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 bgcolor="#BCD2EE" align="center"  Discrete Log problem for the algebraic group PGL_2.   bgcolor="#BCD2EE" align="center"  Discrete Log problem for the algebraic group PGL_2.  
    
 bgcolor="#BCD2EE"    bgcolor="#BCD2EE"  We consider the problem of finding a shortest path between a given pair of vertices in the LPS Ramanujan graphs $X_{p,q}$ where $p$ is a fixed prime number and $q$ is an integer. We give a polynomial time algorithm in $\log(q)$ which returns a shortest path between two diagonal vertices under a standard conjecture on the distribution of integers representable as sum of two squares and assuming one can factor quickly. Numerically, for a typical pair of vertices corresponded to diagonal elements the minimal path has a length about $3\log(q)+ O(1)$ while provably, there are pairs of points with distance at least $4\log(q)+ O(1)$ . For a general pair of vertices, we write it as a product of three Euler angels and as a result for a typical pair we find a path with distance $9 \log(q)+ O(1)$.  
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Revision as of 20:05, 14 September 2016
Return to NTS Spring 2016
Sep 8
Arunabha Biswas 
Limiting values of higher Mahler Measure and cyclotomic polynomials. 
We consider the khigher Mahler measure m_k(P) of a Laurent polynomial P as the integral of log^k P over the complex unit circle. In number theory, Lehmer's conjecture and the appearance of higher Mahler measures in Lfunctions are the main sources of motivation for studying various properties of m_k(P). Beyond number theory, Mahler measure has connections with topological entropies of dynamical systems and polynomial knot invariants. In this talk I shall present (1) an explicit formula for the value of m_k(P)/k! as k approaches infinity, (2) some asymptotic results regarding m_k(P) and (3) a scheme to approximate special values of a class of Lfunctions. 
Sep 15
Naser T. Sardari 
Discrete Log problem for the algebraic group PGL_2. 
We consider the problem of finding a shortest path between a given pair of vertices in the LPS Ramanujan graphs $X_{p,q}$ where $p$ is a fixed prime number and $q$ is an integer. We give a polynomial time algorithm in $\log(q)$ which returns a shortest path between two diagonal vertices under a standard conjecture on the distribution of integers representable as sum of two squares and assuming one can factor quickly. Numerically, for a typical pair of vertices corresponded to diagonal elements the minimal path has a length about $3\log(q)+ O(1)$ while provably, there are pairs of points with distance at least $4\log(q)+ O(1)$ . For a general pair of vertices, we write it as a product of three Euler angels and as a result for a typical pair we find a path with distance $9 \log(q)+ O(1)$.

Sep 22
Alex Smith 
title coming soon 
abstract coming soon 
Sep 29
Steve Lester 
title coming soon 
abstract coming soon 
Oct 6
Nicole Looper 
Arboreal Galois representations of higher degree polynomials and Odoni's Conjecture 
Since the mid1980s, when the study of arboreal Galois representations first began, most results have concerned the representations induced by quadratic rational maps. In the higher degree case, by contrast, very little has been known. I will discuss some recent results pertaining to higher degree polynomials over number fields. This will include a partial solution to a conjecture made by R.W.K. Odoni in 1985. 
Oct 13
Ling Long 
title coming soon 
abstract coming soon 
Oct 20
Jack Klys 
title coming soon 
abstract coming soon 
Oct 27
William Duke 
Nov 3
Nov 10
Nov 17
Dec 1
Dec 8
Dec 15
Efrat Bank 
Primes in short intervals on curves over finite fields 
abstract coming soon
