<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body dir="auto"><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><b><span style="font-size: 14pt; font-family: Helvetica;">Maya Sankar</span></b><o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><b><span style="font-size: 14pt; font-family: Helvetica;">Stanford University</span></b><o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"> <o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 11pt;"> <img width="219" height="219" id="Picture_x0020_1" src="cid:4E38D9E8-3D9F-42D8-9BA5-8F91608CB7A0" alt="image001.jpg" style="width: 2.2812in; height: 2.2812in;"></span><o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 11pt;"> </span> <o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><b><span style="font-size: 11pt;"><span dir="ltr">Tuesday, </span></span></b><b><span style="font-size: 11pt;"><span dir="ltr">January 14</span><span dir="ltr">, 202</span><span dir="ltr">5</span><span dir="ltr"> at 3:30pm</span></span></b><o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><b><span style="font-size: 11pt; background: yellow;">Kent Chemical Laboratory, Room 120</span></b><o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 11pt;"> </span><o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 11pt;"> </span><o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt;"> <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt;"> </span><b><i><span style="font-size: 12pt; color: rgb(33, 33, 33);">Title</span></i></b><b><span style="font-size: 12pt; color: rgb(33, 33, 33);">: </span></b><span style="font-size: 12pt; color: rgb(33, 33, 33);">The Turán Density of 4-Uniform Tight Cycles<o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; color: rgb(33, 33, 33);"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><b><i><span style="font-size: 12pt; color: rgb(33, 33, 33);">Abstract:</span></i></b><span style="font-size: 12pt; color: rgb(33, 33, 33);"> For any uniformity r and residue k modulo r, we give an exact characterization of the r-uniform hypergraphs that homomorphically avoid tight cycles of length k modulo r, in terms of colorings of (r-1)-tuples of vertices. This generalizes the result that a graph avoids all odd closed walks if and only if it is bipartite, as well as a result of Kamčev, Letzter, and Pokrovskiy in uniformity 3. In fact, our characterization applies to a much larger class of families than those of the form C_k^(r)={r-uniform tight cycles of length k modulo r}.<o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; color: rgb(33, 33, 33);"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; color: rgb(33, 33, 33);">We also outline a general strategy to prove that, if C is a family of tight-cycle-like hypergraphs (including but not limited to the families C_k^(r)) for which the above characterization applies, then all sufficiently long cycles in C will have the same Turán density. We demonstrate an application of this framework, proving that there exists an integer L_0 such that for every L>L_0 not divisible by 4, the tight cycle C^(4)_L has Turán density 1/2.<o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; color: rgb(33, 33, 33);"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; color: rgb(33, 33, 33);"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; color: rgb(33, 33, 33);"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; margin: 0in; font-size: 10pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; color: rgb(33, 33, 33);">Bio: Maya is a 5th-year PhD student at Stanford University, advised by Jacob Fox. Her research interests include extremal graph theory and topological combinatorics, with a current focus on understanding the Turán numbers of various families of hypergraphs. Outside of math, she enjoys running, puzzles, and mathematical knitting.</span></p><div dir="ltr"></div></body></html>