Assistant Professor at Iowa State University
Konstantin Slutsky
Research Interests
I study descriptive set theory, ergodic theory, topological dynamics, and motion planning.
Publications
List of publications grouped by topics.
Borel Dynamics and Ergodic Theory
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Equivariant Borel liftings in complex analysis and PDE arXiv:2507.12058
We establish Borel equivariant analogues of several classical theorems from complex analysis and PDE. The starting point is an equivariant Weierstrass theorem for entire functions: there exists a Borel mapping which assigns to each non-periodic positive divisor \(d\) an entire function \(f_{d}\) with divisor of zeros \(\mathrm{div}(f_{d})=d\) and which commutes with translation, \(f_{d-2}(z)=f_{d}(z+w)\). We also examine the existence of equivariant Borel right inverses for the distributional Laplacian, the heat operator, and the \(\bar{\partial}\)-operator on the space of smooth functions. We demonstrate that Borel equivariant inverses for these maps exist on the free part of the range (for the heat operator, this holds up to the removal of a null set with respect to any invariant probability measure). In general, the freeness assumptions cannot be omitted and Borelness cannot be strengthened to continuity. Our positive results follow from a theorem establishing sufficient conditions for the existence of equivariant Borel liftings. Two key ingredients are Runge-type approximation theorems and the existence of Borel toasts, which are Borel analogues of Rokhlin towers from ergodic theory.
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L1 full groups of flows to appear in Mem. Eur. Math. Soc.
We introduce the concept of an \(\mathrm{L}^{1}\) full group associated with a measure-preserving action of a Polish normed group on a standard probability space. Such groups are shown to carry a natural separable complete metric, and are thus Polish. Our construction generalizes \(\mathrm{L}^{1}\) full groups of actions of discrete groups, which have been studied recently by the first author.
We show that under minor assumptions on the actions, topological derived subgroups of \(\mathrm{L}^{1}\) full groups are topologically simple and—when the acting group is locally compact and amenable—are whirly amenable and generically two-generated. \(\mathrm{L}^{1}\) full groups of actions of compactly generated locally compact Polish groups are shown to remember the L1 orbit equivalence class of the action.
For measure-preserving actions of the real line (also known as measure-preserving flows), the topological derived subgroup of an \(\mathrm{L}^{1}\) full groups is shown to coincide with the kernel of the index map, which implies that \(\mathrm{L}^{1}\) full groups of free measure-preserving flows are topologically finitely generated if and only if the flow admits finitely many ergodic components. The latter is in a striking contrast to the case of \( \mathbb{Z} \)-actions, where the number of topological generators is controlled by the entropy of the action.
We also study the coarse geometry of the \(\mathrm{L}^{1}\) full groups. The \(\mathrm{L}^{1}\) norm on the derived subgroup of the \(\mathrm{L}^{1}\) full group of an aperiodic action of a locally compact amenable group is proved to be maximal in the sense of Rosendal. For measure-preserving flows, this holds for the \(\mathrm{L}^{1}\) norm on all of the \(\mathrm{L}^{1}\) full group.
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Katok's special representation theorem for multidimensional Borel flows Ergod. Th. & Dynam. Sys. (2023), 1-18.
Katok's special representation theorem states that any free ergodic measure-preserving \(\mathbb{R}^{d}\)-flow can be realized as a special flow over a \(\mathbb{Z}^{d}\)-action. It provides a multidimensional generalization of the "flow under a function" construction. We prove the analog of Katok's theorem in the framework of Borel dynamics and show that, likewise, all free Borel \(\mathbb{R}^{d}\)-flows emerge from \(\mathbb{Z}^{d}\)-actions through the special flow construction using bi-Lipschitz cocycles.
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Free Borel \( \mathbb{R}^d \)-flows are smoothly equivalent if there is a Borel bijection between the phase spaces that maps orbits onto orbits and is a \(C^\infty\)-smooth orientation preserving diffeomorphism between orbits. We show that all free non-tame Borel \(\mathbb{R}^d\)-flows are smoothly equivalent in every dimension \(d \ge 2\). This answers a question of B. Miller and C. Rosendal.
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This paper addresses the notion of time change equivalence for Borel \(\mathbb{R}^d\)-flows. We show that all free \(\mathbb{R}^d\)-flows are time change equivalent up to a compressible set. An appropriate version of this result for non-free flows is also given.
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Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points. Non smooth flows are proved to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.
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Lebesgue orbit equivalence of multidimensional Borel flows Ergod. Th. & Dynam. Sys. 37 (2017), no. 6, 1966–1996.
The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue Orbit Equivalence, by which we understand an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non smooth Euclidean flows are shown to be Lebesgue Orbit Equivalence if and only if they admit the same number of invariant ergodic probability measures.
Topological Groups
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We construct Graev ultrametrics on free products of groups with two-sided invariant ultrametrics and HNN extensions of such groups. We also introduce a notion of a free product of general Polish groups and prove, in particular, that two Polish groups \( G \) and \( H \) can be embedded into a Polish group \( T \) in such a way that the subgroup of \( T \) generated by \( G \) and \( H \) is isomorphic to the free product \( G*H \).
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Graev metrics on free products and HNN extensions Trans. Amer. Math. Soc. 366 (2014), no. 12, 6353–6395.
We give a construction of two-sided invariant metrics on free products (possibly with amalgamation) of groups with two-sided invariant metrics and, under certain conditions, on HNN extensions of such groups. Our approach is similar to the Graev's construction of metrics on free groups over pointed metric spaces.
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Automatic continuity for homomorphisms into free products J. Symbolic Logic 78 (2013), no. 4, 1288–1306.
A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.
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Non-genericity phenomenon in some ordered Fraïssé classes J. Symbolic Logic 77 (2012), no. 3, 987–1010.
We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationals and in the group of automorphisms of the randomly ordered rational Urysohn space.
Autonomous Systems
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Minimum-Violation Planning for Autonomous Systems: Theoretical and Practical Considerations 2021 American Control Conference (ACC), 2021, pp. 4866-4872.
This paper considers the problem of computing an optimal trajectory for an autonomous system that is subject to a set of potentially conflicting rules. First, we introduce the concept of prioritized safety specifications, where each rule is expressed as a temporal logic formula with its associated weight and priority. The optimality is defined based on the violation of such prioritized safety specifications. We then introduce a class of temporal logic formulas called si-FLTLGX and develop an efficient, incremental sampling-based approach to solve this minimum-violation planning problem with guarantees on asymptotic optimality. We illustrate the application of the proposed approach in autonomous vehicles, showing that si-FLTLGX formulas are sufficiently expressive to describe many traffic rules. Finally, we discuss practical considerations and present simulation results for a vehicle overtaking scenario.
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Hierarchical Multiobjective Shortest Path Problems In: LaValle S.M., Lin M., Ojala T., Shell D., Yu J. (eds) Algorithmic Foundations of Robotics XIV. WAFR 2020. Springer Proceedings in Advanced Robotics, vol 17.
We consider the shortest path problem on graphs with weights taking values in Cartesian products of cost monoids. Such cost structures appear in multiobjective planning including, for instance, the minimum-violation planning framework. It is known that these products often do not satisfy the conditions of a cost monoid. Classical dynamic programming graph search algorithms may therefore fail to find an optimal solution. We isolate the concept of a regular cost monoid and propose an iterative search algorithm that finds an optimal path in graphs weighted by products of such costs. Our algorithm allows this class of multiobjective planning problems to be solved in polynomial time.
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Liability, Ethics, and Culture-Aware Behavior Specification using Rulebooks 2019 International Conference on Robotics and Automation (ICRA), 8536–8542.
The behavior of self-driving cars must be compatible with an enormous set of conflicting and ambiguous objectives, from law, from ethics, from the local culture, and so on. This paper describes a new way to conveniently define the desired behavior for autonomous agents, which we use on the self-driving cars developed at nuTonomy. We define a "rulebook" as a pre-ordered set of "rules", each akin to a violation metric on the possible outcomes ("realizations"). The rules are partially ordered by priority. The semantics of a rulebook imposes a pre-order on the set of realizations. We study the compositional properties of the rulebooks, and we derive which operations we can allow on the rulebooks to preserve previously-introduced constraints. While we demonstrate the application of these techniques in the self-driving domain, the methods are domain-independent.
Miscellaneous
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Riemann Rearrangement Theorem for some types of convergence J. Math. Anal. Appl. 373 (2011), no. 2, 605–613.
We reexamine the Riemann Rearrangement Theorem for different types of convergence and classify possible sum ranges for statistically convergent series and for series that converge along the 2n-filter.
Unpublished Notes
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Given a set of positive reals we provide a necessary and sufficient condition for a free Borel flow to admit a cross section with all distances between adjacent points coming from this set.
Lecture Notes
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This lecture notes form an eight week introductory course to topological groups of Cantor systems. Some of the topics covered by the notes include: Verhsik maps and Bratteli diagrams, flip conjugacy, commutator subgroup, amenability of full groups.
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These are the notes from a semester long course taught at UIC in the Fall of 2015. They cover compressible and hyperfinite relations, including Nadkarni's characterization of compressibility, classfification of hyperfinite relations by Dougherty, Jackson, and Kechris, and some examples of groups that give rise to hyperfinite relations only, after Jackson, Kechris, and Louveau.