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DTSTART:20260308T070000
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DTSTART;TZID=America/Toronto:20260805T140000
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URL:https://uwaterloo.ca/combinatorics-and-optimization/events/masters-thes
 is-presentation-martin-liu
SUMMARY:Master's Thesis Presentation - Martin Liu
CLASS:PUBLIC
DESCRIPTION:SPEAKER:\n Martin Li\n\nSUPERVISOR(S):\n Vijay Bhattiprolu\n\nC
 OMITTEE:\n Jonathan Leake\, Levent Tuncel\n\nLOCATION:\n MC 6483\n\nABSTRA
 CT:\n\nThe $d$-dimensional Grothendieck constant is the smallest constant 
 $K$\nsuch that \\begin{align*} \\sup\\left\\{\\sum_{i\,j=1}^n A_{ij}\\lang
 le\nu_i\,v_j\\rangle:u_i\,v_j\\in S^{d-1}\\right\\}\\le\nK\\cdot\\sup\\lef
 t\\{\\sum_{i\,j=1}^n\nA_{ij}x_iy_j:x_i\,y_j\\in\\{-1\,1\\}\\right\\} \\en
 d{align*}for any\n$n\\in\\mathbb{N}$ and any real $n\\times n$ matrix $A$.
  The inequality\nabove\, called the Grothendieck inequality\, has made a d
 eep impact in a\nvariety of areas such as functional analysis\, quantum in
 formation\ntheory\, and optimization. Determining the $d$-dimensional Grot
 hendieck\nconstant for any $d\\ge 3$ is a long-standing open problem.\n\nI
 n this paper\, we propose a worst operator in dimension 3\, whose\n$\\inft
 y\\to 1$ norm is conjectured to be $1/K_G(3)$. We study a related\nclass o
 f operators with nice geometric interpretations\, and we prove\nthe functi
 on $f:S^{d-1}\\to\\{-1\,1\\}$ corresponding to a hyperplane is\nuniquely o
 ptimal for this class\, with the isoperimetric inequality\nlying at the he
 art of our proof.
DTSTAMP:20260804T153541Z
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