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Foundations of Mathematics and Mathematical Logic Lab
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The lab studies the foundations of mathematics by classifying computation, proof, and definition according to their complexity.
AboutAbout the lab
Studying mathematics itself, using mathematics
Prof. Kihara studies the foundations of mathematics and mathematical logic.
The foundations of mathematics is a field that takes mathematics itself as its object and explores its premises and limits.
Prof. Kihara is interested in theories that classify fundamental mathematical concepts such as computation, proof, and definition by their complexity.
The core of his recent research is the analysis of the internal logic of sheaf toposes through "oracle sites."
Key terms
- Computability theory
- A theory that mathematically studies the boundary between what computers can and cannot compute
- Topos
- In mathematical logic, a structure used as a kind of "mathematical universe" within which mathematics can be developed
- Kolmogorov complexity
- Information complexity measured by the length of the shortest program that outputs a given string
ResearchResearch themes
Within the foundations of mathematics and mathematical logic, the lab studies the whole of computability theory and parts of proof theory, set theory, and categorical logic. It is interested in classifying various mathematical concepts according to their complexity.
*Illustrative imageComputable Analysis and Computability in Topological Spaces
A theory of computation for continuous objects such as real numbers. The lab is particularly interested in computation on infinite-dimensional spaces such as function spaces, and uses topological, continuous approaches to computation to analyze mathematical analysis and descriptive set theory.
*Illustrative imageAlgorithmic Randomness
Explores what "probability" and "randomness" mean from the standpoint of computation. The main perspectives are prediction (martingales), information compression (Kolmogorov complexity), and geometric measure theory (fractal dimension).
*Illustrative imageRealizability Topos Theory
Using toposes as mathematical universes, the lab is interested in analyzing the structure of the intermediate universes lying between the universe of "computable mathematics" and the universe of "set-theoretic mathematics." It is working on problems concerning Lawvere-Tierney topologies on the effective topos.
*Illustrative imageDegrees of Non-computability
The theory of Turing degrees and enumeration degrees, which classifies problems proven to be non-computable while identifying differences that can be handled computably. The lab incorporates topological methods and has recently been exploring the power of a new kind of forcing.
MembersFaculty
- ProfessorTakayuki Kihara
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Machine de Turing pédagogique” Mdtcreator / CC BY-SA 4.0 (cropped)
- Computable Analysis and Computability in Topological Spaces: “Hilbert cube” Claudio Rocchini / CC BY-SA 4.0 (cropped)
- Algorithmic Randomness: “Mandelbrot Set Image 91” Aokoroko / CC0 (cropped)
- Realizability Topos Theory: “Hochhebungseigenschaft” Tensorproduct / CC BY-SA 4.0 (cropped)
- Degrees of Non-computability: “Turing Machine Model Davey 2012” Rocky Acosta / CC BY 3.0 (cropped)
- Related lab (Mathematical Physics and Particle Theory Lab): “Collider Detector at Fermilab (CDF) silicon vertex detector cross section” Z22 / CC BY-SA 4.0 (cropped)
- Related lab (Quantum Information Theory, Theoretical Computer Science, and Algorithms Lab): “Google Sycamore Chip 001” Google (https://www.youtube.com/channel/UCK8sQmJBp8GCxrOtXWBpyEA) / CC BY 3.0 (cropped)
- Related lab (Fourier Analysis and Partial Differential Equations Lab): “Unusual wave at the Gap” Avenue / CC BY-SA 3.0 (cropped)
- Related lab (Algebraic Geometry, Representation Theory, Infinite-Dimensional Integrable Systems, and Mathematical Physics Lab (Shintaro Yanagida)): “Truncated icosahedron cherry model by George W. Hart” George W. Hart / CC BY-SA 4.0 (cropped)
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