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Mingzhi Yang

Publications and source records attributed to Mingzhi Yang.

10 recordsLinked to original sources
arXiv2026

Universal Diffusion Models for Implied Volatility Surfaces: Learning Shared Dynamics Across Stocks

Modeling the dynamics of option implied volatility surface (IVS) is crucial for pricing, hedging, and risk-managing option portfolios. We develop a universal conditional diffusion model that learns to jointly generate next-day IVS increments and the underlying stock's returns. The model is trained on pooled data from 50 stocks and evaluated on a test set comprising 50 in-sample stocks and 50 out-of-sample stocks excluded from training. The training objective is primarily the minimization of MSE, with the variants that jointly impose surface smoothness, and that penalize the presence of static-arbitrage, which are all economically meaningful constraints. Across in-sample and out-of-sample stocks, our diffusion model outperforms the VolGAN benchmark in reducing arbitrage violations, improving stock risk prediction, and aligning explained variance ratios by the first three principal components. The fact that the model extrapolates well to stocks that are excluded from training indicates that the learned dynamics can be shared across stocks. These findings support a universal conditional diffusion as a credible framework for multi-stock implied volatility surface scenario generation.

q-fin.CP↗
arXiv2026

A Peterson program for general Schubert varieties and mirror symmetry

We initiate a `Peterson program' that seeks to extend Dale Peterson's theory for the quantum cohomology of flag varieties to arbitrary Schubert varieties, and that incorporates a mirror-symmetric approach. In this first paper we introduce a generalisation of the (localised) Peterson variety associated to an arbitrary Schubert variety $\check X_{w,P}$ inside a partial flag variety $\check G/\check P$. This generalisation is possibly a non-reduced affine scheme that lies in the Langlands dual full flag variety $G/B$. Additionally, we construct a Lie-theoretic `superpotential' associated to $\check X_{w,P}$, generalising earlier ones for the flag varieties $\check G/\check P$ from [Rie08], and we show that its relative critical point locus recovers the generalised Peterson variety. We make the key conjecture that for smooth Fano Schubert varieties the coordinate ring of the generalised Peterson variety recovers the quantum cohomology ring of $\check X_{w,P}$ localised at the quantum parameters. For smooth Fano Schubert varieties $X_{w,B}$ in $\check G/\check B$ we furthermore map our generalised Peterson variety to the cotangent bundle $\mathcal T^*(T)$ of the maximal torus $T$ of $G$ and construct a partial compactification that we conjecture models the full quantum cohomology ring of $\check X_{w,B}$, in analogy with the Givental-Kim presentation of $QH^*(\check G/\check B)$ via the degenerate leaf of the Kostant Toda lattice of $\check G$. These conjectures are verified for all smooth Schubert divisors in the complete type $A$ flag variety, and we show that our superpotential agrees with that introduced for Grassmannian Schubert varieties by Rietsch and Williams, after a suitable isomorphism of the domains.

math.AG↗
arXiv2026

Quantum Schubert calculus for smooth Schubert divisors of $F\ell_n$

We propose to study the quantum Schubert calculus for Schubert varieties, and investigate the smooth Schubert divisors X of the complete flag variety Fl_n. We provide a Borel-type ring presentation of the quantum cohomology of X. We derive the quantum Chevalley formula for X by geometric arguments. We also show that the quantum Schubert polynomials for X are the same as that for Fl_n introduced by Fomin, Gelfand and Postnikov.

math.AG↗
arXiv2025

CTourLLM: Enhancing LLMs with Chinese Tourism Knowledge

Recently, large language models (LLMs) have demonstrated their effectiveness in various natural language processing (NLP) tasks. However, the lack of tourism knowledge limits the performance of LLMs in tourist attraction presentations and travel planning. To address this challenge, we constructed a supervised fine-tuning dataset for the Chinese culture and tourism domain, named Cultour. This dataset consists of three parts: tourism knowledge base data, travelogues data, and tourism QA data. Additionally, we propose CTourLLM, a Qwen-based model supervised fine-tuned with Cultour, to improve the quality of information about attractions and travel planning. To evaluate the performance of CTourLLM, we proposed a human evaluation criterion named RRA (Relevance, Readability, Availability), and employed both automatic and human evaluation. The experimental results demonstrate that CTourLLM outperforms ChatGPT, achieving an improvement of 1.21 in BLEU-1 and 1.54 in Rouge-L, thereby validating the effectiveness of the response outcomes. Our proposed Cultour is accessible at https://github.com/mrweiqk/Cultour.

cs.CL↗
arXiv2025

An anticanonical perspective on G/P Schubert varieties

We describe a natural basis of the Cartier class group of an arbitrary Schubert variety $X_{w,P}$ in a flag variety $G/P$ of general Lie type. We then characterise when the Schubert variety is factorial/Fano, along with an explicit formula for the anticanonical line bundle in these cases. We also prove that, for Schubert varieties in simply-laced types (only), being factorial is equivalent to being $Q$-factorial, and is equivalent to the equality of the Betti numbers $b_2(X_{w,P})=b_{2\ell(w)-2}(X_{w,P})$. Finally, we give a convenient characterisation of when a simply-laced Schubert variety is Gorenstein and when it is Gorenstein Fano.

math.AG↗
arXiv2024

On Seidel representation in quantum K-theory of Grassmannians

We provide a direct proof of Seidel representation in the quantum K-theory QK(Gr(k, n)) by studying projected Gromov-Witten varieties concretely. As applications, we give an alternative proof of the K-theoretic quantum Pieri rule by Buch and Mihalcea, reduce certain quantum Schubert structure constants of higher degree to classical Littlewood-Richardson coefficients for K(Gr(k, n)), and provide a quantum Littlewood-Richardson rule for QK(Gr(3, n)).

math.AG↗
arXiv2024

QCG-Rerank: Chunks Graph Rerank with Query Expansion in Retrieval-Augmented LLMs for Tourism Domain

Retrieval-Augmented Generation (RAG) mitigates the issue of hallucination in Large Language Models (LLMs) by integrating information retrieval techniques. However, in the tourism domain, since the query is usually brief and the content in the database is diverse, existing RAG may contain a significant amount of irrelevant or contradictory information contents after retrieval. To address this challenge, we propose the QCG-Rerank model. This model first performs an initial retrieval to obtain candidate chunks and then enhances semantics by extracting critical information to expand the original query. Next, we utilize the expanded query and candidate chunks to calculate similarity scores as the initial transition probability and construct the chunks graph. Subsequently, We iteratively compute the transition probabilities based on an initial estimate until convergence. The chunks with the highest score are selected and input into the LLMs to generate responses. We evaluate the model on Cultour, IIRC, StrategyQA, HotpotQA, SQuAD, and MuSiQue datasets. The experimental results demonstrate the effectiveness and superiority of the QCG-Rerank method.

cs.CL↗
arXiv2024

A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus

We construct a Plücker coordinate superpotential $\mathcal{F}_-$ that is mirror to a partial flag variety $\mathbb{ F}\ell(n_\bullet)$. Its Jacobi ring recovers the small quantum cohomology of $\mathbb{ F}\ell(n_\bullet)$ and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues for the action of the first Chern class $c_1(\mathbb{ F}\ell(n_\bullet))$ on quantum cohomology are equal to the critical values of $\mathcal{F}_-$. We achieve this by proving new identities in quantum Schubert calculus that are inspired by our formula for $\mathcal{F}_-$ and the mirror symmetry conjecture.

math.AG↗
arXiv2022

On Frobenius-Perron Dimension

We propose a notion of Frobenius-Perron dimension for certain free $\mathbb{Z}$-modules of infinite rank and compute it for the $\mathbb{Z}$-modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. We also provide a lower bound for the Frobenius-Perron dimension of Schubert classes in the quantum cohomology of complex Grassmannians.

math.AG↗