arXiv research

Researchers find a method to construct projective structures on a specific surface.

Study moduli spaces of elliptic PDEs using derived $C^{\infty}$-geometry.

The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multip…

Characterizes stably elliptic elements in Lie groups and their properties.

We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds wi…

Study character varieties for 3-punctured sphere group representations in PU(2,1).

Extends a theorem for first-order elliptic operators on manifolds.

Survey on a nonlinear d'Alembertian from general relativity.

Deroin and Tholozan's representations are mapped to complex projective space via action-angle coordinates.

Uniform K-homology theory applied to elliptic operators on manifolds with boundary.

Characterizes totally elliptic surface group representations into Lie groups.

A guide for solving first-order elliptic boundary value problems.

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic …

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an …

We study relations between special elliptic isometries in the complex hyperbolic plane. Relations of lengths 2, 3, and 4 are fully classified. Some relative SU(2,1)-character varieties of the quadruply punctured sphere are described and applied to the study of length 5 relations.

We develop an explicit and tractable representation of a twist-grain-boundary phase of a smectic A liquid crystal. This allows us to calculate the interaction energy between grain boundaries and the relative contributions from the bending and compression deformations. We discuss the special stability of the 90 degree g…

We compute the relative zeta-function metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we prove a general formula relating the zeta-determinant to a Fredholm determinant over the boundary for a class of higher-order elliptic boundary value problems.

In this paper, we discuss the following conjecture raised by Baum-Douglas: For any first-order elliptic differential operator $D$ on smooth manifold $M$ with boundary $\p M$, $D$ possesses an elliptic boundary condition if and only if $\partial [D]$ = 0 in $K_1(\partial M)$, where $[D]$ is the relative $K$-cycle in $K_…

Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.

Every element of PU(2,1) can be decomposed into at most 4 special elliptic isometries.

We introduce and study {\it new} relative spectral invariants of {\it two} elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depend on both the eigenvalues and the eigensections of these operators and contain much more information about geometry. We prov…

The paper proves a criterion for L-space knots and their representations.

Elliptical Attention improves transformer performance by focusing on contextually relevant features.

Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.

We give a representation theoretical proof of Branson's classification of minimal elliptic sums of generalized gradients. The original proof uses tools of harmonic analysis, which as powerful as they are, seem to be specific for the structure groups SO(n) and Spin(n). The different approach we propose is based on the r…

Study trisections on rational elliptic surfaces to find new Zariski pairs.

We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fe…

Study shows solutions to certain equations form smooth manifolds.

These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomology. These relations include the Thom isomorphism in equivariant cohomo…

Into this note we collect topics related to homogeneous vector bundles, elliptic adjoint orbits and so forth.

In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author.

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instan…

Let G be a cyclic group of order 3, 5 or 7, and X=E(n) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is nonsmoothable with respect to infinitely many smooth structures on X. This extends the ma…

This paper provides a new proof of the Lefschetz fixed point formula using groupoids.

Proposes vMF distribution for skewed elliptical distributions.

We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators $P$ can be glued in a chosen region in order to obtain a new smooth soluti…

Elliptical processes extend Gaussian models with heavier tails.

Elliptical processes generalize Gaussian and Student-t models with fat tails and computational efficiency.

M. F. Atiyah proved that the index of a transversally elliptic operator relative to a free action can be computed by using indices of elliptic operators on the orbit manifold. In this paper, we derive an explicit formula for the transversal indices on S^1-bundles over complex projective spaces. Using this explicit form…

We show that an Osserman-type inequality holds for spacelike surfaces of constant mean curvature (CMC) 1 with singularities and with elliptic ends in de Sitter 3-space. An immersed end of a CMC 1 surface is an ``elliptic end'' if the monodromy representation at the end is diagonalizable with eigenvalues in the unit cir…

Develops theory of relatively Anosov representations using flow examples.

Uniform estimates for elliptic problems near polygonal domains.

We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attache…

We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.

Theory of relatively Anosov representations using flow methods.

We define analytic torsion of Z_2-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analyti…

This note removes technical assumptions and characterizes relatively dominated representations.

We find that for any n-dimensional, compact, convex subset K of R^{n+1} there is an affinely-spherical hypersurface M in R^{n+1} with center at the relative interior of K, such that the disjoint union of M and K is the boundary of an (n+1)-dimensional, compact, convex set. This so-called affine hemisphere M is uniquely…