Predicts next item in sequential bundles using Transformers.
problem Predicting next item in sequentially consumed bundles.
method Used custom Transformers, GPT-3, LSTM, reinforcement learning, Markov models.
result Custom Transformer with decoder-only architecture most accurate.
The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing rep…
The paper tackles preference prediction from ordinal data.
problem Predicting preferences from ordinal data collected in various forms.
method Solves a convex relaxation of nuclear norm minimization to learn the underlying low-rank model.
result The convex relaxation approach is minimax optimal and provides upper and lower bounds on error.
Decomposes bundle gerbes on supermanifolds into simpler components.
problem Classifying bundle gerbes on supermanifolds.
method Proves decomposition theorem and cohomological classification theorems.
result Bundle gerbes on supermanifolds decompose into a gerbe over the manifold and a 2-form.
Alternative proof of coisotropic embedding theorem for pre-symplectic manifolds.
problem Proving the coisotropic embedding theorem for pre-symplectic manifolds.
method Recast geometric choice of connection as algebraic embedding into cotangent bundle, identify symplectic thickening as submanifold of Hamiltonian momenta conjugate to kernel directions.
result Alternative proof of the coisotropic embedding theorem.
Defines octonion bundles and connections related to G2-structures.
problem Exploring octonion bundles and connections in G2-manifolds.
method Using G2-structures and octonion bundles, defining metric-compatible covariant derivatives and torsions.
result Critical points of the energy functional correspond to divergence-free torsion, an octonionic analog of the Coulomb gauge.
From a certain strongly equivariant bundle gerbe with connection and curving over a smooth manifold on which a Lie group acts, we construct under some conditions a bundle gerbe with connection and curving over the quotient space. In general, the construction requires a choice, and we can consequently obtain distinct st…
New characteristic classes for fiber bundles via flat connections.
problem Constructing new characteristic classes for fiber bundles.
method Using flat connections with infinite-dimensional Lie algebras of derivations and fiberwise metrics.
result Induced map on cohomology groups is independent of choices and gives Morita-Mumford-Miller classes for surface bundles.
For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density fun…
Study holonomy expansions for loops in principal bundles.
problem Holonomy expansions for loops in principal bundles.
method Asymptotic expansion of holonomy in terms of loop length.
result Independent asymptotic formula for holonomy.
Paper proposes a method to estimate consumer valuations from bundle sales data.
problem Estimating consumer valuations from bundle sales data using classical methods is challenging.
method Proposes an approach using EM algorithm and Monte Carlo simulation to estimate consumer valuations from bundle sales data.
result The approach can recover the distribution of consumers' valuations and is robust to unobserved no-purchases and clustered market segments.
Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic L2 torsion, which lies in the determinant line of the twisted L2 Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite…
A method constructs tractor conformal bundles for spacelike submanifolds in Lorentzian manifolds.
problem Characterize conditions for a tractor conformal bundle to be standard and normal.
method Introduce a canonical construction of a tractor conformal bundle and characterize conditions for it to be standard and normal.
result Characterizes conditions for a tractor conformal bundle to be standard and normal.
Researchers calculate the Ray-Singer Torsion for S1 bundles.
problem Few explicit evaluations of path integrals in higher dimensions.
method Algebraic choice of gauge leading to factorization of path integral.
result Explicit calculation of Ray-Singer Torsion for S1 bundles. New tractor geometry derived from asymptotically flat spacetimes.
problem Understanding the geometry of spacetimes near their boundaries.
method Derived null-tractor bundle from interior spacetime geometry, proved connections' uniqueness, and expressed results in BMS coordinates.
result Tractor connection encodes mass and angular momentum in 3D, and asymptotic shear in higher dimensions.
Theory of relatively Anosov representations using flow methods.
problem Developing a theory for relatively Anosov representations.
method Using the contracting flow on a bundle to define and study relatively Anosov representations.
result Definition and study of uniformly relatively Anosov representations and a stability result.
Compact Hermitian manifolds with quasi-negative curvature have ample canonical line bundles.
problem Determining conditions for ample canonical line bundles in Hermitian manifolds.
method Hermitian curvature flow with specific curvature conditions.
result Canonical line bundle is ample under given curvature conditions.
Extends submanifold minimality to multi-index u-minimality using frame bundle analysis.
problem Generalizing submanifold minimality to multi-index settings.
method Analysis of frame bundle and associated vector bundles; variation of σ_u-symmetric function.
result Alternative definition of u-minimality and examples of u-minimal submanifolds.
New equations for geodesics in sub-Riemannian geometry.
problem Finding equations for normal geodesics in sub-Riemannian geometry.
method Developed a new system of equations using a partial connection.
result The new equations split into horizontal and complementary parts.
We show that the heterotic supersymmetry (Killing spinor equations) and the anomaly cancellation imply the heterotic equations of motion in dimensions five, six, seven, eight if and only if the connection on the tangent bundle is an instanton. For heterotic compactifications in dimension six this reduces the choice of …
In this paper, we develop a general study of contributions at infinity of Bochner-Weitzenböck-type formulas on asymptotically flat manifolds, inspired by Witten's proof of the positive mass theorem. As an application, we show that similar proofs can be obtained in a much more general setting as any choice of an irreduc…
Quantizes contact structures using dynamical methods.
problem Quantizing contact structures in a flat connection.
method Constructs a dynamical quantization using a flat connection on a Hilbert tractor bundle.
result Determines a contact tractor connection whose parallel sections determine a distinguished choice of Reeb dynamics.
Explicitly constructs moduli spaces of stable parabolic bundles.
problem Understanding moduli spaces of stable parabolic bundles over the Riemann sphere.
method Explicit construction and quotient of stable parabolic structures by bundle automorphisms.
result Explicit models of moduli spaces as smooth, compact complex manifolds.
Study shows Kähler-Ricci flow singularity type is consistent over time.
problem Understanding singularity types in Kähler-Ricci flow.
method Analyzes the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundles.
result Singularity type at infinity is consistent regardless of initial metric.
Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.
problem Developing Hodge theory on Carrollian bundles with degenerate metrics.
method Constructing Hodge star and Laplacian operators on Carrollian Rimes-bundles. result Hodge--de Rham Laplacian defined on event horizons of Schwarzschild black holes.
New solutions found for G2 system using K3 orbifolds.
problem Finding smooth solutions to the G2 Hull-Strominger system. method Torus fibrations over K3 orbifolds, adapted Serre construction for singular settings.
result Constructed new smooth solutions to the G2 Hull-Strominger system. We extend topological recursion to twisted Higgs bundles with singularities.
problem Computing Taylor expansions of period matrices for twisted Higgs bundles.
method We introduce a twisted topological recursion on the spectral curve of a twisted Higgs bundle, encoding singularities and performing the recursion explicitly.
result The g=0 twisted Eynard-Orantin differentials compute the Taylor expansion of the spectral curve's period matrix, independent of the ambient space. We provide a coordinate-free version of the local classification, due to A. G. Walker [Quart. J. Math. Oxford (2) 1, 69 (1950)], of null parallel distributions on pseudo-Riemannian manifolds. The underlying manifold is realized, locally, as the total space of a fibre bundle, each fibre of which is an affine principal b…
In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation (M,F) with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac op…
Study noncommutative deformations of Calabi-Yau threefolds.
problem Understanding the geometry of Calabi-Yau threefolds under noncommutative deformations.
method Analyzing the influence of Poisson structures on quantum moduli spaces.
result The choice of Poisson structure significantly affects the geometry of quantum moduli spaces.
Study shows zeros of random sections are uniformly distributed.
problem Distribution of zeros in random holomorphic sections.
method Equidistribution and moment assumptions for singular Hermitian line bundles.
result Asymptotic distribution of zeros is independent of probability measure.
We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out tha…
Twists Gromov and Lefschetz invariants for symplectic and fibered 3-manifolds.
problem Defining and relating twisted invariants for symplectic and fibered manifolds.
method Defining twisted Gromov-Taubes invariants and Lefschetz zeta functions for surface bundles over manifolds, proving their equivalence and interpreting them in terms of Reidemeister torsions.
result Twisted Lefschetz zeta functions and Gromov-Taubes invariants are equivalent for certain bundles, leading to new interpretations of Reidemeister torsions.
Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cove…
Extends adjoint representation concept to higher Lie groupoids.
problem Defining adjoint representation for higher Lie groupoids.
method Generalizes standard construction to higher Lie groupoids using simplicial vector bundles.
result Adjoint representation up to homotopy is well-defined and unique.
We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle O⊕O(2). We show that the Newton--Cartan space-times are unstable under the general K…
We study heterotic supergravity at O(α′), first described in detail in 1989 by Bergshoeff and de Roo. In particular, we discuss an ambiguity of a connection choice on the tangent bundle. It is well known that at O(α′) the Hull connection gives a consistent supergravity theory with supersymmetry …
Market bubbles identified via spectral theory of geometric bundles.
problem Identifying asset bubbles in markets with arbitrage opportunities.
method Geometric Arbitrage Theory reformulated as a stochastic principal fibre bundle with a connection Laplacian.
result A market satisfies (NFLVR) if and only if 0 is in the discrete spectrum of the connection Laplacian.
Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding t…
In the jet-bundle description of first-order classical field theories there are some elements, such as the lagrangian energy and the construction of the hamiltonian formalism, which require the prior choice of a connection. Bearing these facts in mind, we analyze the situation in the jet-bundle description of time-depe…
We consider topological T-duality of torus bundles equipped with S^{1}-gerbes. We show how a geometry on the gerbe determines a reduction of its band to the subsheaf of S^{1}-valued functions which are constant along the torus fibres. We observe that such a reduction is exactly the additional datum needed for the const…
We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation is particularly suitable to deal with string connections: it enables us to prove that every string structure admits a string connect…
A simpler definition of diffeological connections on vector pseudo-bundles.
problem Defining connections on diffeological vector pseudo-bundles.
method Adapting standard connection definition to diffeological context.
result Simplified definition of connections is straightforward and uses dual pseudo-bundle as tangent space.
We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter α, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of α, we prove that the limit of a convergent sequence of balanced metrics leads to a Hermitian-Einst…
I begin by explaining how Riemannian geometry can be understood in terms of principal fibre bundles and connections thereon. I then introduce and motivate the definition of a spinor structure in terms of familiar geometrical ideas. The central result of this thesis is a complete and constructive classification of spino…
The paper introduces a new concept of infinitesimal homogeneity for connections on bundles and applies it to prove known theorems and derive new results.
problem Understanding and proving theorems related to connections on bundles and their properties.
method Introducing infinitesimal homogeneity for sections in associated vector bundles and proving the existence of connections satisfying parallelism conditions.
result The existence of connections satisfying parallelism conditions and the ability to classify locally homogeneous and symmetric triples.
This work extends Chern correspondence to higher gauge theory.
problem Generalizing Chern correspondence to higher gauge theory.
method Defined connective structures on multiplicative gerbes and proposed a complexification for 2-groups.
result Established a Chern correspondence for holomorphic principal 2-bundles.
The paper quantizes vortex moduli spaces on compact Kahler surfaces using determinant bundles.
problem Quantizing vortex moduli spaces on compact Kahler surfaces.
method Developed holomorphic determinant bundles and geometric quantization for vortex moduli spaces.
result Quantized vortex moduli spaces on compact Kahler surfaces using determinant bundles.