algebra-and-number-theory

For any irrational $x \in \mathbb{R}$ we define its Lagrange number to be the supremum of real numbers $c$ such that $$ \displaystyle{ \left| \frac{p}{q} - x \right| < \frac{1}{cq^2} }$$ has ...

∞-Lie theory (higher geometry) Background Smooth structure Higher groupoids Lie theory ∞-Lie groupoids ∞-Lie algebroids Formal Lie groupoids Cohomology Homotopy Related topics Examples -Lie groupoids -Lie groups -Lie algebroids -Lie algebras A Lie groupoid is a groupoid internal to smooth manifolds. This is a joint generalization of smooth manifolds and Lie groups to higher differential geometry.…

I am currently learning about finite fields and have encountered several problems that I do not know where to begin. Let $\mathbb{F}_{q^n}$ be the finite extension field of $\mathbb{F}_q$ of degree ...

Let $R$ be a ${\rm PID}$ with fraction field $K$, and let $c\in M_n(R)$ be a square matrix. Suppose that $c$ is similar to a Jordan matrix $J_n$ with eigenvalues $0$ over $K$. Can we find an element ...

Let $R_1, \ldots, R_n$ be rings (fields would also suffice). Under which conditions is every automorphism on the product ring $R = R_1\times\ldots\times R_n$ always induced by individual automorphisms ...

An algebraic pattern is a blueprint for a notion of functors on a fixed category satisfying a Segal condition, suitable for formalizing homotopy-coherent algebra in the Cartesian setting. An algebraic pattern is an (∞,1)-category together with the following data: a pair of wide subcategories , whose morphisms are called inert and active morphisms, and a full subcategory , whose objects are called…

Suppose $G$ is a split connected reductive group over a field $k$ with Weyl group $W$ and root system $\Phi$, with positive roots $\Phi_+$ corresponding to a Borel $B$ with unipotent radical $N$. If ...

We attach Hecke polynomials [Formula: see text]([Formula: see text]) to weak Hecke eigenforms [Formula: see text] of weight 2 – [Formula: see text] and show that, for large [Formula: see text], every zero is simple and lies in [0, 1728]. The construction pulls back a weakly holomorphic Hecke combination of [Formula: see text] along [Formula: see text]; the analysis follows Hecke orbits on the uni…

Recently, Andrews and Hopkins provided a partial extension of the Crank-Mex theorem by refining the equinumerosity among partitions with even mex, with a fixed point, with negative crank and with positive crank in terms of the number of parts greater than one. In this work, we first give a further refinement of Andrews and Hopkins' result by including the 2-measure and the size of the Durfee squa…

Let [Formula: see text] be a [Formula: see text]-torsion free unital *-ring, possessing a non-trivial symmetric idempotent. In the present paper, we discuss that, under certain mild conditions on [Formula: see text], a map [Formula: see text] : [Formula: see text] (not necessarily additive) satisfies [Formula: see text] for all [Formula: see text] if and only if it is an additive *-derivation. Fu…

We consider sequences counting integer partitions in two colors (red and blue) in which the even parts occur only in blue color. We focus on subsequences defined by constraints on the parity and color of the summands. We establish formulas for our sequences and deduce identities of integer partitions.

We prove Conjecture 10 of Cai in A Modern Introduction to Classical Number Theory, World Scientific Publishing Co. Pte., Singapore (2021).

On proof of the (categorical) geometric Langlands conjecture: Dennis Gaitsgory, Sam Raskin: Proof of the geometric Langlands conjecture I: construction of the functor [arXiv:2405.03599, pdf] Dima Arinkin, D. Beraldo, Justin Campbell, L. Chen, Dennis Gaitsgory, J. Faergeman, Kevin Lin, Sam Raskin, Nick Rozenblyum: Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE […

Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts abstract duality: opposite category, concrete duality: dual object, dualizable object, fully dualizable object, dualizing object Examples between higher geometry/higher algebra Langlands duality, geometric Langlands duality, quantum geometric Langlan…

Let $X,Y$ be $G$ spaces, where $G$ is a compact Lie group. Is there a Kunneth (isomorphism) formula for Bredon cohomology with the constant rational Mackey functor $\underline{\mathbb{Q}}$ given by ...

I am writing GAP code to compute conjugacy classes in Galois groups of certain cyclotomic extensions. Here is a simple mock version of the kind of thing I am talking about: galoisGroup := function(n) ...

In Needham's book on Visual Diffential Geometry in chapter 39 exercise 4 he gives a hint: "Recall that each matrix is itself a tensor of valence {1 1}..." I don't understand how a tensor of ...

Objective Given a reduced fraction whose denominator is positive and odd, represent the fraction as a 2-adic integer. Introduction to 2-adic integers Informally Informally, a 2-adic integer is an ...