Chunyin Siu (Alex) is a postdoctoral scholar in the Brain Dynamics Lab, led by Prof. Mannish Saggar, at the Stanford University School of Medicine. He specializes in topological data analysis and its application in analyzing neuroimaging data.

He got his PhD degree in Applied Mathematics at Cornell University under the supervision of Prof. Gennady Samorodnitsky. Before that, he got this MPhil. degree in Mathematics at the Chinese University of Hong Kong under the supervision of Prof. Ronald (Lokming) Lui.

He is a Croucher fellow (2024) and scholar (2019), and a Youde scholar (2018).

• email: siuc at stanford dot edu
• Google scholar
• CV

I love collaboration. If you are a grad student (or an ambitious undergrad) looking for challenging problems to solve in probability, topology, neuroscience, or psychiatry, feel free to reach out! Some of the problems I am thinking about are in the Open Problems section.

Scroll down for the Teaching, Mentorship, Open Problems, and Personal sections.

Research and Publications

Applied Topology

• The Topological Behavior of Preferential Attachment Graphs (accepted by SIAM Journal on Applied Algebra and Geometry)
  • C. Siu
  • (arxiv)
• Betti Numbers of Preferential Attachment Complexes
  • C. Siu, G. Samorodnitsky, C. Yu, and R. He
  • Advances in Applied Probability 2025
  • (arxiv) (video)(poster) (slides) (codes)
• Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration
  • C. Siu, G. Samorodnitsky, C. Yu, and A. Yao
  • Journal of Applied and Computational Topology 2024
  • (doi)(arxiv) (video) (source codes) (poster for ATMCS 10) (slides)

Computational Geometry

• Decomposition of Longitudinal Deformations via Beltrami Descriptors
  • H. Law, C. Siu, and R. Lui.
  • Journal of Scientific Computing 2021
• Image Segmentation with Partial Convexity Shape Prior Using Discrete Conformality Structures
  • C. Siu, H.L. Chan, and R. Lui
  • SIAM Journal on Imaging Sciences 2020

Miscellaneous

• Geometry and Laplacian on Discrete Magic Carpets
  • C. Siu, and R. Strichartz (alphabetical order)
  • Journal of Fractal Geometry 2023
• An Elementary Approach on Left-Orderability, Cables of Torus Knots and Dehn Surgery
  • J. Li, and C. Siu (alphabetical order)
  • arxiv

Talks and Presentations

• Mar 23, 2024: Mid-Atlantic Topology Conference (poster), Northeastern University
  • “Topology of Scale-Free Complexes – Homology and Homotopy”
  • poster
• Mar 8, 2024: Data Science and Applied Topology Seminar (invited), City University of New York
  • “Homology and Homotopy Properties of Scale-Free Complexes”
  • slides
• Feb 16, 2024: Hot Topics in Data Science (invited), University at Buffalo
  • “Detecting Weak Topological Signals in Noisy Environments”
  • slides
• Feb 9, 2024: University of Florida Topological Data Analysis conference (invited). University of Florida
  • “Homology and Homotopy Properties of Scale-Free Networks”
  • slides
• Feb 9, 2023: Topology Seminar (invited), The Chinese University of Hong Kong
  • “Topological Data Analysis and Scale-Free Networks”
  • slides
• Dec 6, 2023: Topology Seminar (invited), The Chinese University of Hong Kong
  • “Topological Data Analysis and Scale-Free Networks”
  • slides
• Nov 17, 2023: Northeast Probability Seminar, New York University
  • “Asymptotics of Expected Betti Numbers of Preferential Attachment Clique Complexes”
  • slides
• Nov 12, 2023: Binghamton University Graduate Combinatorics, Algebra and Topology (BUGCAT 2023), Binghamton University
  • “Betti Numbers of Preferential Attachment Flag Complexes”
  • slides
• Nov 3, 2023: Applied Topology Seminar (invited; virtual), Oxford University
  • “The Expected Betti Numbers of Preferential Attachment Clique Complexes”.
  • slides
• Nov 1, 2023: Applied Algebraic Topology Research Network (AATRN) Online Seminar (invited; virtual)
  • “The Asymptotics of the Expected Betti Numbers of Preferential Attachment Clique Complexes”.
  • slides
• Sep 29, 2023: Computation Persistence Workshop
  • “The Asymptotics of the Expected Betti Numbers of Preferential Attachment Clique Complexes – Theory and Computational Challenges”.
  • slides
• Sep 27, 2023: Probability Seminar (invited), Purdue University
  • The Topology of Preferential Attachment Graphs
  • slides
• Sep 20, 2023: Seminario Doctorado, Actividad del Programa de Doctorado “Mathematicas” (invited), University of Seville
  • The Topology of Preferential Attachment
  • slides
• Sep 14, 2023: Probability and Applications Seminar (invited), Queen Mary University of London
  • The Topology of Preferential Attachment Graphs
  • slides
• Aug 6, 2023: Joint Statistics Meetings 2023
  • Discovery of Small Dense Topological Features from Dataset
  • slides
• Jun 8, 2023: Geometry and Topology meet Data Analysis and Machine Learning 2023
  • The Many Holes of Preferential Attachment”
  • slides
• Mar 22, 2023: Randomness in Topology and its Applications (IMSI workshop)
  • “Betti Numbers of Preferential Attachment Complexes” (poster).
  • poster
• Feb 26, 2023: Finger Lakes Probability Seminar
  • “Expected Betti Numbers of Preferential Attachment Complexes”
  • slides
• Nov 17, 2022: Morse Theory Seminar for Cornell Graduate Students
  • Introduction to Discrete Morse Theory
• Nov 6, 2022: Binghamton University Graduate Combinatorics, Algebra and Topology (BUGCAT 2022)
  • Detection of Small Cycles in Data by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration
  • slides
• Oct 22, 2022: 3rd Upstate New York Topology Seminar (UNYTS 3)
  • Detection of Small Cycles in Data by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration
  • slides
• Jun 20, 2022: Algebraic Topology: Methods, Computation and Science 10 (ATMCS 10)
  • Detection of Small Topological Features by the Scale-FInvariant Robust Density-Aware Distance (RDAD) Filtration (Poster)
  • poster
• Mar 8, 2022: Olivette Club
  • Topological Data Analysis: Klein Bottles and How Cornellians Find Them in Data
  • slides

Teaching

• MATH1920 Spring 2023
• MATH1920 Fall 2022
• MATH2020 Spring 2019
• EPYMT Number Theory and Cryptography, Summer 2018
• MATH4060 Fall 2018
• MATH2010 Spring 2018
• MATH1510 Spring 2018
• MATH1540 Fall 2017

Resources for Learning

• Dr. K. Delp’s unsolicited advice.
• Prof. I. Zakharevich’s homework guide.
• Prof. F. Su’s guidelines for good mathematical writing for lower-division students and upper division students
• D. Zemke’s tips on graph sketching
• G. Sanderson’s cool visualizations of calculus and linear algebra
• The Harvard Physics Department’s Guide to The Hidden Curriculum

Undergraduate Mentorship

If you are interested in doing an undergradate reading / research project, send me an email!

• Fall 2023 Avhan Misra on the topological behavior of preferential attachment graphs
• Fall 2022 Rongyi He on Preferential Attachment Models
• Fall 2021 Luis Hoderlein on Dimension Reduction and UMAP, and Category Theory of Mapper
• Fall 2021 Tom Shi on Ranking of Graph Data
• Spring 2021 Andrey Yao on Computational Topology II: From Persistent Homology to Ripser
• Fall 2020 Andrey Yao on Computational Topology: From Simplicial Complexes to Persistent Homology

Resources for Research

• Mathematical talks
  • How to Avoid Death by PowerPoint by D. Phillips
  • Tips compiled by Prof. D. Margalit
  • Why we choke under pressure – and how to avoid it by Prof. S. Beilock
• Mathematical writing
  • Writing a Math Phase Two Paper by Prof. S. Kleiman and its pdf version.
  • General Instructions for Final Draft of Short Paper by Prof. D. Kleitman
  • More tips compiled by Dr. E. Carberry
  • More tips compiled by Prof. D. Margalit
  • What other undergraduates wrote at Chicago and their Latex templates
  • LaTex Tutorials on Overleaf
  • Mathcha, a graphical interface for drawing Tikz diagrams
  • tikzcd-editor, a graphical interface for drawing commutative diagrams
• Coding skills
  • Some Stackexchange and Stack Overflow discussions on self-documenting codes, code reuse, unit testing, quality codes and the use of debuggers
  • The SOLID principles for writing clean codes
  • C. Williams’ introduction to functional programming
  • Beginners’ guide to programming on Reddit
• Coding tutorials
  • Dr. M. Lee’s Conda tutorial
  • Introduction to Github on Github
• Data management
  • Harvard LMA RDMWG’s tips for naming files
  • The FAIR principle
• Undergraduate Research Opportunities
  • at Williams
  • at Cornell
  • at Chicago (modify the link to see the most up-to-date page)
  • at Lawrence Berkeley National Lab
  • at MSRI
  • through many NSF-funded programs

Open Problems

Questions are arranged roughly in descending order of abstraction. If you are a neuroscientist, I suggest starting from the bottom.

• Probability x Topology: Random Graphs and Simplicial Complexes
  • Background: Combinatorial properties and connectivity of random graphs have been extensively studied, but their algebraic topological properties are poorly understood, unless when they are extremely homogeneous. This poor understanding limits our ability to rigorously model the higher-order connectivity of real-world networks, which have been gaining popularity in various scientific domains. Of course, real-world networks are rarely homogeneous.
  • Question: Investigate the Betti numbers of the clique complex of the stochastic block model. Betti numbers are the simplest topological invariants, and the stochastic block model is the simplest inhomogeneous random graphs.
  • Literature: A lot of work on homogeneous random graphs, and on stochastic block model. I have two papers on the topology of preferentail attachment graphs, which is inhomogeneous. There is a poster due to Darrick Lee on some computational results.
• Probability x Algebra: MCMC for Sampling Random Chain Complexes
  • Background: A random chain complex is a generalization of random matrices. Formally, a chain complex is a sequence of abelian groups (or vector spaces) with homomorphisms between consecutive groups such that adjacent homomorphisms compose to 0. It is an algebraic language that allows us to ``compute” with topological objects. Random chain complexes reveals topological signal in real data by informing us the topological properties of noise. Sampling from the space of chain complexes is difficult, especially when working with matrices with integer entries or entries in finite fields.
  • Questions: Develop an efficient MCMC method to sample on the space of finite-length chain complexes (matrix sequences where adjacent pairs compose to 0) with the following data:
    • matrix entries: in finite fields, distribution: uniformly; or
    • matrix entries: integers, distribution: maximum entropy distribution with given sum of second moments of entries
  • Literature: [Ginzburg and Pasechnik, 2017] deals with a model where all matrices are the same. [Catanzaro and Zabka, 2021] deals with a model for semi-infinite chain complexes. Real and complex chain complexes admit an SVD [Brake, Hausentein, Schreyer, Sommese and Stillman, 2019]. The Macaulay2 package has a random complex package for an inductively built model. The computation can probably be sped up as well.
• Probability x Topology: Percolation and Euler Characteristic (suggested by Gwynne)
  • Percolation traditionally refers to the formation of giant connected components in a random system. In particular, the critical threshold for bond percolation the 3D integer is unknown but is of great interest. Algebraic topology is relevant, as it is known that this threshold coincides with the 1D homological percolation threshold (Cf. Section 3 of [Duncan, Kahle, and Schweinhart, 2023]). See also [Bobrowski and Skraba, 2020].
  • Question: Determine the critical threshold using homological algebra as precisely as possible.
• Probability x Topology: Topology of Random Functions in a High-Dimensional Space (suggested by Auffinger)
  • Background: We are interested in the topological properties of objects in a very high-dimensional space. The motivation comes from machine learning: The multitude of parameters motivates the high-dimensional part, and the nuisance of local minima motivates the topological part, as local minima correspond to 0-cells of a Morse function.
  • Question: Consider a random homogeneous cubic polynomial in a very high dimensional Euclidean space, where the randomness comes from iid Gaussian coefficients of each term. Describe the random persistent diagram at homological dimension 1.
  • Literature: [Auffinger, Arous, and Li], [Auffinger, Lerario, and Lundberg, 2021]
• Statistics x Topology: Stable Mapper Graph
  • Background: The Mapper graph is a data visualization method with rigorous theoretical support. It creates a graph from the input data points, where the nodes are clusters of data points, and nodes corresponding to overlapping clusters have edges between them. The local clustering helps abstractize the dataset to simplify the layout, and the edges break the shackle of low-dimensional Euclidean geometry. However, Mapper is known to have stability issues. This can be overcome in different ways, but the fundamental statistical challenge of partial clutersting in high dimensions has not been addressed.
  • Question: Incorporate thoughtful statistical techniques into the Mapper pipeline to enhance (or gauge) stability
  • Literature: There have been several Mapper stability paper (e.g. [Carriere, Michel, and Oudot, 2018] [Brown, Bobrowski, Munch, and Wang, 2021] [Dey, Memoli, and Wang, 2016]). HDBSCAN is a popular and well studied clustering algorithm. Mode regression (e.g. [Chen, Genovese, Tibshirani, and Wasserman], [Feng, Fan, and Suykens, 2020])
  • Topology x Neuroscience: Persistent Homology of Human Brain Activity in Different Conditions
    • Background: The network model of the brain has been proposed for over a decade, and the main tool is functional conenctivity, i.e. the Pearson correlation of the activation of different brain regions over time. However, the global emergent properties of the whole brain is still poorly understood.
    • Question: Describe brain activity with persistent homology and devise ways to compare across different individuals.
  • Dynamical system x Neuroscience: Dynamics of Psychopathology
    • Background: Psychiatry sets itself apart from the rest of medicine by its emphasis on behavior rather than biology. Since behavior takes place over time, dynamics is a crucial, yet often overlooked component of psychopathology. Modern neuroscience has shedded light on the biological basis of certain mental phenomena (e.g. amygdala is associated with the fear). A general dynamical model for the development of different psychopathology in the brain in response to different environmental factors is a promising direction for discovering the biological underpinnings of psychiatry.
    • Question: Develop a biology-based dynamical system for different psycho-development and pathology
  • Time series analysis x Neuroscience: Stationarity of the Resting Brain
    • The brain has a lot spontaneous activity, but its study is extremely difficult. Basic statistcal questions like the stationarity are still quite open.
    • Question: Investigate the stationarity of brain activity as a functional (time series that takes value in a high-dimensional space)
    • Literature: Use techniques from functional time series, e.g. [van Delft and Holger Dette, 2024], [van Delft, Characiejus and Dette, 2021]
  • Harmonic Analysis x Neuroimaging: Accurate Measurement of Subcortical Neuroactivity
    • functional MRI for subcortical regions are generally noisy, because of (1) the distance from the scalp, (2) physiological movement inside the brain (e.g. fluid flow, breathing), and (3) mechanical imgaging noise.
    • Question: Devise algorithms / techniques to image subcortical neuroactivity more accurately.

Personal

I am from Hong Kong. I would be really happy if you greet me with “zou sun” and “ng on” (Good morning and Good afternoon in Cantonese).

My favorite author is Dostoyevsky; favorite TV show, Taskmaster. Need to balance the heaviness and the lightness.

I am an avid singer. I am a barritone at a choir (University Singers) at Stanford, enjoying the mostly classical reportoire selected by our wonderful director Robert Morgan. I took singing lessons from my vocal coach, Gary Moulsdale, for three years. He was like a kind uncle to me.

I am mostly an indoor guy, but Ithaca has made me more active. Picked up hiking and ice-skating here. Try to catch me at a trail or in the rink.