<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<style type="text/css" style="display:none;"> P {margin-top:0;margin-bottom:0;} </style>
</head>
<body dir="ltr">
<div class="elementToProof"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);">This is an announcement of Jesse Stern's Candidacy Exam.</span></div>
<div id="Signature">
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);">===============================================</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);">Candidate: Jesse Stern</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);"><br>
</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);">Date: Friday, January 12, 2024</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);"><br>
</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);">Time: 9:45 am CST</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);"><br>
</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);">Location: JCL 298</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);"><br>
</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; 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>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);"><br>
</span></div>
<div style="text-align: left; margin: 0px;"><span style="font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; color: rgb(0, 0, 0);">Abstract:
</span><span style="font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);">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 Minkowski 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 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\log^c(n), 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>
<div style="text-align: left; margin: 0px;"><span style="font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);"><br>
</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);">Advisors: David Cash</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);"><br>
</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);">Committee Members: Alexander Razborov, David Cash, and Aloni
Cohen</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 11pt; font-weight: 400; color: rgb(0, 0, 0);"><br>
</span></div>
<div style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Helvetica, serif, EmojiFont; font-size: 12px; color: rgb(0, 0, 0);"><b>Paper Link:</b></span><span style="letter-spacing: normal; font-family: Helvetica, serif, EmojiFont; 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$" target="_blank" id="OWA8b0c950a-952d-d540-1df3-f2b582743864" class="x_OWAAutoLink" rel="noopener noreferrer" data-auth="NotApplicable" data-loopstyle="linkonly" data-linkindex="0" style="margin: 0px;">https://arxiv.org/abs/2303.06717</a></span></div>
<div style="text-align: left; margin: 0px;"><br>
</div>
<div style="text-align: left; margin: 0px;"><br>
</div>
<div style="text-align: left; margin: 0px;"><span class="_Entity _EType_OWALink _EId_OWALink_1 _EReadonly_1" style="display: inline-block;"><span><a href="https://uchicagoedu-my.sharepoint.com/:b:/g/personal/meganwoodward_uchicago_edu/EUlgRE8m2nxJtMxA5ZQeqRwBXQQXCMkoZYVGOZHXbykkhw" rel="noopener noreferrer" id="OLK_Beautified_254e3061-173f-ec44-74f1-0366becbc8eb" class="OWAAutoLink eScj0 none" data-ogsc="" style="padding: 0px 1px; border-radius: 2px; user-select: all; background-color: rgb(244, 244, 244);"><img class="suRDx" alt="" role="presentation" style="width: 16px; height: 16px; vertical-align: middle; padding: 1px 2px 2px 0px;" src="https://res-h3.public.cdn.office.net/assets/mail/file-icon/png/pdf_16x16.png">2303.06717
2.pdf</a></span></span><br>
</div>
<div style="text-align: left; margin: 0px;"><span style="font-family: Calibri, Helvetica, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);"><br>
<br>
</span></div>
</div>
</body>
</html>