# abc and Faster Computation of Isotopy Type

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Suppose f is an n-variate polynomial with integer coefficients and degree d. Many natural computational problems involving the real zero set Z of f are algorithmically difficult. For instance, no known algorithm for computing the number of connected components of Z has complexity polynomial in n+d. Furthermore, no known general algorithm for deciding whether f has root over the p-adic has sub-exponential complexity. So it is worthwhile to seek families of polynomials where these questions are tractable.

Assuming f has n+2 monomial terms, and its exponent vectors do not all lie on an affine hyperplane, we prove that the isotopy type of Z can be determined in time polynomial in log d, for any fixed n. (This family of polynomials --- polynomials supported on circuits ---is highly non-trivial, since it already includes Weierstrass normal forms and several important examples from semi-definite programming.) We also show that a 1998 refinement of the abc-Conjecture (by Baker) implies that our algorithm is polynomial in n as well. Furthermore, the original abc-Conjecture implies that p-adic rational roots for f can be detected in the complexity class NP.

These results were obtained in collaboration with Kaitlyn Phillipson and Daqing Wan.

# Sideways Katz-Sarnak and motivic random variables

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A fundamental observation in Katz-Sarnak's study of the zero spacing of L-functions is that Frobenius conjugacy classes in suitable families of varieties over finite fields approximate infinite random matrix statistics. For example, the normalized Frobenius conjugacy classes of smooth plane curves of degree d over F_q approach the Gaussian symplectic ensemble as we take first q to infinity, then d to infinity. In this talk, we explain a sideways version of this result where the limits in d and q are exchanged, and give a Hodge theoretic analog in characteristic zero.

# Southern California Number Theory Day

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# Unexpected quadratic points on random hyperelliptic curves

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On a hyperelliptic curve over the rationals, there are infinitely many points defined over quadratic fields: just pull back rational points of the projective line through the degree two map. But for a positive proportion of genus g odd hyperelliptic curves, we show there can be at most 24 quadratic points not arising in this way. The proof uses tropical geometry work of Park, as well as results of Bhargava and Gross on average ranks of hyperelliptic Jacobians. This is joint work with Jackson Morrow.

# Analytic aspects in the evalution of the multiple zeta and multiple Hurwitz zeta values

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In this talk, we shall discuss about some new results in the evaluation of some multiple zeta values (MZV). After a careful introduction of the multiple zeta values (Euler-Zagier sums) we point out some conjectures back in the early days of MZV and their combinatorial aspects.

At the core of our talk, we focus on Zagier's formula for the multiple zeta values, $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ and its connections to Brown's proofs of the conjecture on the Hoffman basis and the zig-zag conjecture of Broadhurst in quantum field theory. Zagier's formula is a remarkable example of both strength and the limits of the motivic formalism used by Brown in proving Hoffman's conjecture where the motivic argument does not give us a precise value for the special multiple zeta values $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ as rational linear combinations of products $\zeta(m)\pi^{2n}$ with $m$ odd.

# On a sumset conjecture of Erdos

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Erdos conjectured that a set of natural numbers of positive lower density contains the sum of two infinite sets. In this talk I will describe progress on the conjecture. In particular, I will discuss the truth of the conjecture in the “high density” case and how this implies a “1-shift” version of the conjecture in general. These aforementioned results use nonstandard analysis. Time permitting, I will also discuss the conjecture in model-theoretically tame contexts.

# Existence of certain Weierstrass semigroups

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To any point **p** on a smooth algebraic curve **C**, the Weierstass semigroup is the set of all possible pole orders at **p** of regular functions on **C** \ {**p**}. The question of which sets of integers arise as Weierstass semigroups is a very old question, still widely open. We will describe progress on the question, defining a quantity called the effective weight of a numerical semigroup, and describe a proof that all numerical semigroups of sufficiently small effective weight arise as Weierstrass semigroups. The proof is based on older work of Eisenbud, Harris, and Komeda, based on deformation of certain nodal curves. We will survey some combinatorial aspects of the effective weight, and various open questions regarding both numerical semigroups and algebraic curves.

# On the Gross-Rubin-Stark conjecture

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A special case of the GRS Conjecture predicts a surprising link between values of derivatives of p-adic and global L-functions. Recently, Dasgupta-Kakde-Ventullo have used Hida families of modular forms to make progress towards the proof of a rational form of this special case. In this lecture I will report on an independent approach and progress towards the integral GRS conjecture, building upon my joint work with Greither in equivariant Iwasawa theory.

# Slopes of modular forms and the ghost conjecture

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The topic of this talk will be understanding the p-adic slopes of modular forms. Recently, Bergdall and Pollack, based on computer calculations, raised a very interesting conjecture on the slopes of overconvergent modular forms, which predicts that the Newton polygons of the characteristic power series of U_p are the same as the Newton polygons of another explicit characteristic power series, which they call ghost series. This conjecture would imply many well-known conjectures regarding slopes of modular forms, like Gouvea's conjecture, Gouvea-Mazur conjecture, and etc. The goal of our joint project is to prove this conjecture under some mild hypothesis, and to explore some further application. I will report on the progress so far.