View source for NTS ABSTRACTSpring2019

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== Jan 23 ==

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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yunqing Tang '''
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| bgcolor="#BCD2EE"  align="center" | Reductions of abelian surfaces over global function fields
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| bgcolor="#BCD2EE"  | For a non-isotrivial ordinary abelian surface $A$ over a global function field, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous to the product of two elliptic curves. This result can be viewed as a generalization of a theorem of Chai and Oort. This is joint work with Davesh Maulik and Ananth Shankar.

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== Jan 24 ==

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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hassan-Mao-Smith--Zhu'''
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| bgcolor="#BCD2EE"  align="center" | The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of  $S^{d-2}\subset S^d$
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| bgcolor="#BCD2EE"  | Abstract: Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj},  $d\geq 3,$ and $q$ and $p$ prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$  point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the  diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$. 

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== March 28 ==

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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevitch'''
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| bgcolor="#BCD2EE"  align="center" | Harmonic Analysis on GLn over finite fields
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| bgcolor="#BCD2EE"  | Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters.
For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}:
$$trace (\rho(g))/dim (\rho),$$
for an irreducible representation $\rho$ of G and an element g of G.  For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio.  We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (TAMU).
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