arXiv subjects

Ayush Khaitan

Publications and source records attributed to Ayush Khaitan.

8 recordsLinked to original sources

arXiv2026

Computing renormalized curvature integrals on Poincaré-Einstein manifolds

We describe a general procedure for computing renormalized curvature integrals on Poincaré-Einstein manifolds. In particular, we explain the connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang for the renormalized volume, and explicitly identify a scalar conformal invariant in the latter formula. Our approach constructs scalar conformal invariants of weight $-n$ on $n$-manifolds, $n \geq 8$, that are natural divergences; these imply that the scalar invariant in the Chang-Qing-Yang formula is not unique in dimension $n \geq 8$. Our procedure also produces explicit conformally invariant Gauss--Bonnet-type formulas for compact Einstein manifolds.

math.DG↗

arXiv2026

Local and global conformal invariants of submanifolds

We develop methods for constructing and computing conformal invariants of submanifolds, with a particular emphasis on conformal submanifold scalars and conformally invariant integrals of natural submanifold scalars. These methods include a direct construction of the extrinsic ambient space, a construction of global invariants of conformally compact minimal submanifolds of conformally compact Einstein manifolds via renormalized extrinsic curvature integrals, and the introduction of a large class of conformal submanifold scalars that are easily computed at minimal submanifolds of Einstein manifolds. As an application, we derive an explicit Gauss--Bonnet--Chern-type formula relating the renormalized area of a conformally compact $k$-dimensional minimal submanifold of a conformally compact Einstein manifold to its Euler characteristic and the integral of a conformal submanifold scalar of weight $-k$. As another application, we prove a rigidity result for conformally compact minimal submanifolds of conformally compact hyperbolic manifolds.

math.DG↗

arXiv2026

Weighted GJMS operators on smooth metric measure spaces

We construct weighted GJMS operators on smooth metric measure spaces, and prove that they are formally self-adjoint. We also provide factorization formulas for them in the case of quasi-Einstein spaces and under Gover--Leitner conditions.

math.DG↗

arXiv2025

The weighted ambient metric for manifolds with density

We prove the existence and uniqueness of a weighted analogue of the Fefferman-Graham ambient metric for manifolds with density. We then show that this ambient metric forms the natural geometric framework for the singular Ricci flow: given a singular gradient Ricci flow spacetime in the Kleiner-Lott sense, we construct a unique global ambient half-space from it. We also prove the converse, that every global ambient space contains a singular gradient Ricci flow spacetime, thereby completing the correspondence. Our main application is the construction of infinite families of fully non-linear analogues of Perelman's $\mathcal{F}$ and $\mathcal{W}$ functionals. We extend Perelman's monotonicity result to these two families of functionals under several conditions, including for shrinking solitons and Einstein manifolds. We do so by constructing a "Ricci flow vector field" in the ambient space, which may be of independent research interest. We also prove that the weighted GJMS operators associated with the weighted ambient metric are formally self-adjoint, and that the associated weighted renormalized volume coefficients are variational.

math.DG↗

arXiv2025

O-Forge: An LLM + Computer Algebra Framework for Asymptotic Analysis

Large language models have recently demonstrated advanced capabilities in solving IMO and Putnam problems; yet their role in research mathematics has remained fairly limited. The key difficulty is verification: suggested proofs may look plausible, but cannot be trusted without rigorous checking. We present a framework, called LLM+CAS, and an associated tool, O-Forge, that couples frontier LLMs with a computer algebra systems (CAS) in an In-Context Symbolic Feedback loop to produce proofs that are both creative and symbolically verified. Our focus is on asymptotic inequalities, a topic that often involves difficult proofs and appropriate decomposition of the domain into the "right" subdomains. Many mathematicians, including Terry Tao, have suggested that using AI tools to find the right decompositions can be very useful for research-level asymptotic analysis. In this paper, we show that our framework LLM+CAS turns out to be remarkably effective at proposing such decompositions via a combination of a frontier LLM and a CAS. More precisely, we use an LLM to suggest domain decomposition, and a CAS (such as Mathematica) that provides a verification of each piece axiomatically. Using this loop, we answer a question posed by Terence Tao: whether LLMs coupled with a verifier can be used to help prove intricate asymptotic inequalities. More broadly, we show how AI can move beyond contest math towards research-level tools for professional mathematicians.

cs.AI↗

arXiv2025

Elementary symmetric polynomials under the fixed point measure

We identify a surprising inequality satisfied by elementary symmetric polynomials under the action of the fixed point measure of a random permutation. Concretely, for any collection of $n$ non-negative real numbers $a_1, \dots, a_n \in \mathbb{R}_{\geq 0}$, we prove that \[ \frac{1}{n!} \sum_{π\in S_n} \left[\prod_{\{i:i=π(i)\}} a_i\right] \ge \frac{1}{\binom{n}{2}} \sum_{S \in\binom{[n]}{2}} \left[ \left(\prod_{\{i \in S\}} a_i \right)^{1/2}\right], \] and this bound is sharp. To prove this elementary inequality, we construct a collection of differential operators to set up a monotone flow that then allows us to establish the inequality.

math.CO↗

arXiv2022

The Weighted Ambient Metric

We prove the existence and uniqueness of weighted ambient metrics and weighted Poincaré metrics for smooth metric measure spaces.

math.DG↗

arXiv2022

Weighted renormalized volume coefficients

We define weighted renormalized volume coefficients and prove that they are variational. We also prove that they can be written as polynomials of weighted extended obstruction tensors, the weighted Schouten tensor, and the weighted Schouten scalar.

math.DG↗