Explanipedia

The Tamely Ramified Geometric Quantitative Minimal Ramification Problem Open

Mark Shusterman · 2022

We prove a large finite field version of the Boston--Markin conjecture on counting Galois extensions of the rational function field with a given Galois group and the smallest possible number of ramified primes. Our proof involves a study o…

Hecke's Theorem on the Different for $3$-Manifolds Open

Will Sawin, Mark Shusterman · 2022

Hecke has shown that the different of an extension of number fields is a square in the class group. We prove an analog for branched covers of closed $3$-manifolds saying that the branch divisor is a square in the first homology group.

Finite presentation of the tame fundamental group Open

Hélène Esnault, Mark Shusterman, Vasudevan Srinivas · 2022

Möbius cancellation on polynomial sequences and the quadratic Bateman-Horn conjecture over function fields Open

Will Sawin, Mark Shusterman · 2020

We establish cancellation in short sums of certain special trace functions over $\mathbb{F}_q[u]$ below the Pólya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $\mathbb{F}_q[u…

The Hanna Neumann conjecture for Demushkin groups Open

Andrei Jaikin‐Zapirain, Mark Shusterman · 2019

Virtual retraction and Howson’s theorem in pro-$p$ groups Open

Mark Shusterman, Pavel Zalesskii · 2019

We show that for every finitely generated closed subgroup $K$ of a non-solvable Demushkin group $G$, there exists an open subgroup $U$ of $G$ containing $K$, and a continuous homomorphism $τ\colon U \to K$ satisfying $τ(k) = k$ for every $…

Irreducibility of polynomials with a large gap Open

William F. Sawin, Mark Shusterman, Michael Stoll · 2019

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > \operatorname{deg}(c) + \operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is irreduc…

Balanced presentations for fundamental groups of curves over finite fields Open

Mark Shusterman · 2018

We show that the algebraic fundamental group of a smooth projective curve over a finite field admits a finite topological presentation where the number of relations does not exceed the number of generators.

The Geometric M\"obius function and quadratic character sums Open

Mark Shusterman · 2018

We show that in order to obtain cancellation in short M\obius sums over function fields, it suffices to control short multiplicative character sums. In conjunction with the Burgess bound, this allows us to obtain M\obius cancellation in ce…

On the Chowla and twin primes conjectures over $\mathbb F_q[T]$ Open

Will Sawin, Mark Shusterman · 2018

Using geometric methods, we improve on the function field version of the Burgess bound, and show that, when restricted to certain special subspaces, the Möbius function over $\mathbb F_q[T]$ can be mimicked by Dirichlet characters. Combini…

Semi-free subgroups of a profinite surface group Open

Matan Ginzburg, Mark Shusterman · 2018

We show that every closed normal subgroup of infinite index in a profinite surface group Γ is contained in a semi-free profinite normal subgroup of Γ. This answers a question of Bary-Soroker, Stevenson, and Zalesskii

Groups with positive rank gradient and their actions Open

Mark Shusterman · 2018

We show that given a finitely generated LERF group G with positive rank gradient, and finitely generated subgroups A , B ≤ G of infinite index, one can find a finite index subgroup B 0 of B such that [ G : 〈 A ∪ B 0 〉] = ∞. This generalize…

Squarefree polynomials with prescribed coefficients Open

Amotz Oppenheim, Mark Shusterman · 2017

Gradients of Sequences of Subgroups in a Direct Product Open

Nikolay Nikolov, Zvi Shem‐Tov, Mark Shusterman · 2017

For a sequence $\{U_n\}_{n = 1}^\infty$ of finite index subgroups of a direct product $G = A \times B$ of finitely generated groups, we show that $$\lim_{n \to \infty} \frac{\min\{|X| : \langle X \rangle = U_n\}}{[G : U_n]} = 0$$ once $[A …

Schreier’s formula for prosupersolvable groups Open

Mark Shusterman · 2016

We classify the finitely generated prosupersolvable groups that satisfy Schreier’s formula for the number of generators of open subgroups.

Ascending chains of finitely generated subgroups Open

Mark Shusterman · 2016

Rank gradient of sequences of subgroups in a direct product Open

Nikolay Nikolov, Zvi Shem‐Tov, Mark Shusterman · 2016

For a sequence $\{U_n\}_{n = 1}^\infty$ of finite index subgroups of a direct product $G = A \times B$ of finitely generated groups, we show that $$\lim_{n \to \infty} \frac{\min\{|X| : \langle X \rangle = U_n\}}{[G : U_n]} = 0$$ once $[A …

An axiomatizable profinite group with infinitely many open subgroups of index 2 Open

Or Ben Porath, Mark Shusterman · 2016

We show that a profinite group with the same first-order theory as the direct product over all odd primes p of the dihedral group of order 2 ⁢ p {2p} , is necessarily isomorphic to this direct product.

Free subgroups of finitely generated free profinite groups Open

Mark Shusterman · 2016

We give new and improved results on the freeness of subgroups of free\nprofinite groups: A subgroup containing the normal closure of a finite word in\nthe elements of a basis is free; Every infinite index subgroup of a finitely\ngenerated …

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