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Analytic and Algebraic Number Theory Lab
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Number theory research exploring the connections between multiple zeta values and modular forms.
AboutAbout the lab
Connecting zeta values and modular forms
Prof. Bachmann studies a wide range of topics in number theory related to multiple zeta values and the theory of modular forms.
Multiple zeta values are real numbers that naturally generalize the values of the Riemann zeta function, and they appear in many areas of mathematics and physics.
The lab draws on methods from number theory, the theory of modular forms, and combinatorics, and often incorporates computer experiments using PARI/GP, Mathematica, and Sage into its research.
Key terms
- Multiple zeta values
- Real numbers that generalize the values of the Riemann zeta function. They appear in many areas of mathematics and physics
- Modular forms
- Holomorphic functions that appear in many settings in number theory
- q-analogue
- A holomorphic function or formal power series that behaves similarly to multiple zeta values in certain respects
ResearchResearch themes
The lab studies number theory, focusing on q-analogues of multiple zeta values and their connections with modular forms. Using multiple Eisenstein series and related objects as clues, the research also extends to combinatorial Hopf algebras.
*Illustrative imageMultiple Zeta Values and Modular Forms
The lab investigates the relationship between multiple zeta values and modular forms, which are holomorphic functions that appear in many settings in number theory. Multiple Eisenstein series and q-analogues of multiple zeta values link the two, and this research leads into the study of combinatorial Hopf algebras.
*Illustrative imageVariants of Multiple Zeta Values
The lab is interested not only in classical multiple zeta values but also in variants such as finite multiple zeta values, Schur multiple zeta values, and multiple zeta values of higher level.
*Illustrative imageFormal Multiple Eisenstein Series
Recently, the lab has focused on the formal aspects of these objects and introduced the notion of formal multiple Eisenstein series. It extends formal multiple zeta values and offers a new perspective on the theory of modular forms.
MembersFaculty
The members of Prof. Bachmann's group are faculty, postdocs, and doctoral, master's, and undergraduate students.As of 2026
- Associate ProfessorHenrik Bachmann
VoicesStudent voices
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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: “ZetaFunction” Gaussianer / Public domain (cropped)
- Multiple Zeta Values and Modular Forms: “Uniform tiling 433-t12” Tomruen at English Wikipedia / Public domain (cropped)
- Variants of Multiple Zeta Values: “Red Bead Abacus Display” Aliva Sahoo / CC BY 4.0 (cropped)
- Formal Multiple Eisenstein Series: “Eisenstein 12” Jan Homann / Public domain (cropped)
- Related lab (Ishii Lab): “Surface of degree 7” Jean Constant / CC BY-SA 4.0 (cropped)
- Related lab (Ring Theory and Representation Theory Lab): “Combined group axiom commutative diagram” Struoc / CC BY-SA 4.0 (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)
- Related lab (Combinatorics and Representation Theory Lab): “Line in combinatorics” Tensorproduct / CC BY-SA 4.0 (cropped)
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