Searcharxiv⌕ Search

arXiv subjects

Yevgeny Liokumovich

Publications and source records attributed to Yevgeny Liokumovich.

At least 19 recordsLinked to original sources
arXiv2026

Sky sphere representation in language models

We analyze whether language models of size ~100B have a representation of the night sky map that is decodable from their residual stream. We find that most of the considered open-source models do have such a representation, and it often even surfaces to the top principal components on prompts that ask questions like ``what is close to this object in the night sky''. In all but one model this representation showed significant scores in LOO testing, containing up to 65-85% of variance ($R^2$-score) and having median angular error down to $12^\circ-21^\circ$. We verify that our representation is not a simple leak from a correlated flat representation. To our knowledge, this representation is the first example of a curved high-dimensional irreducible feature manifold. Codes used in the paper are published at https://github.com/l3erdnik/Decodable-sky

cs.LG↗
arXiv2025

On the existence of minimal Heegaard surfaces

Let $H$ be a strongly irreducible Heegaard surface in a closed oriented Riemannian $3$-manifold. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a non-orientable minimal surface with a vertical handle attached. This confirms a long-standing conjecture of J. Pitts and J.H. Rubinstein.

math.DG↗
arXiv2025

The Smale conjecture and min-max theory

We give a new proof of the Smale conjecture for $\mathbb{RP}^3$ and all lens spaces using minimal surfaces and min-max theory. For $\mathbb{RP}^3$, the conjecture was first proved in 2019 by Bamler-Kleiner using Ricci flow.

math.DG↗
arXiv2025

Parametric inequalities and Weyl law for the volume spectrum

We show that the Weyl law for the volume spectrum in a compact Riemannian manifold conjectured by Gromov can be derived from parametric generalizations of two famous inequalities: isoperimetric inequality and coarea inequality. We prove two such generalizations in low dimensions and obtain the Weyl law for 1-cycles in 3-manifolds. We also give a new proof of the Almgren isomorphism theorem.

math.DG↗
arXiv2024

Quantifying stability of non-power-seeking in artificial agents

We investigate the question: if an AI agent is known to be safe in one setting, is it also safe in a new setting similar to the first? This is a core question of AI alignment--we train and test models in a certain environment, but deploy them in another, and we need to guarantee that models that seem safe in testing remain so in deployment. Our notion of safety is based on power-seeking--an agent which seeks power is not safe. In particular, we focus on a crucial type of power-seeking: resisting shutdown. We model agents as policies for Markov decision processes, and show (in two cases of interest) that not resisting shutdown is "stable": if an MDP has certain policies which don't avoid shutdown, the corresponding policies for a similar MDP also don't avoid shutdown. We also show that there are natural cases where safety is _not_ stable--arbitrarily small perturbations may result in policies which never shut down. In our first case of interest--near-optimal policies--we use a bisimulation metric on MDPs to prove that small perturbations won't make the agent take longer to shut down. Our second case of interest is policies for MDPs satisfying certain constraints which hold for various models (including language models). Here, we demonstrate a quantitative bound on how fast the probability of not shutting down can increase: by defining a metric on MDPs; proving that the probability of not shutting down, as a function on MDPs, is lower semicontinuous; and bounding how quickly this function decreases.

cs.AI↗
arXiv2023

Generic density of geodesic nets

We prove that for a Baire-generic Riemannian metric on a closed smooth manifold, the union of the images of all stationary geodesic nets forms a dense set.

math.DG↗
arXiv2022

Geodesic nets on non-compact Riemannian manifolds

A geodesic flower is a finite collection of geodesic loops based at the same point $p$ that satisfy the following balancing condition: The sum of all unit tangent vectors to all geodesic arcs meeting at $p$ is equal to the zero vector. In particular, a geodesic flower is a stationary geodesic net. We prove that in every complete non-compact manifold with locally convex ends there exists a non-trivial geodesic flower.

math.DG↗
arXiv2022

Singular behavior and generic regularity of min-max minimal hypersurfaces

We show that for a generic $8$-dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show that for a dense set of 8-dimensional Riemannian metrics there exists a minimal hypersurface with at most one singular point. This extends previous work on generic regularity that only dealt with area-minimizing hypersurfaces. These results are a consequence of a more general estimate for a one-parameter min-max minimal hypersurface $Σ\subset (M,g)$ (valid in any dimension): $$\mathcal H^{0} (\mathcal{S}_{nm}(Σ)) +{\rm Index}(Σ) \leq 1$$ where $\mathcal{S}_{nm}(Σ)$ denotes the set of singular points of $Σ$ with a unique tangent cone non-area minimizing on either side.

math.DG↗
arXiv2022

Classifying sufficiently connected PSC manifolds in $4$ and $5$ dimensions

We show that if $N$ is a closed manifold of dimension $n=4$ (resp. $n=5$) with $π_2(N) = 0$ (resp. $π_2(N)=π_3(N)=0$) that admits a metric of positive scalar curvature, then a finite cover $\hat N$ of $N$ is homotopy equivalent to $S^n$ or connected sums of $S^{n-1}\times S^1$. Our approach combines recent advances in the study of positive scalar curvature with a novel argument of Alpert--Balitskiy--Guth. Additionally, we prove a more general mapping version of this result. In particular, this implies that if $N$ is a closed manifold of dimensions $4$ or $5$, and $N$ admits a map of nonzero degree to a closed aspherical manifold, then $N$ does not admit any Riemannian metric with positive scalar curvature.

math.DG↗
arXiv2021

Filling metric spaces

We prove a new version of isoperimetric inequality: Given a positive real $m$, a Banach space $B$, a closed subset $Y$ of metric space $X$ and a continuous map $f:Y \rightarrow B$ with $f(Y)$ compact $$\inf_FHC_{m+1}(F(X))\leq c(m)HC_m(f(Y))^{\frac{m+1}{m}},$$ where $HC_m$ denotes the $m$-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps $F:X\longrightarrow B$ such that $F(y)=f(y)$ for all $y\in Y$, and $c(m)$ depends only on $m$. Moreover, one can find $F$ with a nearly minimal $HC_{m+1}$ such that its image lies in the $C(m)HC_m(f(Y))^{1\over m}$-neighbourhood of $f(Y)$ with the exception of a subset with zero $(m+1)$-dimensional Hausdorff measure. The paper also contains a very general coarea inequality for Hausdorff content and its modifications. As an application we demonstrate an inequality conjectured by Larry Guth that relates the $m$-dimensional Hausdorff content of a compact metric space with its $(m-1)$-dimensional Urysohn width. We show that this result implies new systolic inequalities that both strengthen the classical Gromov's systolic inequality for essential Riemannian manifolds and extend this inequality to a wider class of non-simply connected manifolds.

math.DG↗
arXiv2019

On the existence of unstable minimal Heegaard surfaces

We prove that for generic metrics on a 3-sphere, the minimal surface obtained from the min-max procedure of Simon-Smith has index 1. We prove an analogous result for minimal surfaces arising from strongly irreducible Heegaard sweepouts in 3-manifolds. We also confirm a conjecture of Pitts-Rubinstein that a strongly irreducible Heegaard splitting in a hyperbolic three-manifold can either be isotoped to a minimal surface of index at most 1 or else after a neck-pinch is isotopic to a one-sided minimal Heegaard surface.

math.DG↗
arXiv2019

Existence of minimal hypersurfaces in complete manifolds of finite volume

We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. The main tool is the following result of independent interest: if a region $U$ can be swept out by a family of hypersurfaces of volume at most $V$, then it can be swept out by a family of mutually disjoint hypersurfaces of volume at most $V + \varepsilon$.

math.DG↗
arXiv2019

On the existence of closed $C^{1,1}$ curves of constant curvature

We show that on any Riemannian surface for each $0<c<\infty$ there exists an immersed $C^{1,1}$ curve that is smooth and with curvature equal to $\pm c$ away from a point. We give examples showing that, in general, the regularity of the curve obtained by our procedure cannot be improved.

math.DG↗
arXiv2018

Weyl law for the volume spectrum

Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{ω_p(M)\}_{p\in\mathbb{N}}$ satisfies a Weyl law that was conjectured by Gromov.

math.DG↗
arXiv2017

Area of convex disks

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}πR^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound $μ_2(B(p,R)) \leq 2(\fracπ{2 R})^2$ on the first nonzero Neumann eigenvalue $μ_2$ of the Laplacian in terms only of the radius. This has also been conjectured for $R$ up to half the injectivity radius.

math.DG↗