Logic, particularly propositional and first-order logic, is widely used in AI for representing knowledge and reasoning. It allows AI systems to represent facts, rules, and relationships between concepts.

Optimization techniques, such as gradient descent and backpropagation, are extensively used in machine learning to minimize the error between the model's predictions and the actual data.

Linear Algebra is crucial for deep learning neural networks. It is used to represent neurons, layers, and connections within the network, as well as to perform matrix operations during the forward and backward passes.

AI has made significant contributions to mathematics by automating mathematical proofs, generating new mathematical conjectures, and even solving previously unsolved mathematical problems.

Algorithmic Complexity is the mathematical study of AI algorithms, focusing on their efficiency, time complexity, and resource requirements.

Probability Theory, Fuzzy Logic, and Bayesian Networks are all mathematical concepts used in AI for representing and manipulating uncertain information.

AI has contributed to cryptography by developing new encryption algorithms and also by breaking existing encryption algorithms.

Game Theory, Decision Theory, and Information Theory are all mathematical fields used in AI for planning and decision-making under uncertainty.

AI has made significant contributions to natural language processing, including machine translation, text summarization, and sentiment analysis.

Geometry is used in AI for representing and reasoning about spatial relationships, such as in computer vision and robotics.

AI has revolutionized computer vision, enabling tasks such as object detection, image classification, and facial recognition.

Optimization Theory, Linear Programming, and Integer Programming are all mathematical fields used in AI for optimizing the performance of algorithms.

AI has made significant contributions to robotics, including motion planning, robot control, and navigation.

Time Series Analysis, Markov Chains, and Hidden Markov Models are all mathematical concepts used in AI for representing and reasoning about temporal relationships.

AI has made significant contributions to finance, including stock market prediction, fraud detection, and credit scoring.