the modular surface. As an application of our result, along a density 1 subsequence of even Hecke--Maass cusp forms, we obtain a sharp lower bound for the L2-norm of the restriction to a fixed compact geodesic segment of $η=…
Proves smooth torsion invariants match across methods.
problem Matching higher torsion invariants in smooth bundles.
method Explicit comparison of constructions by Igusa/Klein vs. homotopy theory by Badzioch et al.
result Higher torsion invariants agree across methods.
New fundamental domain for Hilbert-Blumenthal cusp shapes.
problem Understanding the geometry of Hilbert-Blumenthal surfaces at cusps.
method Constructing a fundamental domain using Dirichlet domains and deformations of lattices.
result Explicitly describes the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism.
The paper proves the conjecture about the number of cusps in a caustic formed by reflecting rays in a circle.
problem Proving the conjecture about the number of cusps in a caustic formed by reflecting rays in a circle.
method Analyzing the family of rays emanating from a non-focal point inside an elliptic billiard table, focusing on the caustic formed after multiple reflections.
result A proof of the conjecture that a caustic formed by reflecting rays in a circle has exactly four cusps.
The paper constructs a hyperbolic 4-manifold with rational homology sphere cusp sections.
problem Constructing a hyperbolic 4-manifold with rational homology sphere cusp sections.
method Constructing a hyperbolic 4-manifold with specified properties.
result The Laplacian on 2-forms on the constructed manifold has purely discrete spectrum.
An analytic index is defined for a family of cusp pseudodifferential operators, Pb, on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation Qb by a family of cusp operators of…
Study of Quillen metric on Riemann surfaces with cusps and compactification.
problem Behavior of Quillen metric on Riemann surfaces with cusps.
method Continuous extension of Quillen metric over singular curves and explicit universal constant.
result Compatibility of Quillen metric with clutching morphisms and analytic torsion.
We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as…
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…
Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove…
For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of L2 harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the pr…
We prove that for every metric on the torus with curvature bounded from below by -1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem:…
Study shows nontrivial homotopy groups for negatively curved metrics on hyperbolic manifolds.
problem Understanding the homotopy groups of Teichmüller spaces for negatively curved metrics.
method Analyzes Teichmüller spaces of negatively curved metrics on hyperbolic manifolds, proving nontrivial homotopy groups for some dimensions.
result Proves existence of nontrivial rational homotopy groups and elements of infinite order in π_i B Diff(M).
Study on curved metrics on manifolds using smoothing theory.
problem Understanding rational homotopy groups of curved metrics on high-dimensional manifolds.
method Classical results in smoothing theory applied to high-dimensional manifolds.
result Smooth M-bundles with fiberwise negatively curved metrics represent elements of finite order in homotopy groups.
Let p: M -> B be a family of compact manifolds equipped with a unitarily flat vector bundle F -> M. We generalize Igusa's higher Franz-Reidemeister torsion τ(M/B;F) to the case that the fibre-wise cohomology H^*(M/B;F) -> B carries a parallel metric. If moreover M admits a fibre-wise Morse function, we compute the diff…
Sharp asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.
problem Understanding the asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.
method Analyzing the curvature and metric properties of Kähler-Einstein metrics on complex hyperbolic cusps.
result Sharp doubly exponential rate of convergence of metrics to a limiting form.
New invariant cusp crossing density shows cusp densities of hyperbolic knots and links are dense.
problem Densities of cusp invariants for hyperbolic knots and links.
method Defined cusp crossing density as ratio of cusp volume to crossing number; showed densities are dense.
result Cusp crossing density for links is bounded above by 3.1263... and is dense in [0, 2.120...].
4-manifolds show every flat 3-manifold as cusp sections.
problem Realizing flat 3-manifolds as cusp sections of hyperbolic 4-manifolds.
method Transitive action on cusps, dense flat metrics realization.
result Existence of many cusp-transitive 4-manifolds.
Suppose M is a cusped finite-volume hyperbolic 3-manifold and T is an ideal triangulation of M with essential edges. We show that any incompressible surface S in M that is not a virtual fiber can be isotoped into spunnormal form in T . The proof is based directly on ideas of W. Thurston.
Characterizes holonomies of convex projective cusps.
problem Understanding holonomies in strictly convex projective geometry.
method Complete characterization of holonomies for strictly convex and round cusps, building families of generalized cusps.
result Produces the first example of generalized cusps with non-virtually nilpotent fundamental group.
Finite-type k-surfaces in hyperbolic space have finite genus and cusp-like ends.
problem Characterizing the geometry of finite-type k-surfaces in hyperbolic space.
method Analyzing the asymptotic behavior and geometric properties of finite-type k-surfaces.
result Finite-type k-surfaces have well-defined Steiner geodesics and points, leading to new geometric identities.
The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.
problem Classifying symplectic invariants of singularities in integrable Hamiltonian systems.
method Smooth C∞ symplectic classification of Lagrangian fibrations near singularities. result Action variables form complete C∞ symplectic invariants for parabolic orbits and cuspidal tori. Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
Proves methods for creating convex projective 3-manifolds with cusps.
problem Creating convex projective 3-manifolds with generalized cusps.
method Properly convex deformations of hyperbolic structures, controlling cusp types.
result First known example of a 1-cusped hyperbolic 3-manifold with a type 2 cusp.
The paper classifies different types of cusps on plane curves.
problem Investigating various types of cusps on plane curves.
method Examining criteria for (n,n+1) cusps with differential conditions and relations to evolutes of fronts. result Complete classifications for (4,5)-cusps. The paper characterizes links in 3D from divides with cusps.
problem Characterizing links in 3D from divides with cusps.
method Defines and characterizes divides with cusps and their associated links.
result Every strongly invertible link and 2-periodic link can be described as the link of a divide with cusps.
Extends techniques to show existence of all cusp types in convex projective manifolds.
problem Existence of all cusp types in convex projective manifolds.
method Extension of techniques by Ballas-Marquis.
result Existence of all cusp types in all dimensions except diagonalizable.
We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in SL(1,D), where D is a quaternion division algebras defined over a number field E contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbol…
The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.
problem Understanding cusp types in hyperbolic 4-manifolds and their commensurability.
method Criteria for commensurability classes containing specific cusp types.
result Infinitely many examples of commensurability classes without certain cusp types.
The paper classifies cusp types in hyperbolic knot complements.
problem Classifying cusp types in hyperbolic knot complements.
method Analyzing quotients of hyperbolic knot complements.
result All cusp types arise in the quotients of link complements.