# On the proportion of transverse-free curves

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Given a smooth plane curve C defined over an arbitrary field k, we say that C is transverse-free if it has no transverse lines defined over k. If k is an infinite field, then Bertini's theorem guarantees the existence of a transverse line defined over k, and so the transverse-free condition is interesting only in the case when k is a finite field F_q. After fixing a finite field F_q, we can ask the following question: For each degree d, what is the fraction of degree d transverse-free curves among all the degree d curves? In this talk, we will investigate an asymptotic answer to the question as d tends to infinity. This is joint work with Brian Freidin.

# Mod p Galois representations and Abelian varieties

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# The Structure of the Positive Monoid of Integer-Valued Polynomials Evaluated at an Algebraic Number

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# Cokernels of random matrices and distributions of finite abelian p-groups

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We will discuss distributions on finite abelian p-groups that arise from taking cokernels of families of random p-adic matrices. We will explain the motivation for studying these distributions and will highlight several open questions.

# Commensurators of abelian subgroups in CAT(0) groups

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This talk will introduce work in the area of Geometric Group Theory; no prior background in this area will be assumed. The commensurator of a subgroup H of a group G may be seen as a coarse approximation of the normalizer of H. We consider the situation where H is free abelian and G acts properly on a CAT(0) space, that is, a simply connected space of metric non-positive curvature. The structure of the normalizer of H and its action on the space are well understood in this context. However, the commensurator is more mysterious and it contains subtle information about the action which is not seen by the normalizer. For various classes of CAT(0) spaces we obtain structural results about the commensurator and its relation to the normalizer. In this talk, first I will give background on the commensurator and on CAT(0) spaces and groups, and then I will discuss various geometric tools and constructions used in our approach. This is joint work with Jingyin Huang.

# Waring's Problem via generating functions with nonconstant Fourier coefficients

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We'll begin by discussing the history of certain problems in Additive Number Theory. Several problems in Additive Number Theory ask how many ways we can represent the elements of a set A as a sum of s elements of the set B, with the two main examples being Waring's Problem and Goldbach's Conjecture. The Hardy-Littlewood Circle Method is the main tool for attacking these problems and often leads to asymptotic formulas for the number of representations. We'll introduce a variant of the Circle Method that simplifies the arguments involved in finding bounds for when the asymptotic formula holds in Waring's Problem.

# $p$-adic estimates for Artin L-functions on curves

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# How Quickly Can You Approximate Roots?

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How many digits of an algebraic number A do you need to know before you

are sure you know A? This question dates back to the early 20th century (if

not earlier) and work of Kurt Mahler on the minimal spacing between

complex roots of a degree d univariate polynomial f with integer coefficients of

absolute value at most h: One can bound the minimal spacing explicitly as a function

of d and h. However, the optimality of Mahler's bound for sparse polynomials was open

until recently.

We give a unified family of examples, having just 4 monomial terms, showing

Mahler's bound to be asyptotically optimal over both the p-adic complex numbers,

and the usual complex numbers. However, for polynomials with 3

or fewer terms, we show how to significantly improve Mahler's bound, in both

the p-adic and Archimedean cases. As a consequence, we show how certain

sparse polynomials of degree d can be ``solved'' in time (log d)^{O(1)} over certain local fields.

This is joint work with Yuyu Zhu.