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Linda Chen

Publications and source records attributed to Linda Chen.

At least 19 recordsLinked to original sources
arXiv2025

Littlewood--Richardson rules from quivers for two-step flag varieties

Let $\bigwedge_1$ and $\bigwedge_2$ be two symmetric function algebras in independent sets of variables. We define vector space bases of $\bigwedge_1 \otimes_\mathbb{Z} \bigwedge_2$ coming from certain quivers, with vertex sets indexed by pairs of partitions. We use these vector space bases to give a positive tableau formula for Littlewood--Richardson coefficients for the product of Schubert polynomials with certain Schur polynomials in two-step flag varieties, in the spirit of the Remmel-Whitney rule for the product of two Schur polynomials in Grassmannians. This in particular covers the cases considered by the Pieri rule.

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arXiv2024

K-theoretic Gromov-Witten invariants of line degrees on flag varieties

A homology class $d \in H_2(X)$ of a complex flag variety $X = G/P$ is called a line degree if the moduli space $\overline{M}_{0,0}(X,d)$ of 0-pointed stable maps to $X$ of degree $d$ is also a flag variety $G/P'$. We prove a quantum equals classical formula stating that any $n$-pointed (equivariant, K-theoretic, genus zero) Gromov-Witten invariant of line degree on $X$ is equal to a classical intersection number computed on the flag variety $G/P'$. We also prove an $n$-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov-Witten invariants of the variety of complete flags $G/B$. Our formulas make it straightforward to compute the big quantum K-theory ring of $X$ modulo degrees larger than line degrees.

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arXiv2022

Quantum hooks and mirror symmetry for flag varieties

Given a flag variety $Fl(n;r_1, \dots , r_ρ)$, there is natural ring morphism from the symmetric polynomial ring in $r_1$ variables to the quantum cohomology of the flag variety. In this paper, we show that for a large class of partitions $λ$, the image of $s_λ$ under the ring homomorphism is a Schubert class which is described by partitioning $λ$ into a quantum hook (or $q$-hook) and a tuple of smaller partitions. We use this result to show that the Plücker coordinate mirror of the flag variety describes quantum cohomology relations. This gives new insight into the structure of this superpotential, and the relation between superpotentials of flag varieties and those of Grassmannians (where the superpotential was introduced by Marsh--Rietsch).

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arXiv2022

A Neural Network Solves, Explains, and Generates University Math Problems by Program Synthesis and Few-Shot Learning at Human Level

We demonstrate that a neural network pre-trained on text and fine-tuned on code solves mathematics course problems, explains solutions, and generates new questions at a human level. We automatically synthesize programs using few-shot learning and OpenAI's Codex transformer and execute them to solve course problems at 81% automatic accuracy. We curate a new dataset of questions from MIT's largest mathematics courses (Single Variable and Multivariable Calculus, Differential Equations, Introduction to Probability and Statistics, Linear Algebra, and Mathematics for Computer Science) and Columbia University's Computational Linear Algebra. We solve questions from a MATH dataset (on Prealgebra, Algebra, Counting and Probability, Intermediate Algebra, Number Theory, and Precalculus), the latest benchmark of advanced mathematics problems designed to assess mathematical reasoning. We randomly sample questions and generate solutions with multiple modalities, including numbers, equations, and plots. The latest GPT-3 language model pre-trained on text automatically solves only 18.8% of these university questions using zero-shot learning and 30.8% using few-shot learning and the most recent chain of thought prompting. In contrast, program synthesis with few-shot learning using Codex fine-tuned on code generates programs that automatically solve 81% of these questions. Our approach improves the previous state-of-the-art automatic solution accuracy on the benchmark topics from 8.8% to 81.1%. We perform a survey to evaluate the quality and difficulty of generated questions. This work is the first to automatically solve university-level mathematics course questions at a human level and the first work to explain and generate university-level mathematics course questions at scale, a milestone for higher education.

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arXiv2021

Motivic classes of degeneracy loci and pointed Brill-Noether varieties

Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $χ_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.

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arXiv2021

On an equivalence of divisors on $\bar{M}_{0,n}$ from Gromov-Witten theory and conformal blocks

We consider a conjecture that identifies two types of base point free divisors on $\bar{M}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated to simple Lie algebras in type A. Here we reduce this conjecture on $\bar{M}_{0,n}$ to the same statement for $n=4$. A reinterpretation leads to a proof of the conjecture on $\bar{M}_{0,n}$ for a large class, and we give sufficient conditions for the non-vanishing of these divisors.

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arXiv2021

An affine approach to Peterson comparison

The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov's strange duality isomorphism.

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arXiv2021

K-classes of Brill-Noether loci and a determinantal formula

We compute the Euler characteristic of the structure sheaf of the Brill-Noether locus of linear series with special vanishing at up to two marked points. When the Brill-Noether number $ρ$ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $ρ=1$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan-Martín-Pflueger-Teixidor for the arithmetic genus of a Brill-Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the K-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations, and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey-Jockusch-Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux.

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arXiv2020

On the finiteness of quantum K-theory of a homogeneous space

We show that the product in the quantum K-ring of a generalized flag manifold $G/P$ involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the $J$-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.

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arXiv2018

Development of Robust Fractional-Order Reset Control

In this paper, a framework for the combination of robust fractional order CRONE control with non-linear reset is given for both first and second generation CRONE control. General design rules are derived and presented for these CRONE reset controllers. Within this framework, fractional order control allows for better tuning of the open-loop responses on the one hand. On the other, reset control enables a reduction in phase lag and a corresponding increase in phase margin compared to linear control for similar open loop gain profile. Hence, the combination of the two control methods can provide well-tuned open-loop responses that can overcome the fundamental linear control limitation of Bode's gain-phase relationship. Moreover, as established loop-shaping concepts are used in the controller design, CRONE reset can be highly compatible with the industry. The designed CRONE reset controllers are validated on a one degree-of-freedom Lorentz-actuated precision positioning stage. On this setup, CRONE reset control is shown to provide better tracking performance compared to linear CRONE control, which is in agreement with the predicted performance improvement.

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arXiv2018

Beyond the Waterbed Effect: Development of Fractional Order CRONE Control with Non-Linear Reset

In this paper a novel reset control synthesis method is proposed: CRONE reset control, combining a robust fractional CRONE controller with non-linear reset control to overcome waterbed effect. In CRONE control, robustness is achieved by creation of constant phase behaviour around bandwidth with the use of fractional operators, also allowing more freedom in shaping the open-loop frequency response. However, being a linear controller it suffers from the inevitable trade-off between robustness and performance as a result of the waterbed effect. Here reset control is introduced in the CRONE design to overcome the fundamental limitations. In the new controller design, reset phase advantage is approximated using describing function analysis and used to achieve better open-loop shape. Sufficient quadratic stability conditions are shown for the designed CRONE reset controllers and the control design is validated on a Lorentz-actuated nanometre precision stage. It is shown that for similar phase margin, better performance in terms of reference-tracking and noise attenuation can be achieved.

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arXiv2012

Equivariant quantum Schubert polynomials

We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the equivariant quantum cohomology ring, as well as Graham-positivity of the structure constants in equivariant quantum Schubert calculus.

math.AG↗
arXiv2011

Positivity of equivariant Gromov-Witten invariants

We show that the equivariant Gromov-Witten invariants of a projective homogeneous space G/P exhibit Graham-positivity: when expressed as polynomials in the positive roots, they have nonnegative coefficients.

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arXiv2007

The Equivariant Chow rings of quot schemes

We give a presentation for the (integral) torus-equivariant Chow ring of the quot scheme, a smooth compactification of the space of rational curves of degree d in the Grassmannian. For this presentation, we refine Evain's extension of the method of Goresky, Kottwitz, and MacPherson to express the torus-equivariant Chow ring in terms of the torus-fixed points and explicit relations coming from the geometry of families of torus-invariant curves. As part of this calculation, we give a complete description of the torus-invariant curves on the quot scheme and show that each family is a product of projective spaces.

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arXiv2006

Pointed trees of projective spaces

We introduce a smooth projective variety $T_{d,n}$ which compactifies the space of configurations of $n$ distinct points on affine $d$-space modulo translation and homothety. The points in the boundary correspond to $n$-pointed stable rooted trees of $d$-dimensional projective spaces, which for $d = 1$, are $(n+1)$-pointed stable rational curves. In particular, $T_{1,n}$ is isomorphic to $\bar{M}_{0,n+1}$, the moduli space of such curves. The variety $T_{d,n}$ shares many properties with $\bar{M}_{0,n}$. For example, as we prove, the boundary is a smooth normal crossings divisor whose components are products of $T_{d,i}$ for $i < n$, it has an inductive construction analogous to but differing from Keel's for $\bar{M}_{0,n}$ which can be used to describe its Chow groups, Chow motive and Poincaré polynomials, generalizing \cite{Keel,Man:GF}. We give a presentation of the Chow rings of $T_{d,n}$, exhibit explicit dual bases for the dimension 1 and codimension 1 cycles. The variety $T_{d,n}$ is embedded in the Fulton-MacPherson spaces $X[n]$ for \textit{any} smooth variety $X$ and we use this connection in a number of ways. For example, to give a family of ample divisors on $T_{d,n}$ and to give an inductive presentation of the Chow groups and the Chow motive of $X[n]$ analogous to Keel's presentation for $\bar{M}_{0,n}$, solving a problem posed by Fulton and MacPherson.

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arXiv2004

The orbifold Chow ring of toric Deligne-Mumford stacks

Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.

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