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<div class="elementToProof"><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);">This is an announcement of Jesse Stern's Candidacy Exam.</span></div>
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<div><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);">Candidate: Jesse Stern</span></div>
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<div><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);">Date: Friday, January 12, 2024</span></div>
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<div class="elementToProof"><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);">Time: 9:45 am CST</span></div>
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<div><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);">Location: JCL 298</span></div>
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<div><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);">Title: On the Number of Distinct Tilings of Finite Subsets of Z^d With Tiles of Fixed Size</span></div>
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<div><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);">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′.</span></div>
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<div><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);">Advisors: David Cash</span></div>
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<div class="elementToProof"><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);">Committee Members: Alexander Razborov, David Cash, and Aloni Cohen</span></div>
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<div class="elementToProof"><span style="letter-spacing: normal; font-family: Helvetica; font-size: 12px; color: rgb(0, 0, 0);"><b>Paper Link:</b></span><span style="letter-spacing: normal; font-family: Helvetica; font-size: 12px; font-weight: 400; color: rgb(0, 0, 0);"> <a href="https://urldefense.com/v3/__https://arxiv.org/abs/2303.06717__;!!BpyFHLRN4TMTrA!6PdSmcO9yDe68czQF1lHZt6pgzzR1x6QQ6IJ5sWWtXxRfjYPIGmsgA0MtScSbkTFtcjctK_Fa0BjjzXSqOdsfemF7aLKvQuHgA$" id="OWA36d02759-6fc2-07f2-8bce-0400f144a0c8" class="OWAAutoLink" style="text-align: left;" data-loopstyle="linkonly">https://arxiv.org/abs/2303.06717</a> </span></div>
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