Number theory studies the integers and mathematical objects constructed from them. Carl Friedrich Gauss once said, "Mathematics is the queen of the sciences, and number theory is the queen of mathematics". If you've ever heard of a prominent theorem or conjecture in popular media, chances are it is from number theory, whether it be Fermat's Last Theorem, the Riemann Hypothesis, or Goldbach's Conjecture. Number theory has many prominent applications in computer science such as cryptography and error correcting codes.
Algebra studies mathematical structures that behave like but are not limited to the integers. Polynomials that you can add, subtract, multiply, and sometimes divide. Symmetries that you can combine. Algebraic structures are some of the most fundamental objects in mathematics in the sense that they can be described by a small list of conditions. This simplicity makes them difficult to work with, but it also makes them ubiquitous in mathematics and other sciences.
(please see websites of the working group members for more details)
Analytic number theory
Analytic number theory uses the tools of analysis to study number theoretic questions. Such questions include: How are the prime numbers distributed among all integers? How many rational solutions does a system of polynomial equations have? How does the multiplicative structure of integers change under addition?
Staff members: Mangerel
Applied Algebra
Applied algebra uses tools from algebra and related areas, such as algebraic combinatorics and algebraic geometry, to solve problems motivated by applications outside of mathematics. From chemical reaction networks to robot kinematics to neural networks, algebraic structures are hidden in many systems. Understanding them is key to improving existing algorithms and gaining new insights.
Staff members: Ren
Arithmetic Algebraic Geometry
Arithmetic algebraic geometry uses tools from algebraic geometry to solve problems in number theory. Prominent results revolve around rational solutions to polynomial equations such as Fermat's Last Theorem. Galois groups often play a prominent role.
Staff members: Abrashkin, Evans, Funke, Shotton, Stasinski,
L-functions and Modular Forms
Modular forms are highly symmetric classes of functions on the upper complex half-plane. They arise in so many areas of mathematics that they have been called the fifth fundamental operation in mathematics, next to addition, subtraction, multiplication and division. L-functions are meromorphic functions that may be associated to modular forms (and their generalisations) and also arise from Galois groups. They are central to cracking old problems like Fermat's Last Theorem and discovering new conjectures. Their special values contain lots of interesting information, leading to the Birch and Swinnerton-Dyer conjecture and connections with algebraic K-theory.
Staff members: Bouganis, Funke, Gangl, Mangerel, Shotton
Representation Theory
Representation theory studies groups and other algebraic objects by representing their elements as linear transformations. Representation theory is fundamental in several areas of modern mathematics, such as group theory, mathematical physics (via Lie groups and Lie algebras) and number theory (via the Langlands program). Representation theory also has significant overlap with arithmetic algebraic geometry, which can often be used to construct or classify representations
Staff members: Funke, Shotton, Stasinski
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