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<div style="direction: ltr;font-family: Tahoma;color: #000000;font-size: 10pt;">Dear Mel,
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<div>Please post the following:</div>
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<div>Comb/Prob Seminar: Feb 5, 4-5pm, LOM 215. Speaker: A. Lubotzky (Yale) </div>
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<div><span class="Apple-style-span" style="font-family: 'Segoe UI', Helvetica, Arial, sans-serif; font-size: medium; "><font size="1"><span style="font-size: 12.79px; ">title: </span></font><font size="1"><span style="font-size: 12.79px; "> Quantum error correcting
codes and </span></font><font size="1"><span style="font-size: 12.79px; ">4-dimensional hyperbolic manifold</span></font><font size="1"><span style="font-size: 12.79px; ">d</span></font><font size="1"><span style="font-size: 12.79px; ">s an</span></font><br>
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<font size="1"><span style="font-size: 12.79px; "> Abstract:</span></font>
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<div><font size="1"><span style="font-size: 12.79px; "> It is well known that there exist LDPC good error correcting codes (this was proved by Gallager using random methods while explicit constructions were given by Sipser and Spielman ).</span></font></div>
<div><font size="1"><span style="font-size: 12.79px; "> The analogous problem for quantum error correcting codes, in spite being an elementary Z/2Z- linear algebra problem, is still open.</span></font></div>
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<div><font size="1"><span style="font-size: 12.79px; "> Simplical complexes and their homology/cohomology give rise to LDPC quantum error correcting codes ( QECC). But all known examples fail to be "good". </span></font></div>
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<div><font size="1"><span style="font-size: 12.79px; "> In a joint work with Larry Guth ( </span></font><a href="https://urldefense.proofpoint.com/v2/url?u=http-3A__aip-2Dinfo.org_1XPS-2D33V7F-2DC9RS5X-2D1FC1OV-2D1_c.aspx&d=AwMFaQ&c=-dg2m7zWuuDZ0MUcV7Sdqw&r=sJ2GIybLuYHtneA1hCAmEw&m=w32UC_9MjSnqQjStzd12EyeKEZgHaCwxnneTCPYd5EA&s=dN1SqD5y_LxjsxjsatzKmfNrHeB9dj2tAdt_SjL3LLU&e=" target="_blank">J.
Math. Phys.<b> 55</b>, 082202 (2014)</a><font size="1"><span style="font-size: 12.79px; "> ), we constructed a</span></font><font size="1"><span style="font-size: 12.79px; "> family of LDPC QECC out of congruence quotients of the 4- dimensional hyperbolic
space. </span></font><font size="1"><span style="font-size: 12.79px; "> Using methods of systolic geometry over Z/2Z, we evaluate the parameters of these codes and disprove a conjecture of Ze'mor who predicted</span></font><font size="1"><span style="font-size: 12.79px; "> </span></font><font size="1"><span style="font-size: 12.79px; "> that
such homological QECC do not exist.</span></font><font size="1"><span style="font-size: 12.79px; "> The existence of LDPC good QECC is still open.</span></font></div>
<div><font size="1"><span style="font-size: 12.79px; "> </span></font></div>
<div><font size="1"><span style="font-size: 12.79px; "> All notions will be defined and explained.</span></font></div>
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<div><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 13px;">------------------------</span></font></div>
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<div><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 13px;">Feb 12, 4-5pm, LOM 215 Speaker M. Krivelevich (Tel Aviv) </span></font></div>
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<div><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 13px;">Title: The Phase Transition in Site Percolation on Pseudo-Random Graphs<br>
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ABSTRACT<br>
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We establish the existence of the phase transition in site percolation<br>
on pseudo-random d-regular graphs. Let G=(V,E) be an (n,d,lambda)-graph,<br>
that is, a d-regular graph on n vertices in which all eigenvalues of<br>
the adjacency matrix, but the first one, are at most lambda in their<br>
absolute values. Form a random subset R of V by putting every vertex<br>
v in V into R independently with probability p. Then for any small<br>
enough constant epsilon>0, if p=(1-epsilon)/d, then with high<br>
probability all connected components of the subgraph of G induced by R<br>
are of size at most logarithmic in n, while for p=(1+epsilon)/d, if<br>
the eigenvalue ratio lambda/d is small enough as a function of epsilon,<br>
then typically R contains a connected component of size at least<br>
epsilon n/d and a path of length proportional to epsilon^2n/d.</span></font></div>
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