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Ishan Banerjee

Publications and source records attributed to Ishan Banerjee.

14 recordsLinked to original sources
arXiv2026

Monodromy action on character varieties for Lefschetz pencils

Given a Riemann surface {\Sigma}, let {\Gamma} {\subseteq} Mod({\Sigma}) denote the monodromy subgroup of a family of complex curves homeomorphic to {\Sigma}, arising from a sufficently ample Lefschetz pencil. We establish that the group {\Gamma} acts with Zariski dense orbits or ergodically on certain character varieties for {\pi_1}({\Sigma}). This answers a version of a Conjecture of Katzarkov, Pantev, and Simpson appearing in [KPS03].

math.AG↗
arXiv2025

Stable cohomology of universal character varieties

We study the universal PGL_n character variety over M_g whose fiber over a point [C] is the space of PGL_n-local systems on the curve C. We use nonabelian Hodge theory and properties of Saito's mixed Hodge modules to show that the Leray-Serre spectral sequence for the projection to M_g degenerates at E_2. As an application, we prove that the rational cohomology of these varieties stabilizes as g goes to infinity and compute the stable limit. We also deduce similar results for the universal G-character variety over M_{g,1} whose fiber over a punctured curve is the variety of G-local systems with fixed central monodromy around the puncture, for G = GL_n or SL_n. The paper concludes with a computation of the ring structure on the stable cohomology and with an appendix by Anne Larsen and Mirko Mauri proving related results for the intersection cohomology of singular universal character varieties.

math.AG↗
arXiv2025

Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces

We compute the mapping class group-valued monodromy of any sufficiently ample linear system on any smooth simply connected projective surface, identifying this with the r-spin mapping class group associated to a maximal root of the adjoint line bundle. This gives a characterization of the simple closed curves that can arise as vanishing cycles for nodal degenerations in the linear system, as well as other corollaries concerning discriminants, Lefschetz fibrations, and surfaces in 4-manifolds.

math.AG↗
arXiv2025

Slicing Correspondences with High Degree Hypersurfaces

We approximately compute the correspondence degree (as defined by Lazarsfeld and Martin) between two unbalanced complete intersections. This is accomplished by showing that the procedure of taking a subvariety of a product $Y \times Y'$ and intersecting it with $X \times Y'$ (for $X$ a sufficiently ample smooth divisor in $Y$) induces a bijection between two sets of varieties. This may be of independent interest.

math.AG↗
arXiv2024

Survey of Large Multimodal Model Datasets, Application Categories and Taxonomy

Multimodal learning, a rapidly evolving field in artificial intelligence, seeks to construct more versatile and robust systems by integrating and analyzing diverse types of data, including text, images, audio, and video. Inspired by the human ability to assimilate information through many senses, this method enables applications such as text-to-video conversion, visual question answering, and image captioning. Recent developments in datasets that support multimodal language models (MLLMs) are highlighted in this overview. Large-scale multimodal datasets are essential because they allow for thorough testing and training of these models. With an emphasis on their contributions to the discipline, the study examines a variety of datasets, including those for training, domain-specific tasks, and real-world applications. It also emphasizes how crucial benchmark datasets are for assessing models' performance in a range of scenarios, scalability, and applicability. Since multimodal learning is always changing, overcoming these obstacles will help AI research and applications reach new heights.

cs.AI↗
arXiv2024

Generators for the level $m$ congruence subgroups of braid groups

We prove for $m\geq1$ and $n\geq5$ that the level $m$ congruence subgroup $B_n[m]$ of the braid group $B_n$ associated to the integral Burau representation $B_n\to\mathrm{GL}_n(\mathbb{Z})$ is generated by $m$th powers of half-twists and the braid Torelli group. This solves a problem of Margalit, generalizing work of Assion, Brendle--Margalit, Nakamura, Stylianakis and Wajnryb.

math.GR↗
arXiv2024

Monodromy and vanishing cycles for complete intersection curves

We compute the topological monodromy of every family of complete intersection curves. Like in the case of plane curves previously treated by the second-named author, we find the answer is given by the $r$-spin mapping class group associated to the maximal root of the adjoint line bundle. Our main innovation is a suite of tools for studying the monodromy of sections of a tensor product of very ample line bundles in terms of the monodromy of sections of the factors, allowing for an induction on (multi-)degree.

math.AG↗
arXiv2024

Error terms for the motives of discriminant complements and a Cayley-Bacharach theorem

In this paper we prove under some simplifying hypotheses questions of Picoco and Levinson-Ullery on Cayley-Bacharach sets. Our results imply that, under suitable hypotheses Cayley-Bacharach sets lie on curves of low degree. We then use these results to estimate error terms to the normalized motive of the space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$as $d$ grows to infinity. The error term can be expressed in terms of a certain `sum over points' on plane cubic curves and the associated Hodge structure can be expressed in terms of the cohomology of the moduli space of elliptic curves. We also prove convergence of the motive of degree $d$ hypersurfaces in $\mathbb{P}^n$ as $n$ grows to infinity as well as other results on discriminant complements of high dimensional varieties.

math.AG↗
arXiv2024

A $π_1$ obstruction to having finite index monodromy and an unusual subgroup of infinite index in $\textrm{Mod}(Σ_g)$

Let $X$ be an algebraic surface with $\mathcal{L}$ an ample line bundle on $X$. Let $Γ(X, \mathcal{L})$ be the \emph{geometric monodromy} group associated to family of nonsingular curves in $X$ that are zero loci of sections of $\mathcal{L}$. We provide obstructions to $Γ(X, \mathcal{L})$ being finite index in the mapping class group. We also show that for any $k \ge 0$, the image of monodromy is finite index in appropriate subgroups of the quotient of the mapping class group by the $k$th term of the Johnson filtration assuming that $\mathcal{L}$ is sufficiently ample. This enables us to construct several subgroups of the mapping class group with unusual properties, in some cases providing the first examples of subgroups with those properties.

math.GT↗
arXiv2022

Sums of Two Squares Visualized

We provide a geometric interpretation of Brillhart's celebrated algorithm for expressing a prime $p\equiv 1\pmod 4$ as the sum of two squares.

math.NT↗
arXiv2021

Choosing points on cubic plane curves: rigidity and flexibility

Every smooth cubic plane curve has 9 flex points and 27 sextatic points. We study the following question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose $n$ distinct points on every smooth cubic plane curve, for each given positive integer $n$? We give an affirmative answer to the question when $n=9$ and 18 (the smallest open cases), and a negative answer for infinitely many $n$'s.

math.AG↗
arXiv2012

Grey-box GUI Testing: Efficient Generation of Event Sequences

Graphical user interfaces (GUIs), due to their event driven nature, present a potentially unbounded space of all possible ways to interact with software. During testing it becomes necessary to effectively sample this space. In this paper we develop algorithms that sample the GUI's input space by only generating sequences that (1) are allowed by the GUI's structure, and (2) chain together only those events that have data dependencies between their event handlers. We create a new abstraction, called an event-dependency graph (EDG) of the GUI, that captures data dependencies between event handler code. We develop a mapping between EDGs and an existing black-box user-level model of the GUI's workflow, called an event-flow graph (EFG). We have implemented automated EDG construction in a tool that analyzes the bytecode of each event handler. We evaluate our "grey-box" approach using four open-source applications and compare it with the current state-of-the-art EFG approach. Our results show that using the EDG reduces the number of test cases while still achieving at least the same coverage. Furthermore, we were able to detect 2 new bugs in the subject applications.

cs.SE↗