[Colloquium] Jesse Stern Candidacy Exam/Jan 12, 2024

[Megan Woodward]

This is an announcement of Jesse Stern's Candidacy Exam.
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Candidate: Jesse Stern

Date: Friday, January 12, 2024

Time:  9:45 am CST

Location: JCL 298

Title: On the Number of Distinct Tilings of Finite Subsets of Z^d With Tiles of Fixed Size

Abstract: In this work, we study the number of finite tiles A⊂Z^d of size α that translationally tile a finite C⊂Z^d. We consider two tiles A and A′ to be congruent if and only if one can be transformed into the other via some translation. We make several significant contributions to the study of this problem. For any α∈Z^+ and C=[x^1]×[x^2]×…[x^d] where x_1,…,x_d∈Z^+ (which we refer to as a finite contiguous C), we give an efficient method for enumerating all elements of T(α,C), where (A,B)∈T(α,C) if and only if A,B⊂Z^d, the Minkowsji sum of A and B equals C, the size of A equals α, and |C|=α|B|. We then use this to prove a partial order on |T(α,C)| with respect to α for any finite contiguous C. We then study the extremal question as to the the growth rate of max_{α,C}[|T(α,C)|] with respect to |C|. For finite contiguous C, we improve the trivial lower and upper bounds of log(n) and {n \choose n/2} respectively to an upper bound of n^{(1+ϵ)log(n) / log(log(n))} and an infinitely often super-polynomial lower bound such that, for all constants c and some infinite N⊂Z^+, ∀n∈N, ∃α∈Z^+ (|T(α,C)|>n^c), where n=|C|. We conjecture that the number of tilings of any finite contiguous C by tiles of size α is an upper bound on the number of tilings of any finite C′⊂Z^d by tiles of size α. To begin working towards this, we prove that any A of size α that tiles some finite contiguous C itself has at most as many tilings by tiles of size α′ as there are tilings of [α] by tiles of size α′.

Advisors: David Cash

Committee Members: Alexander Razborov, David Cash, and Aloni Cohen

Paper Link: https://arxiv.org/abs/2303.06717<https://urldefense.com/v3/__https://arxiv.org/abs/2303.06717__;!!BpyFHLRN4TMTrA!6PdSmcO9yDe68czQF1lHZt6pgzzR1x6QQ6IJ5sWWtXxRfjYPIGmsgA0MtScSbkTFtcjctK_Fa0BjjzXSqOdsfemF7aLKvQuHgA$>

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