arXiv subjects

Michael Ruofan Zeng

Publications and source records attributed to Michael Ruofan Zeng.

8 recordsLinked to original sources

arXiv2026

An order automorphism of a Dlab group not induced by conjugation

Let $G=D_{\langle2\rangle}([0,1])$ be equipped with either of its two Dlab orders. There exists an order automorphism $α$ of $G$ such that, for every rank-one subgroup $H\leq\mathbb R_{>0}^{\times}$, every one of the six corresponding Dlab groups $A$ listed below, equipped with any of its linear orders, every order-preserving embedding $e:G\hookrightarrow A$, and every $u\in A$, there exists $f\in G$ such that $e(α(f))\neq u^{-1}e(f)u$.

math.GR↗

arXiv2026

Total 3-closure for projective special linear groups

A finite group is totally $3$-closed if every faithful permutation representation of it is $3$-closed. We study this property for the finite simple projective special linear groups. We prove that $\PSL_2(q)$ is totally $3$-closed if and only if $q\geq 7$ is prime, and that $\PSL_3(q)$ is totally $3$-closed if and only if either $q=3$, or $q$ is prime and $q\equiv 2\pmod 3$. We further prove that $\PSL_4(q)$ is never totally $3$-closed and that $\PSL_n(q)$ is not totally $3$-closed whenever $n\geq 5$ and $q>2$. Within the family $\PSL_n(q)$, only the groups $\PSL_n(2)$ with $n\geq 5$ remain unresolved. In particular, this answers Problem~20.2 of the Kourovka Notebook affirmatively.

math.GR↗

arXiv2026

Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements

For any distinct primes $p$ and $q$, we prove that there is a finite group which does not embed into any finite group invariably generated by an element of order $p$ and an element of order $q$. This gives a negative answer to Problem 21.142 of the Kourovka Notebook \cite{kourovka21}. In fact, for every fixed pair $p,q$, the group $A_n$ cannot embed into such a group once $n$ is sufficiently large in terms of $p$ and $q$.

math.GR↗

arXiv2026

Albilich: Steerable Proof-State Orchestration for LLM-Based Mathematical Research with CAS Integration

Large language models can contribute useful ideas to mathematical research, yet long-horizon proof attempts remain difficult to coordinate, evaluate, and reproduce. We present Albilich, an open-source agentic harness for autoresearch in mathematics that combines long-horizon reasoning, computer algebra systems (CAS), literature retrieval, and persistent SQLite-based context management. We evaluate Albilich on the RealMath benchmark (Zhang et al. 2025) and on open problems in group theory from the Kourovka Notebook (Khukhro and Mazurov 2026). It solved 10/10 problems on RealMath with CAS and 9/10 with no CAS. On the Kourovka problems, Albilich produced a counterexample to Problem 21.142 and a proof of a strengthening of Problem20.2. Anablation on Problem 17.91 demonstrates 32.0% token reduction when CAS is enabled. An ablation on Problem 21.142 demonstrates higher verifier-rejection rate and failure to synthesize proof routes in the absence of the advisor agent. These results support Albilich as a human-steerable, CAS-boosted environment for scalable AI-assisted mathematical research.

cs.AI↗

arXiv2026

FactorLibrary: From Polynomials to Circuits via Recursive Subgoals

Finding minimal arithmetic circuits for polynomials over finite fields is a combinatorially hard problem central to algebraic complexity theory. We formulate it as a reinforcement learning problem in two directions, bottom-up and top-down. To address the challenge of a fast-growing combinatorial search space, we introduce FactorLibrary, which stores factorizable subexpressions that serve as reusable subgoals across training episodes. We trained a bottom-up agent with Gumbel-PPO-MCTS and two top-down agents with PPO+MCTS and SAC. The PPO+MCTS top-down agent exhibited the most stable performance, finding certified optimal circuits up to complexity $8$ with a success rate of $91.8\%$.

cs.LG↗

arXiv2026

Oriented Cohomology Rings of Some Moduli Spaces via Blowups

Oriented cohomology theories provide a general framework to perform intersection-theory-type calculus. The Chow ring, algebraic $K$-theory, and Levine--Morel's algebraic cobordism are all instances of such theories satisfying $\mathbb A^1$-invariance. Topological Hochschild homology, topological cyclic homology, and Hodge cohomology are important examples of theories without $\mathbb A^1$-invariance. In this paper, we prove an additive blowup formula for oriented cohomology theories in the non-$\mathbb A^1$-invariant category of motivic spectra, developed by Annala, Hoyois, and Iwasa. Then, we specialize to $\mathbb A^1$-invariant theories and give presentations of oriented cohomology rings of the blowup of a smooth scheme along a smooth center. We compute explicit examples of such presentations for the cases of del Pezzo surfaces, the blowup of $\mathbb P^3$ along the twisted cubic, and the blowup of $\mathbb P^5$ along the Veronese surface, which can be identified with the moduli space of complete conics. We demonstrate that one can recover solutions to classical enumerative geometry problems, such as Steiner's $3264$ conics, using arbitrary oriented cohomology theories. Finally, we give a presentation of oriented cohomology rings of $\overline M_{0,n}$, which generalizes Keel's presentation of the Chow ring.

math.AG↗

arXiv2026

CircuitBuilder: From Polynomials to Circuits via Reinforcement Learning

Motivated by auto-proof generation and Valiant's VP vs. VNP conjecture, we study the problem of discovering efficient arithmetic circuits to compute polynomials, using addition and multiplication gates. We formulate this problem as a single-player game, where an RL agent attempts to build the circuit within a fixed number of operations. We implement an AlphaZero-style training loop and compare two approaches: Proximal Policy Optimization with Monte Carlo Tree Search (PPO+MCTS) and Soft Actor-Critic (SAC). SAC achieves the highest success rates on two-variable targets, while PPO+MCTS scales to three variables and demonstrates steady improvement on harder instances. These results suggest that polynomial circuit synthesis is a compact, verifiable setting for studying self-improving search policies.

cs.LG↗

arXiv2025

The Grothendieck Group of the Variety of Spanning Line Configurations

We study the Grothendieck group of the variety $X_{n,k}$ of spanning line configurations introduced by Pawlowski--Rhoades [arXiv:1711.08301] as a geometric model for the generalized coinvariant algebra $R_{n,k}$. Our first result is a localization statement in $K$-theory for the complements of cell closures in smooth cellular varieties. Combining with the Fulton--Lascoux degeneracy loci formula, we prove that $K_0(X_{n,k})$ is canonically isomorphic to $R_{n,k}$, extending classical isomorphisms for the flag variety. We next identify the classes of the Pawlowski--Rhoades varieties with Grothendieck polynomials associated to words $w \in [k]^n$. Motivated by this identification, we develop models of classical and bumpless pipe dreams for words. We show that Schubert and Grothendieck polynomials of words are monomial-weight generating functions for these pipe dreams, extending the classical story from permutations to words and ordered set partitions.

math.CO↗