Matthew Crawford | Searcharxiv

arXiv2025

Integrals of stable envelopes for cotangent bundles to Grassmannians

We consider cohomological stable envelopes for a natural torus action $\mathsf{T}$ on $X=T^*Gr(k,n)$, introduced by Maulik-Okounkov. We define the $\mathbb{C}^*_\hbar$-equivariant integral of the stable envelope using equivariant localization over the subtorus $\mathbb{C}^*_\hbar\subset\mathsf{T}$, and compute the integral as a non-equivariant limit of the localization over the full torus, $\mathsf{T}$. The integral of such a class is an integer times a power of $\hbar$, and the main result of this paper is a combinatorial formula for these integers. In 3d mirror symmetry, these non-equivariant limits are expected to reflect some curve counting phenomena on the 3d mirror dual, $X^\vee$. When $k=1$, we obtain the binomial coefficients, and we study some of the combinatorics of the integers for higher $k$, which haven't appeared in the literature before. We give some conjectures and interpretations on extending this phenomena to type A quiver and bow varieties.

math.AG↗

arXiv2021

Statically Detecting Adversarial Malware through Randomised Chaining

With the rapid growth of malware attacks, more antivirus developers consider deploying machine learning technologies into their productions. Researchers and developers published various machine learning-based detectors with high precision on malware detection in recent years. Although numerous machine learning-based malware detectors are available, they face various machine learning-targeted attacks, including evasion and adversarial attacks. This project explores how and why adversarial examples evade malware detectors, then proposes a randomised chaining method to defend against adversarial malware statically. This research is crucial for working towards combating the pertinent malware cybercrime.

cs.CR↗