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<div dir="ltr" style="text-decoration: none; color: rgb(33, 33, 33);">Dear all,</div>
<div dir="ltr" style="text-decoration: none; color: rgb(33, 33, 33);"><br>
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<div dir="ltr" style="text-decoration: none; color: rgb(33, 33, 33);">Complementing this week's math colloquium announced below, the following talk by the same speaker on Thursday (in the logic seminar) may also be of interest to people on this list.</div>
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<div dir="ltr" style="text-decoration: none; color: rgb(33, 33, 33);">MM</div>
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<div dir="ltr" style="text-decoration: none; color: rgb(33, 33, 33);">------</div>
<div style="text-decoration: none; color: rgb(33, 33, 33);">F. Manners logic seminar title and abstract:<span class="Apple-converted-space"> </span><span dir="ltr" style="line-height: 1.5; font-size: inherit !important; text-decoration-color: rgb(0, 120, 212) !important; color: currentcolor;"><span dir="ltr" style="text-decoration-color:rgb(0, 120, 212) !important">Thursday
5-6pm</span></span><span class="Apple-converted-space"> </span>Eckhart 202</div>
<div style="text-decoration: none; color: rgb(33, 33, 33);">-------</div>
<div style="text-decoration: none; color: rgb(33, 33, 33);"><b>Title: </b>The "true complexity" of linear equations, and elementary Cauchy--Schwarz proofs</div>
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<div style="text-decoration: none; color: rgb(33, 33, 33);"><b>Abstract</b>: Suppose f : G -> R is a function on a finite abelian group. It is common in additive combinatorics to consider multilinear averages of f, such as</div>
<div style="text-decoration: none; color: rgb(33, 33, 33);"> \[ \mathbb{E}_{a,b in G} f(a) f(a+b) f(a+2b); \]</div>
<div style="text-decoration: none; color: rgb(33, 33, 33);"> if f is the indicator function of a set, this counts 3-term arithmetic progressions (a, a+b, a+2b) in the set. It is also very useful to know when inequalities hold between different multilinear
averages: for instance, the expression above is bounded by the fourth root of the average</div>
<div style="text-decoration: none; color: rgb(33, 33, 33);"> \[ \mathbb{E}_{x,r,s} f(x) f(x+r) f(x+s) f(x+r+s) \]</div>
<div style="text-decoration: none; color: rgb(33, 33, 33);">better known as the Gowers $U^2$-norm, and such arguments (while simple to prove, just using Cauchy--Schwarz) play a crucial role in Gowers' proof of Szemer\'edi's theorem and related results.</div>
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<span style="text-decoration: none; color: rgb(33, 33, 33);">For more complicated averages, the following questions appear to become quite hard: does one multilinear average control another, up to some power? If so, can you prove it just using the Cauchy--Schwarz
inequality lots of times?</span><span dir="ltr" style="text-decoration: none; color: rgb(33, 33, 33);"></span><span style="text-decoration: none; color: rgb(33, 33, 33);">I will try to explain how, in some cases of interest, the answers are yes, but finding
the correct sequence of Cauchy--Schwarz inequalities is (currently) outlandishly complicated and has something to do with encoding first-order proofs.</span><br>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif"><b>From:</b> Theory <theory-bounces@mailman.cs.uchicago.edu> on behalf of Alexander Razborov via Theory <Theory@mailman.cs.uchicago.edu><br>
<b>Sent:</b> Monday, February 3, 2025 17:38<br>
<b>To:</b> Theory@mailman.cs.uchicago.edu <Theory@mailman.cs.uchicago.edu><br>
<b>Subject:</b> [Theory] Fwd: Colloquium: Frederick Manners (UCSD)
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<div dir="ltr">This may be of interest to many on this list.<br>
<br>
Begin forwarded message:<br>
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<blockquote type="cite">
<div dir="ltr"><b>From:</b> Audra Herman <audrah1@uchicago.edu><br>
<b>Date:</b> February 3, 2025 at 4:21:14 PM CST<br>
<b>To:</b> razborov@uchicago.edu<br>
<b>Subject:</b> <b>Colloquium: Frederick Manners (UCSD)</b><br>
<b>Reply-To:</b> Audra Herman <audrah1@uchicago.edu><br>
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<span class="tinyMce-placeholder" style="word-break:break-word">Frederick Manners</span></h1>
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<p style="margin:0; margin-bottom:16px">University of California San Diego</p>
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Marton's Polynomial Freiman--Ruzsa conjecture</h2>
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<strong><span class="tinyMce-placeholder" style="word-break:break-word">Abstract:</span></strong></h3>
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<p style="margin:0; letter-spacing:1px">A function $f : \mathbb{F}_2^n \to \mathbb{F}_2^n$ is linear if $f(x+y)=f(x)+f(y)$ for all pairs $(x,y)$.</p>
<p style="margin:0; letter-spacing:1px">Now suppose $f$ is "a little bit linear" -- say, $f(x+y)=f(x)+f(y)$ for a 1% fraction of pairs $(x,y)$. What can you say about $f$? Must it be closely related to an actually linear function? If so, how closely?</p>
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<p style="margin:0; letter-spacing:1px">This question turns out to be equivalent to asking for good quantitative bounds in the Freiman--Ruzsa theorem, a foundational result in additive combinatorics. Marton gave a formulation, equivalent to the statement above,
which she conjectured should have polynomial bounds. I will discuss this conjecture, and its (comparatively) recent proof (joint with Timothy Gowers, Ben Green and Terence Tao).</p>
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<span class="tinyMce-placeholder" style="word-break:break-word">Details:</span></h1>
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<p style="margin:0">February 5th, 2025</p>
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<p style="margin:0"><span style="word-break: break-word; color: rgb(0, 0, 0);"><strong>Eckhart 202</strong></span></p>
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<p style="margin:0"><span style="word-break: break-word; color: rgb(0, 0, 0);"><u><strong>*3:00-4:00PM*</strong></u></span> Central</p>
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<p style="margin:0"><strong>Date: February 5th, 2025</strong></p>
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<p style="margin:0"><strong>Time</strong>: 3:00-4:00 PM Central</p>
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<p style="margin:0"><strong>Location:</strong> <span style="word-break: break-word; color: rgb(0, 0, 0);">
Eckhart 202</span></p>
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