Réunion d'hiver SMC 2025
Toronto, 5 - 8 decembre 2025
Combinatoire additive et applications
Org: Chi Hoi (Kyle) Yip (Georgia Institute of Technology) et Yifan Jing (Ohio State University)
[PDF]
Org: Chi Hoi (Kyle) Yip (Georgia Institute of Technology) et Yifan Jing (Ohio State University)
[PDF]
- ERNIE CROOT, Georgia Institute of Technology
- ZHENCHAO GE, University of Waterloo
- MARCEL GOH, McGill University
Block complexity and idempotent Schur multipliers [PDF]
- ZHENCHAO GE, University of Waterloo
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We call a matrix blocky if, up to row and column permutations, it can be obtained from an identity matrix by repeatedly applying one of the following operations: duplicating a row, duplicating a column, or adding a zero row or column. Blocky matrices are precisely the boolean matrices that are contractive when considered as Schur multipliers. It is conjectured that any boolean matrix with Schur multiplier norm at most $\gamma$ is expressible as a signed sum
\begin{equation*}A = \sum_{i=1}^L \pm B_i\end{equation*}
for some blocky matrices $B_i$, where $L$ depends only on $\gamma$. This conjecture is an analogue of Green and Sanders's quantitative version of Cohen's idempotent theorem. In this paper, we prove bounds on $L$ that are polylogarithmic in the dimension of $A$. Concretely, if $A$ is an $n\times n$ matrix, we show that one may take $L = 2^{O(\gamma^7)} \log(n)^2$.
- LEO GOLDMAKHER, Williams College
- DAVID GRYNKIEWICZ, University of Memphis
- YIFAN JING, Ohio State University
- YU-RU LIU, U. of Waterloo
Equidistribution Theorems in Additive Combinatorics [PDF]
- DAVID GRYNKIEWICZ, University of Memphis
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We establish a function-field analogue of Weyl’s equidistribution theorem for polynomial sequences and explore its applications to problems in additive combinatorics. This is joint work with Jérémy Champagne, Thái Hoàng Lê and Trevor Wooley.
- COSMIN POHOATA, Emory University
- STEVEN SENGER, Missouri State University
- FERNANDO XUANCHENG SHAO, University of Kentucky
Recent developments on the polynomial Szemeredi theorem [PDF]
- STEVEN SENGER, Missouri State University
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As a special case of the celebrated theorem of Bergelson and Leibman (the polynomial Szemeredi theorem), any positive density subset of the integers must contain a polynomial progression of the form $x, x+y, x+y^2$ with $y$ nonzero. In the last five years since the pioneering work of Peluse and Prendiville, there have been numerous developments on the quantitative aspects of such results. I will give a brief overview of these recent developments, before describing a two-dimensional version and a "popular" version of the polynomial Szemeredi theorem for the pattern $x, x+y, x+y^2$. The talk includes joint works with Sarah Peluse, Sean Prendiville, and Mengdi Wang.
- HUNTER SPINK, University of Toronto
- JONATHAN TIDOR, Princeton University
- STANLEY YAO XIAO, UNBC
Primes of the form $f(p,q)$, $f$ quadratic, and applications [PDF]
- JONATHAN TIDOR, Princeton University
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We capitalize on the breakthrough result of Green and Sawhney proving the infinitude of primes of the form $p^2 + nq^2$, where $n \equiv 0, 4 \pmod{6}$ is a fixed positive integer and $p,q$ are prime variables to arbitrary binary quadratic forms satisfying the obvious non-degeneracy conditions. Notably, our result covers irreducible indefinite binary quadratic forms. This has applications to counting elliptic curves admitting a rational isogeny of prime degree, ordered by conductor.