About

Yuliy Baryshnikov is the Fredric G. and Elizabeth H. Nearing Professor in the Department of Electrical and Computer Engineering at the University of Illinois, with additional professorships in Mathematics, the Coordinated Science Lab, and the Beckman Institute for Advanced Science and Technology. His research interests encompass applied topology, stochastic models, dynamical systems, and analytic combinatorics. Baryshnikov's work integrates mathematical theory with practical applications, focusing on areas such as asymptotics, combinatorics, Gaussian distributions, sensor networks, inverse problems, generating functions, asymptotic normality, and stochastics. He has contributed extensively to the scholarly community, with a significant number of peer-reviewed articles and conference contributions that explore complex mathematical and computational phenomena. His office is located in the Department of Mathematics at the University of Illinois, Urbana, where he continues to advance research at the intersection of mathematics and engineering.

Research topics

• Computer Science
• Artificial Intelligence
• Mathematics
• Speech recognition
• Geometry
• Psychology
• Pure mathematics
• Combinatorics
• Neuroscience
• Physics
• Mathematical analysis

Selected publications

• Multi-floor generalization of TASEP

  arXiv (Cornell University) · 2026-03-13

  preprintOpen access1st authorCorresponding

  We consider an interacting particle system, which generalizes the classical totally asymmetric simple exclusion process (TASEP), in that each site can contain up to a fixed finite number of particles, and the particle movement is governed by a {\em back-pressure} (BP) algorithm (also often called {\em MaxWeight}). There are $N$ sites (with $N$ finite or infinite), each may contain at most $c$ particles, $1 \le c < \infty$. New particles enter the system at the left-most site $1$ as a Poisson process of rate $α\le 1$, unless site $1$ has $c$ particles. Particles (if any) are removed from the right-most site $N$ as a Poisson process of rate $β\le 1$. The left-to-right movement of particles between neighboring sites is governed by the BP rule: one particle moves from site $n$ to $n+1$ at epochs of a rate $1$ Poisson process, as long as the former site has strictly more particles than the latter. When $c=1$, this is the standard TASEP. Our main results address the asymptotics of the stationary distribution of a finite system, and especially the limit of the flux (current) as $N\to\infty$. In particular, we prove that interesting non-trivial phase transitions take place in a system with $c>1$. For example, if $c>1$ and $1/2 \le β\le 1$, the maximum limiting flux $1/4$ is achieved as long as $α\ge α_c^*$, where $α_c^* < 1/2$ is some non-trivial threshold. (For the standard TASEP the threshold is $1/2$.) We also put forward a general conjecture about the stationary distribution asymptotics under an arbitrary parameter setting. We illustrate our formal results and the conjecture by simulations, and identify interesting directions for further research.

  PublisherDOI

• Multi-floor generalization of TASEP

  arXiv (Cornell University) · 2026-03-13

  articleOpen access1st authorCorresponding

  We consider an interacting particle system, which generalizes the classical totally asymmetric simple exclusion process (TASEP), in that each site can contain up to a fixed finite number of particles, and the particle movement is governed by a {\em back-pressure} (BP) algorithm (also often called {\em MaxWeight}). There are $N$ sites (with $N$ finite or infinite), each may contain at most $c$ particles, $1 \le c < \infty$. New particles enter the system at the left-most site $1$ as a Poisson process of rate $α\le 1$, unless site $1$ has $c$ particles. Particles (if any) are removed from the right-most site $N$ as a Poisson process of rate $β\le 1$. The left-to-right movement of particles between neighboring sites is governed by the BP rule: one particle moves from site $n$ to $n+1$ at epochs of a rate $1$ Poisson process, as long as the former site has strictly more particles than the latter. When $c=1$, this is the standard TASEP. Our main results address the asymptotics of the stationary distribution of a finite system, and especially the limit of the flux (current) as $N\to\infty$. In particular, we prove that interesting non-trivial phase transitions take place in a system with $c>1$. For example, if $c>1$ and $1/2 \le β\le 1$, the maximum limiting flux $1/4$ is achieved as long as $α\ge α_c^*$, where $α_c^* < 1/2$ is some non-trivial threshold. (For the standard TASEP the threshold is $1/2$.) We also put forward a general conjecture about the stationary distribution asymptotics under an arbitrary parameter setting. We illustrate our formal results and the conjecture by simulations, and identify interesting directions for further research.

  PublisherOA PDF

• Minimal Unimodal Decomposition is NP-Hard on Graphs

  ArXiv.org · 2025-10-07

  preprintOpen accessSenior author

  A function on a topological space is called unimodal if all of its super-level sets are contractible. A minimal unimodal decomposition of a function $f$ is the smallest number of unimodal functions that sum up to $f$. The problem of decomposing a given density function into its minimal unimodal components is fundamental in topological statistics. We show that finding a minimal unimodal decomposition of an edge-linear function on a graph is NP-hard. Given any $k \geq 2$, we establish the NP-hardness of finding a unimodal decomposition consisting of $k$ unimodal functions. We also extend the NP-hardness result to related variants of the problem, including restriction to planar graphs, inapproximability results, and generalizations to higher dimensions.

  PublisherOA PDFDOI

• Reparametrization of 3D CSC Dubins Paths Enabling 2D Search

  Springer proceedings in advanced robotics · 2025-10-30

  book-chapterOpen access
  PublisherOA PDFDOI

• Universality for Taylor coefficients of rational functions under perturbations

  Proceedings of the National Academy of Sciences · 2025-12-26

  articleOpen access1st authorCorresponding

  We introduce computational methods for analytic combinatorics in several variables. Our findings pertain to rational generating functions whose dominant singularity satisfies either of two conditions, one of which frequently holds for recursions arising from cluster algebras. We show that the coefficients are determined asymptotically by the leading homogeneous term of the denominator of the function near the dominant singularity. Applying this to various statistical physical models, we show their asymptotic behavior to be given by elliptic and hyperelliptic integrals, computation of which is already implemented in computer algebra packages.

  PublisherDOI

• Brownian motions, persistent homology and chirality

  Journal of Applied and Computational Topology · 2025-11-25

  articleOpen access1st authorCorresponding

  Abstract Interactions of the maxima and minima of the univariate functions appear in combinatorics as Dyck paths, and in topological data analysis as persistent homology. We study these descriptors for Brownian motions with drift, deriving the intensity measure and correlation functions for the persistence diagram point process $$\textbf{PH}_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>PH</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , and quantifying the intrinsic asymmetries in the coupling of maxima and minima.

  PublisherOA PDFDOI

• Hemodynamic cortical ripples through cyclicity analysis

  Network Neuroscience · 2024-01-01 · 2 citations

  articleOpen accessSenior author

  A fine-grained understanding of dynamics in cortical networks is crucial to unpacking brain function. Resting-state functional magnetic resonance imaging (fMRI) gives rise to time series recordings of the activity of different brain regions, which are aperiodic and lack a base frequency. Cyclicity analysis, a novel technique robust under time reparametrizations, is effective in recovering the temporal ordering of such time series, collectively considered components of a multidimensional trajectory. Here, we extend this analytical method for characterizing the dynamic interaction between distant brain regions and apply it to the data from the Human Connectome Project. Our analysis detected cortical traveling waves of activity propagating along a spatial axis, resembling cortical hierarchical organization with consistent lead-lag relationships between specific brain regions in resting-state scans. In fMRI scans involving tasks, we observed short bursts of task-modulated strong temporal ordering that dominate overall lead-lag relationships between pairs of regions in the brain that align temporally with stimuli from the tasks. Our results suggest a possible role played by waves of excitation sweeping through brain regions that underlie emergent cognitive functions.

  PublisherOA PDFDOI

• Asymptotics of multivariate sequences in the presence of a lacuna

  Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions · 2024-01-23 · 3 citations

  articleOpen access1st authorCorresponding

  We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value in even dimensions d \geq 4 . This result depends on computations in the homology of the algebraic variety where the generating function has a pole. These computations are similar to, and inspired by, a thread of research in applications of complex algebraic geometry to hyperbolic PDEs, going back to Leray, Petrowski, Atiyah, Bott and Gårding. As a consequence, we give a topological explanation for certain asymptotic phenomena appearing in the combinatorics and number theory literature. Furthermore, we show how to combine topological methods with symbolic algebraic computation to determine explicitly the dominant asymptotics for such multivariate generating functions, giving a significant new tool to attack the so-called connection problem for asymptotics of P-recursive sequences. This in turn enables the rigorous determination of integer coefficients in the Morse–Smale complex, which are difficult to determine using direct geometric methods.

  PublisherOA PDFDOI

• Coefficient Asymptotics of Algebraic Multivariable Generating Functions

  La Matematica · 2024-02-26

  article1st author
  PublisherDOI

• Ising Disks: Topology Preserving Glauber Dynamics

  arXiv (Cornell University) · 2024-10-30

  preprintOpen access1st authorCorresponding

  We introduce a dynamic model where the state space is the set of contractible cubical sets in the Euclidian space. The permissible state transitions, that is addition and removal of a cube to/from the set, are closest to Eden model with topological constraints, and, we show, are locally decidable. We prove that in the planar special case the state space is connected. We then define a continuous time Markov chain with a fugacity (tendency to grow) parameter. Using the correspondence between our model on the plane and the self-avoiding polygons, we prove that the Markov chain is irreducible (due to state connectivity), and is also ergodic if the fugacity is smaller than a threshold.

  PublisherOA PDFDOI

Recent grants

• Collaborative Research: Topological Invariants for Enhanced Data Analysis

  NSF · $190k · 2016–2020

Frequent coauthors

• Robin Pemantle

  19 shared

• Robert Ghrist

  17 shared

• E. G. Coffman

  Columbia University

  14 shared

• Bennett Eisenberg

  Lehigh University

  13 shared

• Stephen Melczer

  École Normale Supérieure de Lyon

  11 shared

• Wolfgang Stadje

  Osnabrück University

  10 shared

• J. E. Yukich

  Lehigh University

  10 shared

• Ed Coffman

  7 shared

Awards & honors

• Arthur B. Coble Memorial Lectures