Jake Levinson
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Fundamental groups of moduli spaces of real weighted stable curves Open
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 a…
Springer fibers and the Delta Conjecture at t = 0 Open
We introduce a family of varieties Yn,λ,s, which we call the Δ-Springer varieties, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring H⁎(Yn,λ,s) and show that there is a symmetric group acti…
Rational Curves in Projective Toric Varieties Open
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…
Products of boundary classes on M_0,n-bar via balanced weights Open
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 int…
Minimal degree fibrations in curves and the asymptotic degree of irrationality of divisors Open
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…
Lazy tournaments and multidegrees of a projective embedding of \(\overline{M}_{0,n}\) Open
We consider the (iterated) Kapranov embedding \(\Omega_n:\overline{M}_{0,n+3} \hookrightarrow \mathbb{P}^1 \times \cdots \times \mathbb{P}^n\), where \(\overline{M}_{0,n+3}\) is the moduli space of stable genus \(0\) curves with \(n+3\) ma…
A proof of a conjecture by Monin and Rana on equations defining $\bar{M}_{0,n}$ Open
Monin and Rana conjectured a set of equations defining the image of the moduli space $\bar{M}_{0,n}$ under an embedding into $\mathbb{P}^1\times \cdots\times \mathbb{P}^{n-3}$ due to Keel and Tevelev and verified the conjecture for $n\leq …
Class groups of open Richardson varieties in the Grassmannian are trivial Open
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 Plücker coordinates. We prove that these Plücker co…
A Cayley–Bacharach theorem and plane configurations Open
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…
Degenerations and multiplicity-free formulas for products of $ψ$ and $ω$ classes on $\overline{M}_{0,n}$ Open
We consider products of $ψ$ classes and products of $ω$ 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…
Springer fibers and the Delta Conjecture at $t=0$ Open
We introduce a family of varieties $Y_{n,λ,s}$, which we call the \emph{$Δ$-Springer varieties}, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring $H^*(Y_{n,λ,s})$ and show that there is a …
Lazy tournaments and multidegrees of a projective embedding of $\overline{M}_{0,n}$ Open
We provide a new geometric interpretation of the multidegrees of the (iterated) Kapranov embedding $Φ_n:\overline{M}_{0,n+3}\hookrightarrow \mathbb{P}^1\times \mathbb{P}^2\times \cdots \times \mathbb{P}^n$, where $\overline{M}_{0,n+3}$ is …
An Analysis of SVD for Deep Rotation Estimation Open
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 alig…
A crystal-like structure on shifted tableaux Open
We introduce coplactic raising and lowering operators , , , and on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals)…
Monodromy and K-theory of Schubert curves via generalized jeu de taquin Open
We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previ…
A Random Matrix Perspective on Mixtures of Nonlinearities for Deep Learning Open
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. Whi…
Class groups of open Richardson varieties in the Grassmannian are trivial Open
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 Plücker coordinates. We prove that these Plücker co…
Latent feature disentanglement for 3D meshes Open
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 AutoEnco…
Axioms for Shifted Tableau Crystals Open
We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed by the authors and Purbhoo. These axioms closely resemble those developed by Stembridge for type A tableau crystals. This axiomatic ch…
Schubert curves in the orthogonal Grassmannian Open
We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(λ_\bullet)$ in the orthogonal Grassmannian $\mathrm{OG}_n$, which are one-dimensional Schubert problems defined with respect to orthogonal flags oscu…
Foundations of Boij–Söderberg theory for Grassmannians Open
Boij–Söderberg theory characterizes syzygies of graded modules and sheaves on projective space. This paper continues earlier work with Sam, extending the theory to the setting of $\text{GL}_{k}$ -equivariant modules and sheaves on Grassman…
Shifted tableaux crystals Open
We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but n…
Monodromy and K-theory of Schubert curves via generalized jeu de taquin Open
We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(λ_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve.…