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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: Matthew Hamil and 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 - 4:15 pm, Khovanov's office): Research (Beck and Nicolas on any research updates, Catie on any research updates).
We will meet additional days.
Thursday October 2 (3:00 - 4:30 pm), Khovanov's office: Research and background
Week of October 6 - 10.
Tuesday October 7 (no meeting): We have a visitor
We will meet additional days
Thursday October 9 (3:00 - 4:30 pm):
Week of October 13 - 17.
Tuesday October 14 (3:00 - 4:30 pm, my office):
We will meet additional days.
Thursday October 16 (3:00 - 4:30 pm):
Week of October 20 - 24.
Tuesday October 21 (3:00 - 4:30 pm, my office):
We will meet additional days.
Thursday October 23 (3:00 - 4:30 pm):
Week of October 27 - 31.
Tuesday October 28 (3:00 - 4:30 pm, my office):
Thursday October 30 (3:00 - 4:30 pm):
Week of November 3 - 7.
Tuesday November 4 (3:00 - 4:30 pm, my office):
Thursday November 6 (3:00 - 4:30 pm):
Week of November 10 - 14.
Tuesday November 11 (3:00 - 4:30 pm, my office):
Thursday November 13 (3:00 - 4:30 pm):
Week of November 17 - 21.
Tuesday November 18 (3:00 - 4:30 pm, my office):
Thursday November 20 (3:00 - 4:30 pm):
Week of November 24 - 28.
Thanksgiving Week
Week of December 1 - 5.
Tuesday December 2 (3:00 - 4:30 pm, my office):
Thursday December 4 (3:00 - 4:30 pm):
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