arXiv research

The paper classifies a specific type of quadratic variety with a small codimension.

We generalize Fujiki relation of Beauville-Bogomolov quadratic form on a projective symplectic variety. As an application, we study a fibre space structure of a projective symplectic variety.

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…

The paper proves a quadratic formality for Sasakian manifolds' representation varieties.

Extends classification of hyperbolic Dehn fillings in quadratic fields.

The paper proves properties of strata of differentials, showing they are affine and extremal.

Abstract perspective on quadratic programming for optimal portfolio allocation.

We prove compactification theorems for some complete Kähler manifolds with nonnegative Ricci curvature. Among other things, we prove that a complete noncompact Kähler Ricci flat manifold with maximal volume growth and quadratic curvature decay is a crepant resolution of a normal affine algebraic variety. Furthermore, s…

We study the limits of holonomy representations of complex projective structures on a compact Riemann surface in the Morgan-Shalen compactification of the character variety. We show that the dual R-trees of the quadratic differentials associated to a divergent sequence of projective structures determine the Morgan-Shal…

Study of non-Archimedean Hitchin map for SL2(F) characters.

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of the fundamental group of U with certain restrictions related to the divisor. We show that the possible singularities of this variety as well as of the corresponding modul…

In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …

Partition functions arise in a variety of settings, including conditional random fields, logistic regression, and latent gaussian models. In this paper, we consider semistochastic quadratic bound (SQB) methods for maximum likelihood inference based on partition function optimization. Batch methods based on the quadrati…

This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces …

Policy gradient converges to globally optimal policy in nearly linear-quadratic systems.

We show how to price and replicate a variety of barrier-style claims written on the $\log$ price $X$ and quadratic variation $\langle X \rangle$ of a risky asset. Our framework assumes no arbitrage, frictionless markets and zero interest rates. We model the risky asset as a strictly positive continuous semimartingale w…

A new sampler for complex discrete distributions efficiently updates all variables in parallel.

We characterize a certain neck-pinching degeneration of (marked) $CP^1$- structures on a closed oriented surface S of genus at least two. Namely, we consider a path $C_t$ of $CP^1$-structures on S leaving every compact subset in the deformation space of (marked) $CP^1$-structures on S, such that its holonomy converges …

We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional…

In this paper, we classify three-dimensional complex Abelian varieties isogenous to a product $A_1 \times A_2$, where one of the factors admits real multiplication by a real quadratic order $\mathcal{O}_D$ of discriminant $D$. We show that the moduli space $X_D^{(3)}$ of these varieties essentially is the disjoint unio…

Graph matching aims at finding the vertex correspondence between two unlabeled graphs that maximizes the total edge weight correlation. This amounts to solving a computationally intractable quadratic assignment problem. In this paper we propose a new spectral method, GRAph Matching by Pairwise eigen-Alignments (GRAMPA)…

For any symmetric collection of natural numbers h^{p,q} with p+q=k, we construct a smooth complex projective variety whose weight k Hodge structure has these Hodge numbers; if k=2m is even, then we have to impose that h^{m,m} is bigger than some quadratic bound in m. Combining these results for different weights, we so…

We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and…

Direct policy gradient methods for reinforcement learning and continuous control problems are a popular approach for a variety of reasons: 1) they are easy to implement without explicit knowledge of the underlying model 2) they are an "end-to-end" approach, directly optimizing the performance metric of interest 3) they…

Computable Stein discrepancies have been deployed for a variety of applications, ranging from sampler selection in posterior inference to approximate Bayesian inference to goodness-of-fit testing. Existing convergence-determining Stein discrepancies admit strong theoretical guarantees but suffer from a computational co…

The paper studies geometric loci and their invariants in complex dynamics.

We show that Thurston's skinning maps of Teichmuller space have finite fibers. The proof centers around a study of two subvarieties of the SL_2(C) character variety of a surface, one associated to complex projective structures and the other associated to a 3-manifold. Using the Morgan-Shalen compactification of the cha…

Study of symplectic Monge-Ampère equations using moment maps and contact structures.

Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3-surface and that the family of such varieties is locally complete for deformations …

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose a new, algebraic-geometric approach to the classification problem - based on a p…

Paper connects geometric and analytic aspects of Higgs bundles and pleated surfaces.

The paper explores the geometry of algebraic numbers and their roots.

The critical locus of the loss function of a neural network is determined by the geometry of the functional space and by the parameterization of this space by the network's weights. We introduce a natural distinction between pure critical points, which only depend on the functional space, and spurious critical points, …

Most of the empirical studies on stochastic volatility dynamics favor the 3/2 specification over the square-root (CIR) process in the Heston model. In the context of option pricing, the 3/2 stochastic volatility model is reported to be able to capture the volatility skew evolution better than the Heston model. In this …

We introduce a class of quadratic support (QS) functions, many of which play a crucial role in a variety of applications, including machine learning, robust statistical inference, sparsity promotion, and Kalman smoothing. Well known examples include the l2, Huber, l1 and Vapnik losses. We build on a dual representation…

Statistical inference and information processing of high-dimensional data often require efficient and accurate estimation of their second-order statistics. With rapidly changing data, limited processing power and storage at the acquisition devices, it is desirable to extract the covariance structure from a single pass …

A meromorphic projective structure on a punctured Riemann surface $X\setminus P$ is determined, after fixing a standard projective structure on $X$, by a meromorphic quadratic differential with poles of order three or more at each puncture in $P$. In this article we prove the analogue of Thurston's grafting theorem for…

A recent line of work has shown that an overparametrized neural network can perfectly fit the training data, an otherwise often intractable nonconvex optimization problem. For (fully-connected) shallow networks, in the best case scenario, the existing theory requires quadratic over-parametrization as a function of the …

This paper considers the recovery of a rank $r$ positive semidefinite matrix $X X^T\in\mathbb{R}^{n\times n}$ from $m$ scalar measurements of the form $y_i := a_i^T X X^T a_i$ (i.e., quadratic measurements of $X$). Such problems arise in a variety of applications, including covariance sketching of high-dimensional data…

We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metr…

Quantum algorithm speeds up MIP solving by a near-quadratic factor.

This paper introduces an acceleration structure for hyperbolic embeddings.

We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special…

Given a manifold M with an action of a quadratic Lie algebra d, such that all stabilizer algebras are co-isotropic in d, we show that the product M\times d becomes a Courant algebroid over M. If the bilinear form on d is split, the choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector field on M, w…

We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily algebraic and are homogeneous quadratic in each of the distinguished holomorphic coordinate charts determined by the open …

We construct a finitely presented group $G$ with non-quadratic Dehn function $f$ majorizable by a quadratic function on arbitrary long intervals.

The paper classifies biharmonic quadratic maps between spheres, proving their energy density properties.