Hi! I'm Victor Wang, a Marie Skłodowska-Curie Fellow at IST Austria (June 2023–May 2025). During the 2022–2023 academic year I was a Courant Instructor at NYU; and before that, a graduate student of Peter Sarnak. You can contact me at vywang (at) alum.mit (dot) edu. I am broadly interested in number theory (esp. analytic), geometry (esp. algebraic), and discrete math (esp. with arithmetic, topological, or other structure).
See also arXiv and Google Scholar Citations.
Pairs of commuting integer matrices (with Tim Browning and Will Sawin), 13 pages.
Appendix to the paper Integral Hasse principle for Markoff type cubic surfaces by Hrishabh Mishra. My intern, Hrishabh, in the Browning Group, improved on work of Ghosh and Sarnak. His paper produces new quantitative insights and applications to sparse sequences of log K3 surfaces. A poster summarizing his work can be found here.
Optimal sums of three cubes in Fq[t] (with Tim Browning and Jakob Glas), 23 pages.
A nonabelian circle method (with Nuno Arala, Jayce R. Getz, Jiaqi Hou, Chun-Hsien Hsu, and Huajie Li), 60 pages. A 6-page supplement, entitled Nonabelian circle method supplements, is attached to the arXiv PDF.
Harper's beyond square-root conjecture (with Max Wenqiang Xu), 24 pages.
Sums of three cubes over a function field (with Tim Browning and Jakob Glas), 57 pages.
Notes on zeta ratio stabilization, 24 pages. This extends a breakthrough of Bergström–Diaconu–Petersen–Westerland and Miller–Patzt–Petersen–Randal-Williams from moments to ratios. We also give a general axiomatization of the argument. The appendix discusses some other geometric families one might want to look at.
Asymptotic growth of translation-dilation orbits, 32 pages. This overcomes difficulties in some adelic harmonic analysis of Tanimoto–Tschinkel, and thereby establishes Manin's conjecture for equivariant compactifications of the ax+b group in "geometric generality over Q", although we leave delicate secondary terms unresolved in general. We introduce new local coordinates at all places, based on new geometric observations along the boundary divisor of the compactification. Leading-order computations in the spirit of arXiv:2108.03396 then pave the way for further progress.
Sums of cubes and the Ratios Conjectures, 61 pages, to be split further for practical reasons; the first part, Zeta statistics at the square-root barrier for cubes (46 pages), is available here. This uses harmonic analysis, algebraic geometry, and mean-value predictions of Random Matrix Theory type, over the complement of a dual variety, to conditionally establish a square-root level randomness-structure dichotomy conjecture of Hooley, Manin, et al.
Special cubic zeros and the dual variety, J. Lond. Math. Soc. (2) 110 (2024), no. 3, Paper No. e12975, 35 pages. This uses algebraic geometry and harmonic analysis over a dual variety to detect in the delta method a square-root level "structured" term in an asymptotic dichotomy conjecture of Hooley, Manin, et al. concerning rational points on a cubic fourfold.
Diagonal cubic forms and the large sieve, 23 pages. In a GRH application of Hooley highlighted elsewhere by Bombieri, this paper replaces GRH with an average large-sieve hypothesis. The structure of the proof informs some of our later work, such as arXiv:2108.03396 and arXiv:2108.03398. On the other hand, the application itself is of much classical interest, and weakening GRH is the main focus of the paper and its appendices.
Prime Hasse principles via Diophantine second moments, 22 pages, accepted to JAMR. In the setting of sums of three cubes, this crystallizes some insights of Ghosh, Sarnak, and Diaconu, and pushes them to some natural limits (in the absence of higher-than-second moments). This paper uses Poisson summation, many local calculations, positivity, and the Selberg upper-bound sieve.
Paucity phenomena for polynomial products (with Max Wenqiang Xu), Bull. Lond. Math. Soc. 56 (2024), no. 8, 2718–2726. We resolve a natural Diophantine paucity problem inspired by arXiv:2207.11758.
Partial sums of typical multiplicative functions over short moving intervals (with Mayank Pandey and Max Wenqiang Xu), Algebra Number Theory 18 (2024), no. 2, 389–408. We make some progress on a question of Harper, obtaining in some sense near-optimal results, but leave a critical epsilon-sized range open.
Families and dichotomies in the circle method, Ph.D. Thesis, Princeton University, 2022. URL: http://arks.princeton.edu/ark:/88435/dsp01rf55zb86g (in Princeton's DataSpace). An unofficial single-spaced version (131 pages), differing mostly in the page numbering, can be found here (with main source here and full source here).
Dichotomous point counts over finite fields, J. Number Theory 250 (2023), 1–34. This establishes a near dichotomy between randomness and structure for the point counts of projective cubic threefolds over finite fields.
1-color-avoiding paths, special tournaments, and incidence geometry (with Jonathan Tidor and Ben Yang); written during MIT SPUR 2016. We thought about a recent question of Loh (2015): must a 3-colored transitive tournament on N vertices have a 1-color-avoiding path of vertex-length at least N^(2/3)? This question generalizes the classical Erdos–Szekeres theorem on monotone subsequences (1935). To me this problem seems natural and surprisingly rich. Note: Gowers and Long just uploaded a very nice preprint (September 2016) on this problem and its natural generalizations.
On Hilbert 2-class fields and 2-towers of imaginary quadratic number fields, J. Number Theory 160 (2016), 492–515; written at Duluth REU 2015. I thought about a question of Martinet (1978): must every imaginary quadratic number field K/Q have an infinite Hilbert 2-class field tower when the discriminant of K has 5 prime factors? Naturally one tries to use class-field-theoretic constructions and inequalities, but difficulties arise from the combinatorics of certain 5 x 5 (or smaller) binary matrices of quadratic symbols. I proved some new cases and found some precise reasons and examples explaining some of the difficulties.
Simultaneous core partitions: parameterizations and sums, Electron. J. Combin. 23 (2016), no. 1, Paper 1.4, 34 pages; written at Duluth REU 2015. This showed me a beautiful side of partitions and hook lengths that I had not seen before, and gave me a nice opportunity to interact with the vibrant core partitions community. Specifically, I re-interpreted some number-theoretic stabilizer sizes of Fayers in a friendlier way, which led to proofs of some enumerative-combinatorial conjectures of Fayers, as well as a simpler proof of a foundational structural result of Fayers.
A nonabelian circle method, Twelfth Bucharest Number Theory Days (7/2024).
Number theory inequalities, USA Math Olympiad Program (6/2024).
Nonabelian delta, Stanford Joint Analytic Number Theory Seminar (5/2024).
Sums of three cubes, a flash talk at IST Austria for Austrian Math Olympiad students (3/2024); handwritten notes available here.
Sums of three cubes over a function field, Number Theory Web Seminar (2/2024; virtual); video recording available here; slides available here.
Sums of three cubes over a function field, Graz-ISTA Number Theory Days (2/2024).
Zeta ratio averages at the square-root barrier for cubes, Göttingen Oberseminar on Analytic Number Theory (11/2023).
Paucity and polynomial products, Browning Group Working Seminar (10/2023).
Harmonic analysis towards Manin–Peyre, Seminar on Arithmetic Geometry and Algebraic Groups (10/2023; virtual); slides available here.
Diophantine equations, IST Austria Math ISTern Day (7/2023).
Continuation under translation-dilation equivariance, Browning Group Working Seminar (6/2023).
Sums of cubes and the Ratios Conjectures, Brown Algebra and Number Theory Seminar (3/2023).
Sums of cubes and the Ratios Conjectures, Joint Columbia-CUNY-NYU Number Theory Seminar (3/2023).
Families and dichotomies in the circle method, Stanford Student Analytic Number Theory Seminar (11/2022; virtual); slides available here.
Sums of cubes and Random Matrix Theory, Courant Postdoc Seminar Day (11/2022); slides available here.
Special subvarieties over finite and infinite fields, L-function and Stratification FRG Grad Seminar (4/2022; virtual); slides available here.
Randomness and structure for sums of cubes, Seoul National University Number Theory Seminar (4/2022; virtual); slides available here.
Biased point counts over finite fields, Princeton Graduate Student Seminar (3/2022).
Some perspectives on cubic Diophantine equations, Duke Number Theory Seminar (3/2022; virtual); slides available here.
Dichotomous point counts, Northwestern Number Theory Seminar (3/2022).
Conditionally around the square-root barrier for cubes, Copenhagen Number Theory Seminar (12/2021; virtual); slides available here.
Conditional approaches to sums of cubes, Bristol Linfoot Number Theory Seminar (12/2021; virtual); slides available here.
Conditional approaches to sums of cubes, Joint IAS/Princeton University Number Theory Seminar (11/2021); video recording available here (missing a few minutes of video/audio at the beginning and in the middle); notes available here with beamer notes, and here without.
Conditional approaches to sums of cubes, Browning Group Working Seminar (10/2021; virtual); slides available here with beamer notes, and here without.
Conditional approaches to sums of cubes, Purdue Analytic Number Theory and Harmonic Analysis Seminar (9/2021; virtual); slides available here with beamer notes, and here without.
L, 1/L, and L'/L, Princeton Graduate Student Seminar (4/2021; virtual); very rough handwritten notes available here.
Statistics of random hypersurfaces (mod p), Princeton Number Theory Graduate Student Tea (1/2021; virtual); slides available here.
Two talks on graphs and surfaces, GIANT International Internship Program, Grenoble, France; one to a general math audience (6/2017), and one to a general scientific audience (7/2017); slides available here and here, respectively; further notes on this topic available in the first section here (with some details here and a picture here regarding a suspected-to-intervene fundamental group).
Local-global principles in number theory (Hasse–Minkowski for quadratic equations), MIT Student Colloquium for Undergraduates in Math (11/2016).
Riemann–Roch with Hodge theory, MIT Curves Learning Seminar (9/2016).
1-color-avoiding paths, special tournaments, and incidence geometry (presented with Jonathan Tidor), MIT SPUR Conference (8/2016); slides available here; further notes on this topic available in the second section here.
Martinet's question on Hilbert 2-class field towers, AMS Contributed Papers Session in Number Theory, Joint Mathematics Meetings, Seattle, WA (1/2016); slides available here.
Roots of unity filter (finite Fourier analysis), MIT Student Colloquium for Undergraduates in Math (10/2015).
18.100p (p-adic analysis), MIT Student Colloquium for Undergraduates in Math (11/2014).
Informal notes on the Cohen–Lenstra heuristics: PDF.
Notes and problems from USA MOP 2018 (Math Olympiad Summer Program):
Some good books I've read:
See here.
I've enjoyed collaborating (as a math consultant) with many great teammates on Expii, an online education project seeking to break down topics while providing fun, relatable content (such as the theme-based Expii Solve) and a curated problem stream (organized and scaffolded through concept maps in the background). I hope that Expii's resources and outreach will complement the existing infrastructure in education, helping to give teachers the freedom to spend less time on tasks that can be automated, and more on the deepest human aspects that cannot be. The current focus is on high school math and science, but there are also other topics, like elementary number theory and advanced plane geometry.
If people played Codenames with only math terms, would we get better at finding connections in math? How about other topics (academic or not)?
Larger amsart fonts to reduce eyestrain while drafting.
A guide I've found helpful on whether to trash, recycle, or compost (in Princeton): campus guidelines; municipality webpage; municipality PDF; old municipality webpage; old municipality PDF.
Food for thought: if socialism is 0-dimensional (in terms of purely economic incentives), and modern capitalism is 1-dimensional, what lies beyond? Related: time banking; Digital Social Credits idea of Andrew Yang (too bad it will automatically be superficially compared with China's system).
