About

Qixia Luo

A picture of Qixia Luo

I am currently a PhD student in the Department of Mathematics and the Institute of Applied Mathematics at The University of British Columbia in beautiful Vancouver, British Columbia. I am working with Prof. Elina Robeva on algebraic statistics and mathematical questions related to data science.

In July, 2025, I obtained my MSc from the Department of Mathematics and Statistics at Queen's University at Kingston, where I worked with Prof. James A. Mingo on free probability and random matrix theory.

In June, 2022, I obtained my Honours BSc from the University of Toronto, where I graduated with honours and high distinction in mathematics and statistics.

Summer Reading Program

Finite Frames SRP 2026

This page is reserved for material, schedule, and other information for the Finite Frames SRP 2026 project.

The Summer Reading Program (SRP) at UBC is a program for undergraduate students who are interested in learning a topic in math beyond the standard undergraduate curriculum. Each summer, selected undergraduate students are matched with mentors who are graduate students or postdoctoral researchers at UBC. For more information, see UBC MATH SRP.

This summer, I have decided to create a group to learn about finite frames, which is a fascinating area of mathematics that lies at the intersection of several areas of math that is currently undergoing active research, with exciting applications to many areas such as signal processing, quantum information theory, and data science, to name a few.

The reason for the focus on finite frames is twofold.

  1. In practical applications, we always work with finite-dimensional spaces.
  2. Since functional analysis is not a prerequisite for participation in this reading program (participants are in years 1-3 of their undergraduate studies), studying frames in the finite dimensional setting is the accessible way to introduce them to this subject.

Despite the restriction to the simplified finite-dimensional setting, we will still be able to go surprisingly far. The participants will write a report or do a presentation on a cutting-edge topic at the end of the program.

The basic content of the reading program will be guided by the following two books:

  • Frames for Undergraduates (Han, Kornelson, Larson, Weber, 2007)
  • Finite Frames: Theory and Applications (Casazza, Kutyniok, 2013)
We will also look at some material from the first chapter of:
  • An Introduction to Frames and Riesz Bases (Christensen, 2003)

Schedule

  • Week 1
    Readings: Sections 3.1, 3.2, and 3.3 in Frames for Undergraduates.
    • June 1: Introductions. Overview and expectations of the reading program. Review of linear algebra concepts (hilbert spaces, bases, orthonormal bases, orthogonal complement, rank). Frames as objects that generalize orthonormal bases. First definition of a finite frame. Equivalent deinfitions of a finite frame. Parseval frames.
    • June 5: Reconstruction formula for Parseval frames. Uniform frames. Property of the norm of vectors in a Parseval frame. Mercedez-Benz frames. Definition of a general frame (works for finite or infinite dimensions). (Optimal) frame bounds. Triviality of the upper frame bound.
    Week 2
    Readings: 3.4 and 3.5 in Frames for Undergraduates and Section 1.9 in An Introduction to Frames and Riesz Bases.
    • June 8: Equivalence of the general frame with finite frames in finite dimensions (future participant presentation). Tight frames. Review of the adjoint of an operator. Operators associated with a frame and their properties: Analysis operator, synthesis operator, and frame operator. Reconstruction formula for frames and how this generalizes the reconstruction formula for Parseval frames. Dual frames and the canonical dual frame.
    • June 12: Redundancy. Eigenvalues of the frame operator. Application of tight frames for noise supression in signal transmission.
    Week 3
    Readings: Sections 3.5, 3.6, 5.1 in Frames for Undergraduates
    • June 15: Defintiion and properties of postive operators and matrices. Definition and properties of similar operators and matrices. Sqaure root of a positive operator or matrix. Similarity and unitary equivalence of frames. Every frame is similar to a Parseval frame. Two parseval frames are similar iff they are unitarily equivalent. Two frames are similar iff their analysis operators have the same range (proof will be a future participant presentation).
    • June 19: Participant presentation on the equivalence of R^n frames and general frames in finite dimensions.
    Week 4
    Readings: Sections 5.2, 5.3, 6.1 in Frames for Undergraduates
    • June 22: Participant presentation on orthogonal compression of frames.
    • June 26: Participant presentation on the proof of two frames being similar iff the range of their analysis operators being the same.
    Week 5
    Readings: Sections 6.2, 6.3 in Frames for Undergraduates
    • June 29: Participant presentation on the proof of tight frames achieving minimal frame potential.
    • July 3: Participant presentation on dilation of frames.
    Week 6
    Readings: Sections 6.4, 6.5 in Frames for Undergraduates
    • July 6:
    • July 10:
    Week 7
    Readings: Section 6.6, 7.1 in Frames for Undergraduates
    • July 13:
    • July 17:
    Week 8
    Readings: Section 7.2, 7.3 in Frames for Undergraduates
    • July 20:
    • July 24:
    Week 9
    Readings: None
    • July 27: Final projects
    • July 31: Final projects