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 bgcolor="#F0A0A0" align="center" style="fontsize:125%"  '''Alex Smith'''   bgcolor="#F0A0A0" align="center" style="fontsize:125%"  '''Alex Smith'''  
    
 bgcolor="#BCD2EE" align="center"    bgcolor="#BCD2EE" align="center"  Statistics for 8class groups and 4Selmer groups  
    
 bgcolor="#BCD2EE"    bgcolor="#BCD2EE"  Assuming the grand Riemann hypothesis, we verify that the set of quadratic imaginary fields has the distribution of 8class groups predicted by the CohenLenstra heuristic. To do this we prove that, in families of quadratic fields parameterized by a single prime p, the 8class rank is determined by the Artin symbol of p in a certain extension of the rationals. Using Chebotarev's density theorem, we find that the distribution of 8class ranks in most of these small families is given by the CohenLenstra heuristic. We can bundle these small families together to get the full result, with GRH necessary to control error bounds in this process. By analogous means, we also find the distribution of 4Selmer groups in the quadratic twist family of an elliptic curve with full 2torsion.  
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Revision as of 02:37, 20 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 the 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 the 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 
Statistics for 8class groups and 4Selmer groups 
Assuming the grand Riemann hypothesis, we verify that the set of quadratic imaginary fields has the distribution of 8class groups predicted by the CohenLenstra heuristic. To do this we prove that, in families of quadratic fields parameterized by a single prime p, the 8class rank is determined by the Artin symbol of p in a certain extension of the rationals. Using Chebotarev's density theorem, we find that the distribution of 8class ranks in most of these small families is given by the CohenLenstra heuristic. We can bundle these small families together to get the full result, with GRH necessary to control error bounds in this process. By analogous means, we also find the distribution of 4Selmer groups in the quadratic twist family of an elliptic curve with full 2torsion. 
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
