Constantin Kogler

I'm a member at the School of Mathematics of the Institute for Advanced Study. My research is generously funded by a Postdoc Mobility Fellowship of the Swiss National Science Foundation (Grant number 235409).

Email address: kogler@ias.edu

I recently completed my PhD at the University of Oxford under the supervision of Emmanuel Breuillard and also mentored by Péter Varjú. During the first two years of my doctoral studies, I was based at the University of Cambridge as part of the CCIMI cohort. Prior to my doctorate, I earned a master’s degree in mathematics from ETH Zürich in 2020.

My research is at the intersection of probability, analysis, dynamics and number theory. In pure mathematics, I focus broadly on random walks on Lie groups, particularly self-similar measures, and since very recently also on understanding random polynomials. I am increasingly interested in applying mathematics to other topcis and have worked on mathematical aspects of machine learning, particularly random neural networks. From a certain viewpoint, these are closely related to random walks on Lie groups.

I am excited about using AI to make progress on questions I find interesting. I will only publish work that I can understand and write about. I am very open to discussing where the field is going and to hearing constructive criticism of my research practice. Please get in touch if you want to chat.

Image Copyright: Maria O'Leary, Institute for Advanced Study.

Research

  1. The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line (with Samuel Kittle).
    (arxiv, github, lean)
  2. The Odlyzko-Poonen Conjecture on Irreducibility of Random Polynomials.
    (arxiv, lean)
  3. Optimal Initialization in Depth: Lyapunov Initialization and Limit Theorems for Deep Leaky ReLU Networks (with Tassilo Schwarz and Samuel Kittle).
    (arxiv)
  4. Permutation-Invariant Spectral Learning via Dyson Diffusion (with Tassilo Schwarz et al.).
    (arxiv)
  5. On absolute continuity of inhomogeneous and contracting on average self-similar measures (with Samuel Kittle).
    (arxiv)
  6. Dimension of contracting on average self-similar measures (with Samuel Kittle).
    (arxiv)
  7. Entropy theory for random walks on Lie groups (with Samuel Kittle).
    Journal of Theoretical Probability 2025. (arxiv, journal)
  8. Polynomial tail decay for stationary measures (with Samuel Kittle).
    To appear Proceedings of the AMS. (arxiv)
  9. Local limit theorem for random walks on symmetric spaces.
    Journal d'Analyse Mathématique 2025. (arxiv, journal)
  10. Effective density of non-degenerate random walks on homogeneous spaces (with Wooyeon Kim).
    IMRN 2024. (arxiv, journal)
Theses:
  1. Topic in Random Walks on Lie Groups.
    PhD Thesis. (205 pages, pdf)
  2. Effective p-adic ergodic theory, Diophantine approximation and spectral gap.
    Master Thesis. (204 pages, pdf)
  3. Closed geodesics on compact hyperbolic surfaces.
    Bachelor Thesis. (80 pages, pdf)

Notes

Talks