<div dir="ltr"><div class="gmail_default" style="font-family:georgia,serif;font-size:small"><font color="#000000" face="georgia, serif"><b>When:</b>         Friday, March 24th<b> at <span style="background-color:rgb(255,255,0)">12:30pm CT  </span></b></font><div dir="ltr"><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#000000" face="georgia, serif"> <br><b>Where:</b><b>  </b>      Talk will be given<span style="background-color:rgb(255,255,0)"> </span><span style="background-color:rgb(255,255,0)"><font style="font-weight:bold"><u>live, in-person</u></font><font style="font-weight:bold"> </font></span>at</font></p><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#000000" face="georgia, serif">                       <span>TTIC</span>, 6045 S. Kenwood Avenue</font></p><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#000000" face="georgia, serif">                        5th Floor, Room 530<b>              </b>   </font></p><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#000000" face="georgia, serif"><br></font></p><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font face="georgia, serif" color="#000000"><b><span class="gmail_default"></span>Virtually:</b>     via Panopto (<a href="https://uchicago.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=40cc785c-ef33-484a-a4ad-af3f01413b95" target="_blank">Livestream</a>)</font></p><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#000000" face="georgia, serif"> </font></p><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#000000" face="georgia, serif"><b>Who:</b><b> </b>          Aaron Potechin, University of Chicago</font></p><table border="0" cellspacing="0" cellpadding="0" width="0" style="font-family:Arial,Helvetica,sans-serif;width:0in;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;border-collapse:collapse"><tbody><tr><td style="padding:0in"></td></tr></tbody></table><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#000000" face="georgia, serif"> <span class="gmail_default"></span></font></p><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font face="georgia, serif"><font color="#000000"><b>Title:</b><b> </b>          </font></font><span style="font-family:georgia,serif">Graph Matrices and the Ellipsoid Fitting Conjecture</span></p><p class="MsoNormal" style="font-family:Arial,Helvetica,sans-serif;margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><span style="font-family:georgia,serif"><br></span></p><p class="MsoNormal" style="margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><b>Abstract:   </b><font face="georgia, serif">Graph matrices are a type of matrix whose entries are functions of a random input such as a G(n, 1/2) graph or a random m × n matrix. Graph matrices have entries which are random but are generally not independent, which makes their behavior different than the behavior of random matrices with independent random entries. Graph matrices are crucial for analyzing the sum of squares hierarchy on average case problems and are also useful for analyzing other methods which use higher moments of the input.<br><br>The ellipsoid fitting problem is as follows. Given n random vectors v1, . . . , vn ∈ Rd, is there a matrix X such that X ⪰ 0 and viTMvi = 1 for all i ∈ [n]? Saunderson, Parrilo, and Willsky conjectured that for all ϵ > 0, for all sufficiently large d, the answer to the ellipsoid fitting conjecture is yes with high probability when n ≤ (1−ϵ)d24and no with high probability when n ≥ (1 + ϵ)d24.<br><br>In this talk, I will describe graph matrices and describe how they can be used to resolve the ellipsoid fitting conjecture up to a polylogarithmic factor.</font></p><p class="MsoNormal" style="margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font face="georgia, serif"><br></font></p><p class="MsoNormal" style="margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font face="georgia, serif"><br></font></p><div class="gmail_default"><div dir="ltr"><p class="MsoNormal" style="margin:0in;font-family:Arial,Helvetica,sans-serif;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><b><font color="#000000" face="georgia, serif">*********************************************************************************************</font></b></p><p class="MsoNormal" style="margin:0in 0in 8pt;font-family:Arial,Helvetica,sans-serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;line-height:15.6933px"><font face="georgia, serif" color="#000000"><b>Masks are optional in all common areas. </b><b>Full visitor guidance is available at <a href="http://ttic.edu/visitors" target="_blank">ttic.edu/visitors</a>.</b></font></p><p class="MsoNormal" style="margin:0in 0in 12pt;font-family:Arial,Helvetica,sans-serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;line-height:15.6933px"><b><font color="#000000" face="georgia, serif">*********************************************************************************************</font></b></p><p class="MsoNormal" style="margin:0in;font-family:Arial,Helvetica,sans-serif;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><b><i><font color="#000000" face="georgia, serif">Research at TTIC Seminar Series</font></i></b></p><p class="MsoNormal" style="margin:0in;font-family:Arial,Helvetica,sans-serif;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#000000" face="georgia, serif"> </font></p><p class="MsoNormal" style="margin:0in;font-family:Arial,Helvetica,sans-serif;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#000000" face="georgia, serif">TTIC is hosting a weekly seminar series presenting the research currently underway at the Institute. Every week a different TTIC faculty member will present their research.  The lectures are intended for students seeking research topics and advisors and for the general TTIC and University of Chicago communities interested in hearing what their colleagues are up to.</font></p></div></div><p class="MsoNormal" style="margin:0in;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><font color="#888888"></font></p><div><br style="color:rgb(136,136,136);font-family:Arial,Helvetica,sans-serif;font-size:small;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial"></div><div><br></div></div></div><div><br></div><span>-- </span><br><div dir="ltr" data-smartmail="gmail_signature"><div dir="ltr"><b style="background-color:rgb(255,255,255)"><font color="#3d85c6">Brandie Jones </font></b><div><div><div><font color="#3d85c6"><b><i>Executive </i></b></font><b style="color:rgb(61,133,198)"><i>Administrative Assistant</i></b></div></div><div><span style="background-color:rgb(255,255,255)"><font color="#3d85c6">Toyota Technological Institute</font></span></div><div><span style="background-color:rgb(255,255,255)"><font color="#3d85c6">6045 S. Kenwood Avenue</font></span></div><div><span style="background-color:rgb(255,255,255)"><font color="#3d85c6">Chicago, IL  60637</font></span></div></div><div><span style="background-color:rgb(255,255,255)"><font color="#3d85c6"><a href="http://www.ttic.edu" target="_blank">www.ttic.edu</a> </font></span></div><div><font color="#3d85c6"><div style="background-color:rgb(238,238,238)"><div style="color:rgb(32,33,36)"><span style="color:rgb(61,133,198)">Working Remotely on Tuesdays</span></div></div></font></div></div></div></div>