Related papers: An Introduction to Motivic Integration
We present modular implicits, an extension to the OCaml language for ad-hoc polymorphism inspired by Scala implicits and modular type classes. Modular implicits are based on type-directed implicit module parameters, and elaborate…
The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals.…
Engaging students in teaching foundational Computer Science concepts is vital for the student's continual success in more advanced topics in the field. An idea of a series of Jupyter notebooks was conceived as a way of using Bloom's…
This is a short introduction to categories with some emphasis on coalgebras. We start from introducing basic notions (categories, functors, natural transformations), move to Kleisli tripels and monads, with a short discussion of monads in…
We put forward in this paper a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related…
Modular exponentiation is crucial to number theory and cryptography, yet remains largely unexplored from a mechanistic interpretability standpoint. We train a 4-layer encoder-decoder Transformer model to perform this operation and…
Lecture-notes: introducing and discussing basic kp concepts related to bands interaction at an elementary level to beginners
In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure $L$ of a finite field with coefficients in a global function field over this finite…
Considering a (co)homology theory $\mathbb{T}$ on a base category $\mathcal{C}$ as a fragment of a first-order logical theory we here construct an abelian category $\mathcal{A}[\mathbb{T}]$ which is universal with respect to models of…
This work is dedicated to the study of the discrete version of the Maier-Saupe model in the presence of competing interactions. The competition between interactions favoring different orientational ordering produces a rich phase diagram…
Reasoning, as an essential ability for complex problem-solving, can provide back-end support for various real-world applications, such as medical diagnosis, negotiation, etc. This paper provides a comprehensive survey of cutting-edge…
This text is an introduction to the study of NIP (or dependent) theories. It is meant to serve two purposes. The first is to present various aspects of NIP theories and give the reader the background material needed to understand almost any…
These notes are based on lecture courses I gave to third year mathematics students at Cambridge. They could form a basis of an elementary one--term lecture course on integrable systems covering the Arnold-Liouville theorem, inverse…
These are the notes of the three lectures I delivered at the mini-workshop "Knot Theory and Number Theory around the A-Polynomial" at the Instituto Superior Tecnico (IST) in Lisbon in January 2014. The goal of the lectures was to…
Inspired from human cognition, machine learning systems are gradually revealing advantages of sparser and more modular architectures. Recent work demonstrates that not only do some modular architectures generalize well, but they also lead…
We propose an action of a certain motivic cohomology group on the coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo $p$ and…
Interpretable machine learning offers insights into what factors drive a certain prediction of a black-box system. A large number of interpreting methods focus on identifying explanatory input features, which generally fall into two main…
Imitation learning often assumes that demonstrations are close to optimal according to some fixed, but unknown, cost function. However, according to satisficing theory, humans often choose acceptable behavior based on their personal (and…
The aim of this text is to provide a linguistically accessible, but comprehensive introduction into a variety of topics in dynamical systems and its applications. Whilst preliminary knowledge of dynamical systems is useful, it is not…
Can language models (LMs) learn to faithfully describe their internal computations? Are they better able to describe themselves than other models? We study the extent to which LMs' privileged access to their own internals can be leveraged…