Matilde Marcolli, a mathematician at Caltech, has developed a new mathematical framework for Noam Chomsky's model of language, using ideas from theoretical physics. Chomsky's latest model, the minimalist program, simplifies language into a single computational operation called "merge," which combines elements of a sentence. Marcolli used Hopf algebras, a mathematical tool used in theoretical physics, to describe this merge operation. She worked with Chomsky and Bob Berwick to create a forthcoming book explaining this mathematical approach to language. Marcolli also taught a course and hosted workshops at Caltech on this topic, bringing together experts from various fields. This research supports Chomsky's philosophy that language should be studied using the methods and tools of the physical sciences.
Pham Tiep, a professor at Rutgers University, has made significant advancements by solving two long-standing problems in the representation theory of finite groups, which could enhance our understanding of symmetries and random processes in various scientific fields.
Height Zero Conjecture: Tiep solved the 1955 Height Zero Conjecture posed by Richard Brauer. This proof is published in the Annals of Mathematics and is considered one of the most significant challenges in the representation theory of finite groups.
Deligne-Lusztig Theory: In a second achievement, Tiep addressed a difficult problem in the Deligne-Lusztig theory, a foundational area in the representation theory of finite groups. This breakthrough is detailed in two papers published in Inventiones mathematicae and Annals of Mathematics.
Height Zero Conjecture: Tiep solved the 1955 Height Zero Conjecture posed by Richard Brauer. This proof is published in the Annals of Mathematics and is considered one of the most significant challenges in the representation theory of finite groups.
Deligne-Lusztig Theory: In a second achievement, Tiep addressed a difficult problem in the Deligne-Lusztig theory, a foundational area in the representation theory of finite groups. This breakthrough is detailed in two papers published in Inventiones mathematicae and Annals of Mathematics.
Mathematicians have made a groundbreaking discovery that sheds light on the mysterious world of prime numbers. This breakthrough could lead to new insights into the fundamental building blocks of mathematics.
This article explains the concept of abstraction in neural networks and its connection to generalization. It also discusses how different components in neural networks contribute to abstraction and reveals an interesting duality between abstraction and generalization.
analogue I-K of the Laplacian in potential theory
