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Jake Levinson

Publications and source records attributed to Jake Levinson.

At least 19 recordsLinked to original sources
arXiv2026

Degenerations in tropical compactifications and tropical intersection theory of $\overline{M}_{0,n}$

The main result of this paper is a formula for the limit cycle of a 1-parameter family of subvarieties of a tropical compactification, expressed in terms of tropical intersections. Our theorem generalizes results of Dickenstein-Feichtner-Sturmfels and Katz to the case of tropical compactifications. In the second part of the paper, we apply our formula to the moduli space $\overline{M}_{0, n}$ of stable marked rational curves. We describe the tropicalization of the Kapranov maps $\overline{M}_{0, n}\to\mathbb{P}^{n-3}$, whose hyperplane pullbacks are the $\psi$-classes, with respect to a suitable choice of torus. We introduce tropical $\psi$-hypersurfaces (in genus zero). These are different from the standard definition of Mikhalkin and Kerber-Markwig, and may be of independent interest. We demonstrate our main result by giving a "firework algorithm" that computes limits of intersections of $\psi$-hypersurfaces.

math.AG↗
arXiv2025

Sign-reversing involutions in moduli spaces of curves

We use sign-reversing involutions to solve two computational problems that arise naturally in the geometry of moduli spaces of curves. In particular, we give an explicit combinatorial formula for arbitrary $\psi$ class intersection products on the genus zero multicolored spaces $\overline{M}_{0,[r_1,\ldots,r_m]}$ using a novel sign reversing involution on decorated diagrams. As an application, we give a necessary and sufficient condition for when these intersection products are nonzero in terms of matchings on graphs. We also calculate the analog of the tropical Euler characteristic for the graphical moduli spaces $\overline{M}_{0,\Gamma}$ for graphs with two dominant vertices $P, Q$, by constructing two new sign-reversing involutions to simplify the sum. We show that (up to sign) it is the number of acyclic orientations of $\Gamma \smallsetminus \{P, Q\}$.

math.CO↗
arXiv2025

Fundamental groups of moduli spaces of real weighted stable curves

The ordinary and $S_n$-equivariant fundamental groups of the moduli space $\overline{M_{0,n+1}}(\mathbb{R})$ of real $(n+1)$-marked stable curves of genus $0$ are known as \emph{cactus groups} $J_n$ and have applications both in geometry and the representation theory of Lie algebras. In this paper, we compute the ordinary and $S_n$-equivariant fundamental groups of the Hassett space of weighted real stable curves $\overline{M_{0,\mathcal{A}}}(\mathbb{R})$ with $S_n$-symmetric weight vector $\mathcal{A} = (1/a, \ldots, 1/a, 1)$, which we call \emph{weighted cactus groups} $J_n^a$. We show that $J_n^a$ is obtained from the usual cactus presentation by introducing braid relations, which successively simplify the group from $J_n$ to $S_n \rtimes \mathbb{Z}/2\mathbb{Z}$ as $a$ increases. Our proof is by decomposing $\overline{M_{0,\mathcal{A}}}(\mathbb{R})$ as a polytopal complex, generalizing a similar known decomposition for $\overline{M_{0,n+1}}(\mathbb{R})$. In the unweighted case, these cells are known to be cubes and are `dual' to the usual decomposition into associahedra (by the combinatorial type of the stable curve). For $\overline{M_{0,\mathcal{A}}}(\mathbb{R})$, our decomposition instead consists of products of permutahedra. The cells of the decomposition are indexed by weighted stable trees, but `dually' to the usual indexing.

math.AG↗
arXiv2023

Rational Curves in Projective Toric Varieties

We study embedded rational curves in projective toric varieties. Generalizing results of the first author and Zotine for the case of lines, we show that any degree $d$ rational curve in a toric variety $X$ can be constructed from a special affine-linear map called a degree $d$ Cayley structure. We characterize when the curves coming from a degree $d$ Cayley structure are smooth and have degree $d$. We use this to establish a bijection between the set of irreducible components of the Hilbert scheme whose general element is a smooth degree $d$ curve, and so-called maximal smooth Cayley structures. Furthermore, we describe the normalization of the torus orbit closure of such rational curves in the Chow variety, and give partial results for the orbit closures in the Hilbert scheme.

math.AG↗
arXiv2023

Products of boundary classes on M_0,n-bar via balanced weights

In this note, we give a simple closed formula for an arbitrary product, landing in dimension 0, of boundary classes on the Deligne--Mumford moduli space M_0,n-bar. For any such boundary strata $X_{T_1}, \ldots, X_{T_\ell}$, we show the intersection product $\int \prod_{i=1}^\ell [X_{T_i}]$ is either a signed product of multinomial coefficients, or zero, and provide a simple criterion for determining when it is nonzero. We do not claim originality for our product formula, but to our knowledge it does not appear elsewhere in the literature.

math.AG↗
arXiv2023

Minimal degree fibrations in curves and the asymptotic degree of irrationality of divisors

In this paper we study the degrees of irrationality of hypersurfaces of large degree in a complex projective variety. We show that the maps computing the degrees of irrationality of these hypersurfaces factor through rational fibrations of the ambient variety. As a consequence, we give tight bounds on the degree of irrationality of these hypersurfaces in terms of a new invariant of independent interest: the minimal fibering degree of a projective variety with respect to an effective divisor. As a corollary we show that the degree of irrationality of a complete intersection of sufficiently large and unbalanced degrees is roughly the product of the degrees. This gives a partial answer to a question of Bastianelli, De Poi, Ein, Lazarsfeld, and the third author.

math.AG↗
arXiv2022

Degenerations and multiplicity-free formulas for products of $\psi$ and $\omega$ classes on $\overline{M}_{0,n}$

We consider products of $\psi$ classes and products of $\omega$ classes on $\overline{M}_{0,n+3}$. For each product, we construct a flat family of subschemes of $\overline{M}_{0,n+3}$ whose general fiber is a complete intersection representing the product, and whose special fiber is a generically reduced union of boundary strata. Our construction is built up inductively as a sequence of one-parameter degenerations, using an explicit parametrized collection of hyperplane sections. Combinatorially, our construction expresses each product as a positive, multiplicity-free sum of classes of boundary strata. These are given by a combinatorial algorithm on trees we call 'slide labeling'. As a corollary, we obtain a combinatorial formula for the $\kappa$ classes in terms of boundary strata. For degree-$n$ products of $\omega$ classes, the special fiber is a finite reduced union of (boundary) points, and its cardinality is one of the multidegrees of the corresponding embedding $\Omega_n: \overline{M}_{0,n+3}\to \mathbb{P}^1\times \cdots \times \mathbb{P}^n$. In the case of the product $\omega_1\cdots \omega_n$, these points exhibit a connection to permutation pattern avoidance. Finally, we show that in certain cases, a prior interpretation of the multidegrees via tournaments can also be obtained by degenerations.

math.AG↗
arXiv2021

Springer fibers and the Delta Conjecture at $t=0$

We introduce a family of varieties $Y_{n,\lambda,s}$, which we call the \emph{$\Delta$-Springer varieties}, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring $H^*(Y_{n,\lambda,s})$ and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induction products of Specht modules with trivial modules. The $\lambda=(1^k)$ case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at $t=0$. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety $Y_{n,\lambda}$ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud--Saltman rank variety and diagonal matrices.

math.AG↗
arXiv2021

Lazy tournaments and multidegrees of a projective embedding of $\overline{M}_{0,n}$

We provide a new geometric interpretation of the multidegrees of the (iterated) Kapranov embedding $\Phi_n:\overline{M}_{0,n+3}\hookrightarrow \mathbb{P}^1\times \mathbb{P}^2\times \cdots \times \mathbb{P}^n$, where $\overline{M}_{0,n+3}$ is the moduli space of stable genus $0$ curves with $n+3$ marked points. We enumerate the multidegrees by disjoint sets of boundary points of $\overline{M}_{0,n+3}$ via a combinatorial algorithm on trivalent trees that we call a lazy tournament. These sets are compatible with the forgetting maps used to derive the recursion for the multidegrees proven in 2020 by Gillespie, Cavalieri, and Monin. The lazy tournament points are easily seen to total $(2n-1)!!=(2n-1)\cdot (2n-3) \cdots 5 \cdot 3 \cdot 1$, giving a natural proof of the fact that the total degree of $\Phi_n$ is the odd double factorial. This fact was first proven using an insertion algorithm on certain parking functions, and we additionally give a bijection to those parking functions.

math.CO↗
arXiv2021

A Cayley-Bacharach theorem and plane configurations

In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley-Bacharach condition. In particular, by bounding the number of points satisfying the Cayley-Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.

math.AG↗
arXiv2020

An Analysis of SVD for Deep Rotation Estimation

Symmetric orthogonalization via SVD, and closely related procedures, are well-known techniques for projecting matrices onto $O(n)$ or $SO(n)$. These tools have long been used for applications in computer vision, for example optimal 3D alignment problems solved by orthogonal Procrustes, rotation averaging, or Essential matrix decomposition. Despite its utility in different settings, SVD orthogonalization as a procedure for producing rotation matrices is typically overlooked in deep learning models, where the preferences tend toward classic representations like unit quaternions, Euler angles, and axis-angle, or more recently-introduced methods. Despite the importance of 3D rotations in computer vision and robotics, a single universally effective representation is still missing. Here, we explore the viability of SVD orthogonalization for 3D rotations in neural networks. We present a theoretical analysis that shows SVD is the natural choice for projecting onto the rotation group. Our extensive quantitative analysis shows simply replacing existing representations with the SVD orthogonalization procedure obtains state of the art performance in many deep learning applications covering both supervised and unsupervised training.

cs.CV↗
arXiv2019

A Random Matrix Perspective on Mixtures of Nonlinearities for Deep Learning

One of the distinguishing characteristics of modern deep learning systems is that they typically employ neural network architectures that utilize enormous numbers of parameters, often in the millions and sometimes even in the billions. While this paradigm has inspired significant research on the properties of large networks, relatively little work has been devoted to the fact that these networks are often used to model large complex datasets, which may themselves contain millions or even billions of constraints. In this work, we focus on this high-dimensional regime in which both the dataset size and the number of features tend to infinity. We analyze the performance of random feature regression with features $F=f(WX+B)$ for a random weight matrix $W$ and random bias vector $B$, obtaining exact formulae for the asymptotic training and test errors for data generated by a linear teacher model. The role of the bias can be understood as parameterizing a distribution over activation functions, and our analysis directly generalizes to such distributions, even those not expressible with a traditional additive bias. Intriguingly, we find that a mixture of nonlinearities can improve both the training and test errors over the best single nonlinearity, suggesting that mixtures of nonlinearities might be useful for approximate kernel methods or neural network architecture design.

stat.ML↗
arXiv2019

Class groups of open Richardson varieties in the Grassmannian are trivial

We prove that the divisor class group of any open Richardson variety in the Grassmannian is trivial. Our proof uses Nagata's criterion, localizing the coordinate ring at a suitable set of Pl\"ucker coordinates. We prove that these Pl\"ucker coordinates are prime elements by showing that the subscheme they define is an open subscheme of a positroid variety. Our results hold over any field and over the integers.

math.AG↗
arXiv2019

A topological proof of the Shapiro-Shapiro conjecture

We prove a generalization of the Shapiro-Shapiro conjecture on Wronskians of polynomials, allowing the Wronskian to have complex conjugate roots. We decompose the real Schubert cell according to the number of real roots of the Wronski map, and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is given as an evaluation of a symmetric group character. In the case where all roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro-Shapiro conjecture.

math.AG↗
arXiv2019

Latent feature disentanglement for 3D meshes

Generative modeling of 3D shapes has become an important problem due to its relevance to many applications across Computer Vision, Graphics, and VR. In this paper we build upon recently introduced 3D mesh-convolutional Variational AutoEncoders which have shown great promise for learning rich representations of deformable 3D shapes. We introduce a supervised generative 3D mesh model that disentangles the latent shape representation into independent generative factors. Our extensive experimental analysis shows that learning an explicitly disentangled representation can both improve random shape generation as well as successfully address downstream tasks such as pose and shape transfer, shape-invariant temporal synchronization, and pose-invariant shape matching.

cs.LG↗
arXiv2019

Schubert curves in the orthogonal Grassmannian

We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(\lambda_\bullet)$ in the orthogonal Grassmannian $\mathrm{OG}_n$, which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A. First, using the type B Wronski map, we show that the real locus of the Schubert curve has a natural covering map to $\mathbb{RP}^1$, with monodromy operator $\omega$ defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute $\omega$ without rectifying the skew tableau. The first uses recently-developed shifted tableau crystal operators, while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yong's shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves.

math.CO↗
arXiv2018

Axioms for shifted tableau crystals

We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed in a previous paper by Gillespie, Levinson, and Purbhoo. These axioms closely resemble those developed by Stembridge for type A tableau crystals. This axiomatic characterization gives rise to a new method for proving and understanding Schur $Q$-positive expansions in symmetric function theory, just as the Stembridge axiomatic structure provides for ordinary Schur positivity.

math.CO↗