 | “My research interests are mainly on polynomial automorphisms and the Jacobian conjecture. My earlier research also involves Vertex Operator Algebra and Conformal Field Theory”. Wenhua worked on a new notion of Mathieu subspaces, which he introduced in 2009. |
| April 2, 1968 – Dec. 23, 2023 | For updates: signup to the WZ-AGS-Newsletter (email organizers). |
This Algebraic-Geometry Seminar is a traditional seminar with a Theme: Belyi Theorem, as a central result bridging Algebraic Number Theory (Number Fields) and Algebraic Geometry (Riemann Surfaces and Function Fields). It is a Mathematical Model for baryons, as an application to Elementary Particle Physics.
… and a Team: Lucian M. Ionescu, ISU (organiser); Charlotte Ure, ISU; Prof. Gabriel Pripoae, Bucharest University, Romania (Diff. Geometry and General Relativity specialist); Prof. Dr. Bertrand Eynard, I. Ph. T. CES, Saclay, France (Math-Physicist, author: Lecture Notes on Riemann Surfaces, Counting Surfaces).
Spring 2024
Meeting times: Th. 10:00 a.m.-12:00 pm, STV 120 – please ask for a Zoom link if interested (recordings are stored for 30 days in the Cloud: see the corresponding link after the run of the seminar).
- 2/15 Organizational Meeting, plan and resources: ISU Milnor Library Counting Surfaces; keywords: covering maps, branched covers, function fields (for algebraic RS), number fields etc.
- Goals: study AG-Tools and Methods (R&D in Math) with applications in Chemistry/Physics;
- Format: 1st hour presentations; 2nd hour discussions (“Lecture-Recitation”);
- Plan: presentations from “Counting Surfaces” (B.E.), relating topics (LI), formal aspects (CU) etc;
- 2/22 L. M. Ionescu, “Belyi Maps: a brief Math-Physics motivation and overview” (see also [1] for additional details); we will watch John Voight’s YT survey (10-20 min; pdf here) followed by some explanations and background. Discussions are encouraged.
- 2/29 B. Eynard, “Maps and Discrete Surfaces; Riemann surfaces” (I) from Counting Surfaces, Ch. 1,2 & 6 (see [2]).
- 3/7 C. Ure, “Function fields in number theory and geometry”, Abstract: Function fields have many applications in various areas of mathematics. Often, they can be employed to translate a geometric problem to a purely algebraic one. In this talk, we will discuss the basic notions of function fields in the context of Riemann surfaces and arithmetic curves. If time permits, connections to zeta functions will be discussed. This talk is accessible to undergraduate and graduate students. Rec/Pwd.
- 3/21 L. M. Ionescu “Covering maps, branched covers and Riemann-Hurwitz Formula; on function/number fields connection”, Ramified coverings of Riemann surfaces over number fields encode topologically and algebraically a Gauge Theory with singularities (sources). We will recall covering spaces, ramification and interpret as dynamics: “cars flowing on a network” (Poincare-Hopf Theorem) and intuitively as “traveling in a multi-level building” to help our intuition about a very rich Math framework.
- 3/28 L.M. Ionescu, The connection between Matrix Integrals and Belyi’s Theorem will be reviewed, motivating the importance of understanding the connection between function fields and number fields, via Hodge-de Rham Theory and periods.
- 4/4 SPECIAL EVENT – Conference Catherine Goldstein and Clemence Perronnet at IHES, 10:30 am ->?, via Zoom. Viewers of the event in STV 120, are invited to an open discussion afterwords :
- Title: “The role that female role models can play in advancing equality in science”
- Abstract: “After its visit to the Lumen of the University of Paris-Saclay, the exhibition Just Do Maths! will travel to IHES and will be inaugurated by a conference by Catherine Goldstein and Clémence Perronnet on the role that female role models can play in advancing equality in science. This conference is based on the book “Matheuses” which has just been published by the CNRS. “
- Language: French (probably);
- Event online via Zoom.
- Post-event YT recording.
- 4/11 Break – no seminar; preparing a “new series” …
New Series
Rethinking the targeted audience and format: we will try presentations of a subject / topic (AG/Belyi Map theme), including a student oriented learning component, presentation of an article with several segments (presentations) … (details will follow).
- 4/18 (Special event) L. M. Ionescu, Rethinking the Real Numbers, presented in the Pure & Applied Math Seminar @12:00, STV 136B.
- 4/25 No seminar
- 5/2 (Special event) Anurag Kurumbail, Title: “A Modular Group construction of Real Numbers“, MAT 490 Master Project Presentation;
- Abstract: The Continued Fractions representations of Real Numbers provides a deeper understanding of many classes of numbers, including quadratic irrational numbers. A norm-free construction of real numbers is presented, using the modular group PSL(2;Z). It defines a Sequential Space extending the rational numbers. The theory of extensions of Sequential Spaces is developed to include the former construction. An application of the modular group representation of real numbers to quadratic irrational numbers, yields new results in the theory of Number Fields.
- Presentation will be @12:00 in STV 136B.
[1] L. M. Ionescu, The Beauty and The Beast (preliminary motivation); additional details: Algebraic-Geometric Tools for Particle Physics, https://vixra.org/abs/2305.0001
[2] B. Eynard, Counting Surfaces, Progress in Mathematical Physics (PMP, volume 70).
[3] John Voight, “Belyi Maps in Number Theory: a survey“, YT presentation, 2021; pdf.
Fall 2024
The Seminar will take place in STV 310 Thursdays 11-11:50. UG & Graduate students are welcomed (bring a laptop, coffee etc. if needed). Zoom Link (ID: 930 2972 9488 ask for password LI).
Note: we’ll restart Oct. 10, with a one hour seminars 11:00 – 11:50 am. Participants: G. Seelinger, L. M. Ionescu … (Faculty & GS welcome!). Article to study: https://webspace.science.uu.nl/~oort0109/EigArt-RHurwitz-2016.pdf “The Riemann-Hurwitz Formula” by F. Oort.
For some Lecture Notes, “minutes” and references see WZ-AG-Seminar 2024-2025.
[1] 8/29 Organizational meeting: suggest topics to study, article / book to follow and present, learn from. Suggested topics: Morse Theory and AG-updates, connections with Ramification Theory (Poincare-Hopf Theorem, Belyi maps etc.), applications to Physics (e.g. Witten’s work) and other topics you would like to learn/research etc.
New Plan:
Ramification Theory (Alg. NT & AG geometric applications: visual!) is essential in expanding Gauge Theory (Connection Theory applied to Physics, e.g. EM, QFT, QCD etc.) to incorporate “sources” as singularities (electron: E-charge, fluxon: magnetic charge as monodromy & Pi1-representations from a connection in a principle bundle etc.).
The focus of the Seminar will be on Ramification Th. Alg. Number Theory & Alg. Geom., with just comments from my part on the interface with Physics.
Bibliography for Fall 2024
[Oort] F. Oort, “The Riemann-Hurwitz Formula”
[2] 10/10 Introduction: the Physics / Mathematics interface (Gauge Theory / Bundles, with ramification to account for “sources” and break of symmetry etc.; also from “differential/smooth category” / Real Numbers, to “complex / conformal algebraic category” without Reals: rather algebraic closure of Q) … and we’ll run into “Belyi Theorem” as a central result for this topic.
– L. M. Ionescu, [Oort] Introduction. Note: Come at the Undergraduate Colloquium Series 2-3pm, STV 314, L.M. Ionescu, “Evariste Galois and Felix Klein Programs in Mathematics; Emmy Noether bridging Math and Physics”.
[3] 10/17 L. M. Ionescu, “Introduction to Ramification Theory: motivation and relations to number field/function field analogy”.
[4] 10/24 Talk in Pure and Applied Mathematics, 12pm-1pm, STV 314, “On some Programs in Math-Physics, Part II: On Langlands Program and Beyond” (same Google slides doc; Zoom rec.). It is recommended to glance over Part I: Galois, Klein, Lie and Noether.
[5] 10/31 L. M. Ionescu, “Introduction and Results” ([Oort], pp.2-5).
[6] 11/7 L. M. Ionescu, “Main results (Part 2)”, Oort pp.4-5.
[7] 11/14 …
For updates and comments, email LMI.