Ryan E. G. Bushling

Teaching

Office

Padelford C-8E

Fall 2024 Office Hours

By appointment

Courses

Spring 2024: MATH 224 - Advanced Multivariable Calculus*
Winter 2024: MATH 525 - Real Analysis II
Fall 2023: MATH 524 - Real Analysis I, MATH 399 B - Undergraduate Research
Spring 2023: MATH 428 - Complex Analysis, MATH 398 B - Special Topics
Winter 2023: MATH 525 - Real Analysis II, MATH 398 B - Special Topics
Fall 2022: MATH 207 DA/DB/DC - Differential Equations
Summer 2022: MATH 207 B - Differential Equations*, MATH 530 - Seminar in Analysis*
Winter 2022: MATH 525 - Real Analysis II
Fall 2021: MATH 125 DD - Calculus with Analytic Geometry II
Summer 2021: MATH 126 D - Calculus with Analytic Geometry III*
Spring 2021: MATH 424 A/B - Fundamental Concepts of Analysis I
Winter 2021: MATH 327 A / 574 A - Introductory Real Analysis I
Fall 2020: MATH 125 FA - Calculus with Analytic Geometry II
Summer 2020: MATH 126 B/E - Calculus with Analytic Geometry III
Spring 2020: MATH 126 DC/DD - Calculus with Analytic Geometry III
Winter 2020: MATH 125 AC/AD - Calculus with Analytic Geometry II
Fall 2019: MATH 124 DA/DB - Calculus with Analytic Geometry I

* Primary instructor

Qualifications

UW REU in Mathematical Analysis, Co-Organizer (Summer 2024)
Walter J. Michaelis Graduate Fellow (Fall 2023)
WXML Graduate Mentor (Fall 2023)
MATH 597 B - Seminar on Teaching Math (Summer 2022)
MS Mathematics, University of Washington (Spring 2022)
TA Mentor (Fall 2021)
MATH 597 A - Seminar on Teaching Math (Summer 2021)
GRDSCH 501 - TA Training (Fall 2019)
BS Mathematics, University of Washington (Spring 2019)

GSAS

I established the UW Math Department's Graduate Student Analysis Seminar (GSAS)

"as a low-stakes forum organized by and for graduate students to share their interests and professional experiences in the field of mathematical analysis. Among its primary goals are (1) to provide a safe space for graduate students to expound their knowledge of special topics in analysis with minimal pressure and no judgment, and (2) to build community among those interested in analysis by providing them a forum in which to learn about their peers' research interests."

GSAS provides graduate students interested in mathematical analysis the opportunity to grow as mathematicians by engaging new material in diverse subfields. A founding tenet of the organization is that it should afford its members resources to explore and experiment—to find their identities as professional mathematicians. Please contact me to request additional information or to subscribe to our mailing list.

Contact

Research

Figure. (Above left) An illustration of Falconer's digital sundial theorem. (Above right) A union of s-dimensional subsets of lines and its s-Hausdorff extension. (Below) Examples of (n–1)-rectifiable sets satisfying the orientation cancellation condition.

My research focuses on problems in geometric measure theory and harmonic analysis that lend themselves to study through the lense of linear and nonlinear projections. My results on projections include a generalization of a result of Orponen concerning exceptional sets of orthogonal projections in Rⁿ, and a nonlinear projection theorem generalizing a result of M. Järvenpää. The latter is currently seeing use in joint work with Iqra Altaf and my adviser Bobby Wilson on the packing dimension of distance sets. This is follow-up work to our previous paper on the Hausdorff dimension of distance sets with respect to general norms, in which we used Kolmogorov complexity to prove a sharp bound on the dimension of distance sets for polyhedral norms.

Complexity methods were also used in joint work with Jacob Fiedler on a packing dimension analogue of Keleti's line segment extension conjecture and a sharp generalized Kakeya set estimate in the plane.

Finally, in a rather different vein, I study the geometry of rectifiable sets with global structure. This has led to a singular integral identity with applications including an analytic characterization of convex domains (without regularity assumptions). Max Goering and I are currently finishing work on an anisotropic analogue.

Publications

(April 2024) Bounds on the dimension of lineal extensions, joint work with Jake Fiedler, to appear in the Journal of Fractal Geometry
(February 2024) A study guide to "Kaufman and Falconer estimates for radial projections", joint work with Paige Bright, Caleb Marshall, and Alex Ortiz
(December 2023) A decomposition theorem for balanced measures, joint work with Gregory Baimetov, Ansel Goh, Raymond Guo, Owen Jacobs, and Sean Lee
(May 2023) Distance sets bounds for polyhedral norms via effective dimension, joint work with Iqra Altaf and Bobby Wilson
(April 2023) A singular integral identity for surface measure, The Journal of Geometric Analysis
(November 2022) Exceptional projections of sets exhibiting almost dimension conservation, Annales Fennici Mathematici
(October 2022) Exceptional sets of orthogonal projections: From classical results to current research

Elements

i = 0;

while (!deck.isInOrder()) {
    print 'Iteration ' + i;
    deck.shuffle();
    i++;
}

print 'It took ' + i + ' iterations to sort the deck.';