[Csdmsemo] Special Number Theory Mathematics Seminars--Wednesday, January 9

[Kristina Phillips]

INSTITUTE FOR ADVANCED STUDY

School of Mathematics

Princeton, NJ 08540

 

Mathematics Seminars

Week of January 7, 2019

 

 

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**All are welcome to this week's Special Number Theory Seminars**

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Wednesday, January 9

 

Mathematics Seminar

Topic:                    Distribution of the integral points on quadrics

Speaker:              Naser Talebi Zadeh Sardari, University of Wisconsin
Madison

Time/Room:       2:00pm - 3:00pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=141782>
http://www.math.ias.edu/seminars/abstract?event=141782

 

Mathematics Seminar

Topic:                    The Sup-norm Problem on $S^3$

Speaker:              Raphael Steiner, Member, School of Mathematics

Time/Room:       3:30pm - 4:30pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=142301>
http://www.math.ias.edu/seminars/abstract?event=142301

 

Mathematics Seminar

Topic:                    Ramanujan complexes and golden gates in PU(3).

Speaker:              Shai Evra, Member, School of Mathematics

Time/Room:       4:30pm - 5:30pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=142307>
http://www.math.ias.edu/seminars/abstract?event=142307

 

1 Distribution of the integral points on quadrics 
   Naser Talebi Zadeh Sardari 



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.

http://www.math.ias.edu/seminars/abstract?event=141782

 



2 The Sup-norm Problem on $S^3$ 
   Raphael Steiner 



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 $\sum_j |\phi_j(z)|^4$.

http://www.math.ias.edu/seminars/abstract?event=142301

 



3 Ramanujan complexes and golden gates in PU(3). 
   Shai Evra 



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. 

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. 

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). 

This is a joint work with Ori Parzanchevski.

http://www.math.ias.edu/seminars/abstract?event=142307

 



IAS Math Seminars Home Page:
http://www.math.ias.edu/seminars

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