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Katy Loyd

Department of Mathematics
Northwestern University
Office: Locy Hall 212
E-mail: loydka at math.northwestern.edu

I am a sixth year Ph.D. candidate in Mathematics, working under the supervision of Bryna Kra. My research interests lie in dynamical systems, specifically ergodic theory and symbolic dynamics. I like to think about pointwise ergodic theorems with connections to analytic number theory.

Here is a copy of my CV (Last updated: Oct 2023).

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Publications and Preprints      Show all   Hide all   Show abstracts   Hide abstracts

Pointwise ergodic theorems

• A dynamical approach to the asymptotic behavior of the sequence \(\Omega(n)\), Ergodic Theory and Dynamical Systems 2023, 43(11), 3685–3706
  

  We study the asymptotic behavior of the sequence \( \{\Omega(n) \}_{ n \in \mathbb{N} } \) from a dynamical point of view, where \( \Omega(n) \) denotes the number of prime factors of \( n \) counted with multiplicity. First, we show that for any non-atomic ergodic system \( (X, \mathcal{B}, \mu, T) \), the operators \( T^{\Omega(n)}: \mathcal{B} \to L^1(\mu) \) have the strong sweeping-out property. In particular, this implies that a pointwise ergodic theorem does not hold along \( \Omega(n) \). Second, we show that the behaviors of \( \Omega(n) \) captured by the Prime Number Theorem and Erdos-Kac Theorem are disjoint, in the sense that their dynamical correlations tend to zero.

• Ergodic averaging along the sequence \( \Omega(n) \), joint with S. Mondal, In preparation
  

  Abstract TBA

Symbolic dynamics

• New points in the Heinis spectrum, arXiv preprint coming soon in 2023!
  

  Abstract TBA

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Invited Talks        Show all   Hide all   Show titles   Hide titles

Research Talks

• Analysis Seminar, Virginia Tech, Oct 2023
  

  Title: Convergence of ergodic averages along the sequence \( \Omega(n) \)

• Low Complexity Dynamical Systems Conference, University of Maryland, Oct 2023
  

  Title: New points in the Heinis spectrum

• Ergodic Theory and Analysis Seminar, Rutgers University, Sep 2023
  

  Title: Convergence of ergodic averages along the sequence \( \Omega(n) \)

• Chicago Action Now Conference, UIC, Apr 2023
  

  Title: Convergence of ergodic averages along the sequence \( \Omega(n) \)

• IAS Special Year Seminar, Institute for Advanced Study, Feb 2023
  

  Title: Convergence of ergodic averages along the sequence \( \Omega(n) \)

• Midwest Dynamical Systems Conference, IUPUI, Nov 2022
  

  Title: A dynamical approach to the asymptotic behavior of the sequence \( \Omega(n) \)

• AMS Spring Western Sectional Meeting, University of Denver, May 2022
  

  Title: A dynamical approach to the asymptotic behavior of the sequence \( \Omega(n) \)

• Midwest Dynamical Systems Early Career Conference, Notre Dame, May 2022
  

  Title: Convergence of ergodic averages along the sequence \( \Omega(n) \)

• Dynamical Systems Seminar, Northwestern University, Nov 2022
  

  Title: A dynamical approach to the asymptotic behavior of the sequence \( \Omega(n) \)

Expository Talks

• Dynamical Systems REU Colloquium, Northwestern University, Aug 2023
  

  Title: Interactions between Ergodic Theory and Number Theory

• Undergraduate Math Society, Northwestern University, Apr 2022
  

  Title: Dynamical Systems via Cats (and Ghosts?)

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Teaching Experience        Show all   Hide all

Continuing Studies Instructor, Northwestern University School of Professional Studies

• Summer 2022:
  • MATH 240-0: Linear Algebra, First Course

Graduate Teaching Assistant, Northwestern University

• Fall 2021:
  • MATH 230-2: Multivariable Integral Calculus
• Winter 2021:
  • MATH 230-1: Multivariable Differential Calculus (2 Sections)
• Fall 2020:
  • MATH 354-0: Chaotic Dynamical Systems
  • MATH 220-2: Single Variable Integral Calculus
• Winter 2020:
  • MATH 230-1: Multivariable Differential Calculus (2 Sections)
• Fall 2019:
  • MATH 220-2: Single Variable Integral Calculus (2 Sections)

Research Group Assistant, Northwestern Dynamical Systems REU

• Summer 2023: Coordinated with faculty leaders to run three undergraduate research projects.
• Project topics:
  • Dynamics and Representation Theory: Two projects on dynamics of \( Aut(F_n) \)
  • Dynamics and Graph Theory: Spectra of Periodic Schrodinger Operators

Teaching Assistant, Bridge Program, Northwestern University

• Summer 2021: "Introduction to Quantitative and Scientific Reasoning"

Teaching Assistant, Northwestern Prison Education Program

• Winter 2020:
  • MATH 110: Introduction to Mathematics

For further information on my teaching philosophy and practices, please see my teaching e-Portfolio
PW: 'LoydTeach'

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Other

More About Me

• I have an orange cat named Dumpling.
  
  Cat tax: Dumpling and his littermate Toast
  
  
  
• I train at a competitive boxing gym. My favorite boxers are GGG and Lomachenko.

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Last updated: October 21, 2023