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Fall 2025
Undergraduate Research in Advanced Topics
Introduction to 𝒟-modules
Johns Hopkins University, Baltimore, MD
Krieger 204
Description: How does one generalize the operation of differentiation on a function? What does it mean to study differential operators on an algebraic variety? As the representation theory of Lie algebras, differential equations, singularity theory, mathematical physics, and number theory have been influenced by 𝒟-modules, they play an important and active role in advanced mathematics. This course consists of introductory lectures on and research in algebraic-analytic approach to 𝒟-modules and related areas.
You must be registered for Math 585 in order to participate.
Course Number: AS.110.585.01
Room: Krieger 204
Dates: August 25, 2025 (Monday) - December 5, 2025 (Friday)
Time: Please be flexible on when we meet (we may meet in the morning on some of the days). Note that there are 15 minute breaks (4:00 - 4:15 pm) on the meeting days for coffee, tea, and food.
Postdoc Assistants: Haihan Wu
Course assessment: This is a 3-hour research course on a topic chosen by the professor. Throughout the semester, each student will present related topics, including write ups for the presentations, which may be merged as a contribution to a collaborative paper.
Required write-up: I will assign parts of collaborative paper(s), which will be your required write-up.
Required presentation: Give a talk at the Math Club or a local conference. More information will be provided after the midpoint of the semester. When you have scheduled to give a talk, please let me know.
Regular attendance and participation are expected. If you plan to be absent for a significant duration of the semester, see me.
Week of August 25 - 29.
Tuesday August 26 (5:30 - 6:30 pm): Background (Chapter 1): Weyl algebras, canonical basis.
Thursday August 28 (3:00 - 7:10 pm): Background (Chapter 1 and 2): Free algebra K{z_1, ..., z_{2n}}, discussion of when is an endomorphism of a Weyl algebra an automorphism (related to Jacobian conjecture), a discussion of a specific endomorphism and how to prove that it is an automorphism, Chapter 1 exercises, the ideal structure of Weyl algebra, degree of an operator (deg (D+D') ≤ max{deg(D), deg(D')}, deg(DD') = deg(D)+deg(D'), deg([D,D']) ≤ deg(D)+deg(D')-2), n-th Weyl algebra over an algebraically closed field is a domain.
Week of September 1 - 5.
Tuesday September 2 (8:00 - 9:00 am, Zoom): Background (Chapter 2): commutative simple rings form a field, Weyl algebras are simple but they are not a division ring, Weyl algebras do not have nontrivial 2-sided ideals but they do have nontrivial left (or right) ideals (if an element in the Weyl algebra has an inverse, then it must be an element from the base field), if I is a left ideal of a Weyl algebra then I is generated by 2 elements, if I is a left ideal of a ring of differential operators over a smooth variety then I is generated by 2 elements, Weyl algebras over positive characteristic is not a domain and has a nontrivial center, Weyl algebras over positive characteristic are not simple.
Tuesday September 2 (3:00 - 4:15 pm): Faculty meeting.
Tuesday September 2 (4:15 - 5:30 pm, with Professor Khovanov): Properties of D-modules and Research Problem 2.
Research Problem 1 has been slightly postponed.
Thursday September 4 (3:00 - 4:00 pm, with Professor Khovanov): Research Problem 3.
Thursday September 4 (4:15 - 5:15 pm): Background (Chapter 3): show that a certain ideal generated by 2 elements is not a cyclic left ideal, ring of differential operators of a commutative ring R, an operator P ∈ End_K(R) of order n, derivations D^1(R) are linear operators satisfying Leibniz's rule, operators of order ≤ 1 are of the form Der_K(R) + R, P + Q and PQ have finite order if P and Q have finite order, derivations of a polynomial ring must be of the form ∑ f_i ∂_i where f ∈ K[X], if P ∈ D(K[X]) and [P, x_i] = 0 for all i then P ∈ K[X], the notion of a potential function for a polynomial vector field satisfying a certain property, the ring of differential operators of the algebra K[X] of polynomials is precisely the (n-th) Weyl algebra, the ring of differential operators of a commutative ring R is not always generated by Der_K(R) + R (this is true only if R is regular or if R is the coordinate ring of a nonsingular irreducible affine variety).
Homework: starting typing (assignments were given during today's meeting). If you have questions about what you are typing up, see Prof. Khovanov or me. We will also help you type.
Homework: Nicolas will present a certain paper next week related to Research Problem 2, but everyone should read the paper in advance.
Week of September 8 - 12.
Tuesday September 9 (no meeting): We have a visitor.
Tuesday September 9 (3 - 4:15pm, Khovanov's office): Research related to Problems 2 and 4 (Nicolas will present the paper mentioned from Thursday September 4).
Thursday September 11 (3:00 - 4:30 pm, Khovanov's office): Research related to Problems 2 and 4 (Nicolas will resume his research discussion today).
Homework: Many research problems. Check email. Type them as you obtain the results.
Week of September 15 - 19.
Tuesday September 16 (no meeting): We have a visitor.
Tuesday September 16 (3 - 4:15 pm, Khovanov's office): Research related to Problem 4 (Catie will present any progress made so far and then Beck and Nicolas will present any progress made so far).
Thursday September 18 (3:00 - 5:00 pm, Khovanov's office): Research (Catie will present first, then back to Beck and Nicolas on any research updates).
Week of September 22 - 26.
Tuesday September 23 (3:00 - 4:00 pm): Faculty meeting.
Tuesday September 23 (4:00 - 4:30 pm, Khovanov's office): Research (Beck and Nicolas on any research updates, Catie on any research updates).
Tuesday September 23 (5:30 - 6:30 pm, Khovanov's office): Research (Beck and Nicolas on any research updates, Catie on any research updates).
Thursday September 25 (3:00 - 5:15 pm, Khovanov's office): Research (Beck and Nicolas on any research updates, Catie on any research updates).
Week of September 29 - October 3.
Tuesday September 30 (no meeting): We have a visitor.
Tuesday September 30 (3:00 - 4:20 pm, Khovanov's office): Research (Beck and Nicolas on any research updates, Catie on any research updates).
Thursday October 2 (3:00 - 5:30 pm), Khovanov's office): Research
Week of October 6 - 10.
Tuesday October 7 (no meeting): We have a visitor.
Tuesday October 7 (3:00 - 4:25 pm, Khovanov's office): Research (Beck and Nicolas on any research updates, Catie on any research updates).
Wednesday October 8 (on your own, Khovanov's office): Beck or Nicolas meets with Prof. Khovanov and update him with your proof.
Thursday October 9 (3:00 - 4:30 pm, Khovanov's office): Research (Nicolas on any research updates, Catie on any research updates).
Homework: Nicolas will type up the proof that was discussed during Thursday's meeting with Khovanov. Beck will help as needed. Catie continues to update the paper with her findings.
Week of October 13 - 17.
Tuesday October 14 (3:00 - 4:30 pm, Khovanov's office): Research discussions. Representations of quivers and path algebras (by Khovanov).
Thursday October 16 (3:00 - 4:30 pm, Khovanov's office): Fall Break
Week of October 20 - 24.
Tuesday October 21 (3:00 - 3:50 pm, Khovanov's office): Research discussions (Sections 8 and 9 in our paper).
Tuesday October 21 (3:50 - 4:25 pm, Khovanov's office): Beck will present the first half of Chapter 5 in A Primer of Algebraic D-modules. Include all proofs. Modules over the Weyl algebra: polynomial ring, twisting by the Weyl algebra.
Tuesday October 21 (4:25 - 5:00 pm, Khovanov's office): Research meeting with Catie.
Thursday October 23 (3:00 - 4:00 pm, Khovanov's office): Research discussions (Sections 8 and 9 in our paper) and Catie's work.
Thursday October 23 (4:00 - 5:30 pm, Khovanov's office): Beck will finish where he left off. Nicolas will present the second half of Chapter 5 in A Primer of Algebraic D-modules. Include all proofs. Modules over the Weyl algebra: holomorphic functions.
Let's do a few exercises together when we meet.
Homework: Find closed formulae for Sections 8 and 9 in our paper. Prove them.
Week of October 27 - 31.
Wednesday October 29 (3:30 - 5:30 pm, Khovanov's office): Research discussions (Section 8 by Catie, Section 9 by Nicolas. Postpone discussing Sections 10 and up by Catie.
Week of November 3 - 7.
Monday November 3 (3:00 - 4:15pm, Khovanov's office): Research discussions with Catie.
Tuesday November 4 (3:00 - 4:15 pm): Faculty meeting.
Tuesday November 4 (3:00 - 5:30 pm, Khovanov's office): Research discussions with Fan Zhou. Research discussions with Nicolas and Beck. Beck reviews Chapter 5 topics in Coutinho to Prof. Khovanov. Stop by to talk about anything (homotopy theory, topics in algebraic geometry, étale topology, etc.).
Thursday November 6 (3:00 - 4:30 pm, Khovanov's office): Research discussions. Stop by to talk about anything (homotopy theory, topics in algebraic geometry, étale topology, etc.).
Week of November 10 - 14.
Monday November 10 (after 12:00pm?, Khovanov's office): TBC.
Tuesday November 11 (3:00 - 4:30 pm, Khovanov's office): Research discussions. Chapter 5 exercises by Beck, Chapter 6 (Differential Equations) by Beck).
Thursday November 13 (3:00 - 4:30 pm, Khovanov's office): Research discussions.
Week of November 17 - 21.
Tuesday November 18 (3:00 - 4:30 pm, Khovanov's office):
TBA (6:00 pm - ): Research Group Dinner (this week or after the Thanksgiving week)
Thursday November 20 (3:00 - 4:30 pm, Khovanov's office):
Week of November 24 - 28.
Thanksgiving Week
Week of December 1 - 5.
Tuesday December 2 (3:00 - 4:30 pm, Khovanov's office):
TBA (6:00 pm - ): Research Group Dinner
Thursday December 4 (3:00 - 4:30 pm, Khovanov's office):
December 5 - 7, 2025 (George Washington University): Knots in Washington 52 Conference
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