Sofia Torelli

Week 1: 5/27-5/30

Reading: Spielman's Spectral and Algebraic Graph Theory §1.1-1.4, §4.4, as well as the Introduction to McMullen's notes on ergodic theory.

Exercises.

Attended orientation and met this year's lovely REU cohort. Moved into my office at CoRE 411. Met with Cecelia in-person for the first time, we reviewed the proof of Wilf's Theorem and the construction of a non-Lebesgue-measurable set using the Axiom of Choice (meeting notes). Went to the Busch Student Center for the first time with Cecelia, Esmé, and Sofia A for lunch. Began building this site.

Week 2: 6/01-6/05

Reading: Kechris's Classical Descriptive Set Theory §1, §3.A, §10-11 and Higgins, Spaas, Tenenbaum's "Spectral Theory for Borel PMP Graphs" (2026) sections 1, 2.

Exercises.

Met with Cecelia Tues morning: We reviewed how the Hoffman (and Wilf) bounds on \(\chi (G) \) were extended to bounded-degree Borel pmp graphs in her recent paper. Cecelia believes this can also be done for Nikiforov's bound; this will be one of my goals for this summer. We defined the adjacency operator \(T_{\mathcal G}:L^2(V) \to L^2(V)\) and proved that it is bounded for \(V = \mathbb R\) (meeting notes).

After the meeting, I presented to my peers a 5 minute overview of my project and its goals and learned about their work in turn (slides). Finished building out my website and uploaded it. On Wednesday, I went with Lucy to the College Ave campus to attend a DIMACS Computational Geometry Week seminar; afterwards, we enjoyed a coffee at Hidden Grounds.

Met with Cecelia Thurs morning: Defined ordinals and discussed the Borel hierarchy constructed through transfinite recursion in Kechris §11. Outlined proof of a theorem stating: if \(\mathcal G\) is locally finite and Borel, then \(\chi_B(\mathcal G)\) (the Borel chromatic number) is countable (meeting notes).

Week 3: 6/08-6/12

Reading: Spielman SAGT §19.3, 19.5, 19.6. HST (2026) sections 2, 3.1, 6. Nikiforov's "Chromatic number and spectral radius" (2018).

Exercises.

Met with Cecelia Tues morning: Discussed operator spectra (discrete, continuous, and residual). Bounded + self-adjoint operators on Hilbert spaces have no residual spectrum, moreover their spectra are nonempty compact subsets of \(\mathbb R\). Outlined a proof of the measurable version of Hoffman's bound, \(\chi_\mu(\mathcal G) \ge \left \lceil 1 - \frac{M(T_\mathcal G)}{m(T_\mathcal G)}\right\rceil \) (meeting notes).

Proved a measurable version of Hoffman's bound.

Met with Cecelia Thurs morning: Discussed the proof of Nikiforov's bound and how to potentially generalize it for the measurable case (meeting notes).

Explored the Rutgers gardens and the weekly farmers' market with Esmé and Sofia A. Over the weekend, we also had a chance to visit Princeton thanks to Filippo and his wife Dima Sinapova, who graciously hosted us for a day trip + dinner.

Week 4: 6/15-6/19

Exercises.

Met with Cecelia and Filippo Mon afternoon: Presented progress on a measurable version of Nikiforov's paper.

Proved a measurable version of Nikiforov's bound, \(\chi_\mu(\mathcal G) \ge \left \lceil 1 - \frac{M(T_\mathcal G)}{M(L_\mathcal G) - M(T_\mathcal G)}\right\rceil \). Along the way, found an alternative proof of the measurable Hoffman bound. Thank you to Filippo for hosting a visit to the local Tavern on George, with live jazz music!

Met with Cecelia, Filippo, and Dr. Pieter Spaas Wed morning: Presented proof of approximate measurable Nikiforov's bound.

On Friday, accompanied the Czech students to visit the NYC Museum of Modern Art; got the chance to view works from Frida Kahlo to Andy Warhol.

Week 5: 6/22-6/26

Reading: Kolotilina's "Inequalities for the extreme eigenvalues of block-partitioned Hermitian matrices with applications to spectral graph theory" (2011). HST (2026) section 4.

Exercises.

Gave a presentation on the history and culture of salsa dancing for Culture Day.

Met with Cecelia and Filippo Tues morning: Discussed next steps and papers to read after concluding work on Nikiforov's bound.

Wrapped up some small results for sharpness re: Nikiforov's bound. Generalized Kolotilina's bound to the measurable case.

Met with Cecelia, Filippo, and Pieter Fri morning: Confirmed results on Nikiforov sharpness and measurable Kolotilina. Outlined a proof of HST Theorem 4.9 generalized to the non-d-regular case.

Week 6: 6/29-7/02

Week 7: 7/6-7/10

Week 8: 7/13-7/17

Week 9: 7/20-7/24 (Prague)

Week 10: 7/27-7/31 (Prague)