2020/2021

Research Seminar "Introduction to Frobenius Algebras and Mirror Symmetry"

Category 'Best Course for Career Development'

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 3, 4 module

Instructors: Alexey Basalaev, Petr Dunin-Barkowski

Language: English

ECTS credits: 6

Contact hours: 72

Course Syllabus

Abstract

Frobenius algebra is just an associative algebra with a unit equipped with a bilinear form and satisfying a certain simple condition called the Frobenius condition. Despite the notion of Frobenius algebras being very simple, it turns out that these objects play a profound role in many interesting areas and examples. One particular area where they naturally arise is the singularity theory which arises from studying cusps with behavior similar to the one of the curve y^2 = x^3 near (x,y) = (0,0) (compare this to the behavior of the curve y^2 = x^3 + x^2 near the same point). In the present course we will start with introducing the basic concept of Frobenius algebra and discuss some examples and properties of these objects. Then we will discuss the basics of singularity theory and the relation of Frobenius algebras to this theory. Along the way we will mention the relation of all this to physics (in particular, we will discuss the so-called «two dimensional topological quantum field theories» which sound much scarier than they really are). Towards the end of the course we will touch the concept of F-manifolds, providing all the necessary preliminaries. This course is aimed to be completely accessible for all second-year students and above.

Learning Objectives

• Students will know how to distinguish Frobenius algebras in the set of all associative, commutative algebras and how to use the special properties of them.

Expected Learning Outcomes

• Familiarity with the basic theory of Frobenius algebras.
• Familiraity with basic examples of Frobenius algebras.
• Familiarity with examples of Frobenius algebras coming from the singularity theory.
• Familiarity with root systems, coxeter groups and Frobenius algebra structure on the space of invariant polynomials.
• Familiarity with examples of Frobenius algebras coming from the cohomology of manifolds.
• Familiarity with relation of Frobenius algebras to physics.
• Familiarity with relation of Frobeius algebras to mirror symmetry.
• Familiarity with F-manifolds.
• Familiarity with examples of F-manifolds arising from the singularity theory.

Course Contents

• Frobenius algebras
  Algebras with a pairing, Frobenius algebras. Equivalent formulations, uniqueness of the pairing, restrictions arising from the Frobenius property.
• Examples of Frobenius algebras
  Examples of non-Frobenius associative commutative algebras. Formal description of Frobenius algebras in terms of the unit and the counit and the multiplication tensor. Corresponding graphs and cobordisms description.
• Frobenius algebras coming from the singularity theory
  Frobenius algebras coming from the singularity theory: ADE examples.
• Frobenius algebras on spaces of invariant polynomials
  Root systems of the ADE type, Coxeter groups, Frobenius algebra structure on the space of invariant polynomials.
• Frobenius algebras arising from the cohomology of manifolds
  Frobenius algebras arising from the cohomology of manifolds. CP^n examples.
• Relation of Frobenius algebras to physics
  Atiyah’s axioms of 2D TQFTs. Relation to physics.
• Frobenius algebras and mirror symmetry
  Mirror symmetry as an isomorphism of Frobenius algebras, certain simple examples.
• Manifolds with a product
  Manifolds with a product. Associative and commutative case, F-manifolds.
• Examples of F-manifolds
  F-manifolds arising from deformations of singularities: ADE examples.

Assessment Elements

• Problem sheets
• Written tests
• Talks at seminars
• Written exam
  The exam will happen in online mode, in written form (solving problems on subjects of this course). The exam will be carried out via the corporate email and through Telegram (either option is available). At the predetermined start time of the exam, the students have to inform one of the teachers of their readiness to take the exam via the corporate email or via Telegram. Technical requirements: the student needs to have a device with Internet connection (PC, tablet or smartphone); device which allows to take a photo or a scan of the sheets with solutions (smartphone or tablet camera or a separate digital camera or a scaner) or a device which allows to take written notes in electronic form (digitizer tablet). In order to take part in the exam, the student has to: inform of their readiness via the means mentioned above, receive the problems and submit the photos/scans/electronic form of solutions via the corporate email or via Telegram to one of the teachers. During the exam it is forbidden to: discuss the problems with other students or any other people. It is allowed to: consult personal notes or literature. A connection disruption shorter than 10 minues is regarded as a short-term disruption. A connection disruption of 10 minutes or longer is regarded as a long-term disruption. If a student had a long-term connection disruption which prevented them to submit the solutions before the predetermined end time of the exam, this student cannot continue with this exam. the retake procedure is analogous to the main take.

Interim Assessment

• Interim assessment (4 module)
  Grading is based on the following four marks: S – the total mark for the problem sheets, a real number between 0 and 4; C – the total mark for the tests (small written tests given every few seminars), a real number between 0 and 4; T – the mark for a 30-minute talk given at one of the seminars, a real number between 0 and 3; E – the final oral exam mark, a real number between 0 and 5. The total score for the course is computed according to the following formula: min(10,ceil(S + C + T + E)), where ceil() stands for rounding up. If for any student min(10,ceil(S+C +T)) >= 8 already before the final exam, this student can take this value as his or her final score and skip the exam.

Bibliography

Recommended Core Bibliography

• Dubrovin, B. (1994). Geometry of 2d topological field theories. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.hep-th%2f9407018

Recommended Additional Bibliography

• Dubrovin, B. (1998). Painleve’ transcendents and two-dimensional topological field theory. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f9803107