In the ring $\mathbb{Q}[x]$ of polynomials with coefficients in the rational numbers, it is interesting to consider the subring of all integer-valued polynomials, i.e. polynomial $p(x)$ such that $p(n)$ is an integer for every integer $n$. This ring is known as the most natural and simple example of a non-Noetherian ring. One may wonder whether this is not just the set of all polynomials with integer coefficients. However, e.g. the polynomial $(x^2+x)/2$ is integer-valued. It turns out that this ring consists of exactly the polynomials with integer coefficients in the basis of binomial coefficients $\binom{x}{n}$. Motivated by the characterization of symmetric monoidal functors between Deligne categories, we examine the set $R_{+}(x)$ of polynomials which have nonnegative integer coefficients in this basis. More precisely, we study the set of values of these polynomials at a fixed number $\alpha$. It turns out that this set has a fascinating algebraic structure, explicitly determined by the $p$-adic roots of the minimal polynomial of $\alpha$, which we will fully describe in this talk. This work is joint with Daniil Kalinov, MIT.
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.
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.
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.
Abstract: Let $C$ be a curve over a finite field and let $\rho$ be a
nontrivial representation of $\pi_1(C)$. By the Weil conjectures, the
Artin $L$-function associated to $\rho$ is a polynomial with algebraic
coefficients. Furthermore, the roots of this polynomial are
$\ell$-adic units for $\ell \neq p$ and have Archemedian absolute
value $\sqrt{q}$. Much less is known about the $p$-adic properties of
these roots, except in the case where the image of $\rho$ has order
$p$. We prove a lower bound on the $p$-adic Newton polygon of the
Artin $L$-function for any representation in terms of local monodromy
decompositions. If time permits, we will discuss how this result
suggests the existence of a category of wild Hodge modules on Riemann
surfaces, whose cohomology is naturally endowed with an irregular
Hodge filtration.
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.
An Artin-Schreier curve is a $\mathbb{Z}/p$-branched cover of the projective line over a field of characteristic $p>0$. A unique aspect of positive characteristic is that there exist flat deformations of a wildly ramified cover that change the number of branch points but fix the genus. In this talk, we introduce the notion of Hurwitz tree. It is a combinatorial-differential object that is endowed with essential degeneration data of a deformation. We then show how the existence of a deformation between two covers with different branching data equates to the presence of a Hurwitz tree with behaviors determined by the branching data. One application of this result is to prove that the moduli space of Artin-Schreier covers of fixed genus g is connected when g is sufficiently large. If time permits, we will describe a generalization of the technique for all cyclic covers and the lifting problem.
We will discuss questions about the geometry of Chenevier's eigenvarieties for automorphic forms on definite unitary groups. For example, we will give bounds on the eigenvalues of the $U_p$ Hecke operator that appear in these eigenvarieties. These bounds generalize ones of Liu-Wan-Xiao for rank 2, which they used to prove the Coleman-Mazur-Buzzard-Kilford conjecture in that setting, to all ranks. If time permits, we will discuss possible avenues for recovering additional information not obtainable from these bounds and coming closer to fully generalizing Liu-Wan-Xiao's results.
In the early 1990s Ribet observed that the classical mod l multiplicity one results for modular curves, which are a consequence of the q-expansion principle, fail to generalize to Shimura curves. Specifically he found examples of Galois representations which occur with multiplicity 2 in the mod l cohomology of a Shimura curve with discriminant pq and level 1.
I will describe a new approach to proving multiplicity statements for Shimura curves, using the Taylor-Wiles-Kisin patching method (which was shown by Diamond to give an alternate proof of multiplicity one in certain cases), as well as specific computations of local Galois deformation rings done by Shotton. This allows us to re-interpret and generalize Ribet's result. I will prove a mod l "multiplicity 2^k" statement in the minimal level case, where k is a number depending only on local Galois theoretic data. This proof also yields additional information the Hecke module structure of the cohomology of a Shimura curve, which among other things has applications to the study of congruence modules.
The torsion order elliptic curves over $\mathbb{Q}$ with prime conductor have been well studied. In particular, we know that for an elliptic curve $E/\mathbb{Q}$ with conductor $p$ a prime, if $p > 37$, then $E$ has either no torsion, or is a Neumann-Setzer curve and has torsion order 2. In this talk we examine similar behavior for elliptic curves of prime conductor defined over totally real number fields.