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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" vlink="#954F72"><div class=WordSection1><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>INSTITUTE FOR ADVANCED STUDY<o:p></o:p></p><p class=MsoNormal>School of Mathematics<o:p></o:p></p><p class=MsoNormal>Princeton, NJ 08540<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Mathematics Seminars<o:p></o:p></p><p class=MsoNormal>Week of January 7, 2019<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>--------------<o:p></o:p></p><p class=MsoNormal><b>**All are welcome to this week’s Special Number Theory Seminars**<o:p></o:p></b></p><p class=MsoNormal>--------------<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><b>Wednesday, January 9<o:p></o:p></b></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Mathematics Seminar<o:p></o:p></p><p class=MsoNormal>Topic: Distribution of the integral points on quadrics<o:p></o:p></p><p class=MsoNormal>Speaker: Naser Talebi Zadeh Sardari, University of Wisconsin Madison<o:p></o:p></p><p class=MsoNormal>Time/Room: 2:00pm - 3:00pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=141782"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=141782</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>Mathematics Seminar<o:p></o:p></p><p class=MsoNormal>Topic: The Sup-norm Problem on $S^3$<o:p></o:p></p><p class=MsoNormal>Speaker: Raphael Steiner, Member, School of Mathematics<o:p></o:p></p><p class=MsoNormal>Time/Room: 3:30pm - 4:30pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=142301"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=142301</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>Mathematics Seminar<o:p></o:p></p><p class=MsoNormal>Topic: Ramanujan complexes and golden gates in PU(3).<o:p></o:p></p><p class=MsoNormal>Speaker: Shai Evra, Member, School of Mathematics<o:p></o:p></p><p class=MsoNormal>Time/Room: 4:30pm - 5:30pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=142307"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=142307</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'>1 Distribution of the integral points on quadrics <br> Naser Talebi Zadeh Sardari <br><br><o:p></o:p></p><p class=MsoNormal>Motivated by questions in computer science, we consider the problem of approximating local points (real or p-adic points) on the unit sphere S^d optimally by the projection of the integral points lying on R*S^d, where R^2 is an integer. We present our numerical results which show the diophantine exponent of local point on the sphere is inside the interval [1, 2-2/d]. By using the Kloosterman's circle method, we show that the diophantine exponent is less than 2-2/d for every d>3. By using the theta-lift and Ramanujan bound on the Fourier coefficients of the holomorphic modular forms we prove that the diophantine exponent is 1+o(1) for almost all local points and odd d>=3 and even d>=2 by assuming R is an integer. This generalizes the result of Sarnak for d=3 to higher dimensions.<o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=141782">http://www.math.ias.edu/seminars/abstract?event=141782</a><o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal style='margin-bottom:12.0pt'><br><br>2 The Sup-norm Problem on $S^3$ <br> Raphael Steiner <br><br><o:p></o:p></p><p class=MsoNormal>We consider the problem of bounding the sup-norm of $L^2$-normalised Hecke-Laplace eigenforms $\phi_j$ on $S^3$. Along the way, we overcome the difficulty of possibly small eigenvalues in the Iwaniec-Sarnak amplifier by taking a whole space of amplifiers, given by the theta-series, on which we may use Parseval. As a consequence, we can even say something about the fourth moment $\sum_j |\phi_j(z)|^4$.<o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=142301">http://www.math.ias.edu/seminars/abstract?event=142301</a><o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal style='margin-bottom:12.0pt'><br><br>3 Ramanujan complexes and golden gates in PU(3). <br> Shai Evra <br><br><o:p></o:p></p><p class=MsoNormal>In their seminal works from the 80's, Lubotzky, Phillips and Sarnak proved the following two results: (i) An explicit construction of Ramanujan regular graphs. (ii) An explicit method of placing points on the sphere uniformly equidistributed. These two seemingly unrelated problems, were solved by applying deep number theoretic Theorems (Deligne proof of the Ramanujan conjecture, Jacobi four square Theorem) on a single group form of PGL_2 over the field of rationals. <br><br>In recent years these two results have seen the following generalizations and developments: (i+) The explicit construction of Ramanujan complexes by Lubotzky, Samuels and Vishne. (ii+) The explicit construction of super golden gates for PU(2) by Parzanchevski and Sarnak. This time, the two results are unrelated, since the construction of LSV is over a field of positive characteristic and not over the rationals. <br><br>In this talk I will describe a recent new construction of both golden gates for PU(3) as well as explicit constructions of new Ramanujan complexes. Moreover, we shall see that these constructions are actually 'local' consequences coming from analyzing a single 'global' group (just like in LPS). <br><br>This is a joint work with Ori Parzanchevski.<o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=142307">http://www.math.ias.edu/seminars/abstract?event=142307</a><o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><br><br>IAS Math Seminars Home Page:<br><a href="http://www.math.ias.edu/seminars">http://www.math.ias.edu/seminars</a><o:p></o:p></p></div></body></html>