Room:  Bloomberg Center for Physics and Astronomy  178,  Johns Hopkins University

Dates:  August 26, 2024 - December 6, 2024  on  Mondays and Wednesdays 

Time:  12:00 - 1:15 pm

Office:  Krieger Hall 419

Office Hours:  Mondays and Wednesdays after class,  and  via appointment.

Email Address:  meeseong@jhu.edu

This is a graduate course.  We will be using  Lectures on Riemann Surfaces by Otto Forster.

We will be switching over and using  Algebraic Curves and Riemann Surfaces by Rick Miranda.

Teaching Assistant (TA):  Elham Matinpour

Elham's Office Hours:  By appointment

Course description.  Abstract Riemann surfaces. Examples: algebraic curves, elliptic curves and functions on them. Holomorphic and meromorphic functions and differential forms, divisors and the Mittag-Leffler problem. The analytic genus. Bezout's theorem and applications. Introduction to sheaf theory, with applications to constructing linear series of meromorphic functions. Serre duality, the existence of meromorphic functions on Riemann surfaces, the equality of the topological and analytic genera, the equivalence of algebraic curves and compact Riemann surfaces, the Riemann-Roch theorem. Period matrices and the Abel-Jacobi mapping, Jacobi inversion, the Torelli theorem. Uniformization (time permitting).

Expectations.  All questions and thoughts are valuable.  Please respect one another in and out of class.

Structure of the course.  We will aim to cover at least one section each day.

Grades.  

Choice 1.  I will assign several homework problems at the end of each class.  These homework problems will be posted here.  The previous week's homework will be due on the Monday of each week.  We will give you feedback on your submissions, but they will be graded on completion.  Collaboration is encouraged.

Selected solutions will be provided on the course calendar below.

Choice 2.  Alternatively, rather than submitting homework each week, you can collaborate and work on a publishable research problem related to Riemann surfaces and submit your drafts to me.  I will give you feedback regularly.  A list of open problems will be posted soon.  You can also propose a research problem to me.  Furthermore, I am perfectly fine if all of you want to work on a research problem together. 

Please let me know which you would like to do.

There will be no exams.

Research Problems.

Problem 1:  Higgs bundles.  Read  Branes and moduli spaces of Higgs bundles on smooth projective varieties  and investigate modular setting.

Problem 2:  Chern--Simons theory and matrix factorizations.  Extend  Towards ℛ-matrix construction of Khovanov-Rozansky polynomials I. Primary T-deformation of HOMFLY  to the super-setting.

Problem 3:  Fukaya categories.  Investigate a topological quantum field theory (TQFT) perspective for  Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions.

Problem 4:  Exotic Springer resolutions and delta Springer varieties.  Generalize  Knot homology via derived categories of coherent sheaves I, sl(2) case  to the exotic Springer resolution or two-row delta Springer varieties.

Problem 5:  Springer resolutions.  Construct an explicit bijection between 3-row Springer fibers using Nakajima or bow varieties and certain web basis, analogous to  Irreducible components of two-row Springer fibers for all classical types.

Problem 6:  Tropical semirings and automata.  Extend  Topological theories and automata  for tropical semirings.

Sheaves  (also talk to M. Khovanov).  Take a sufficiently nice topological space X  (just think of an interval [0,1]).  Pick a field k and consider the category Sh(X) of sheaves of k-vector spaces on X.  This category is abelian and has enough injective objects  (every object is a subobject of an injective object), allowing to construct injective resolutions.

Problem 7.  Show that this category does not have enough projective objects.  Apparently the only projective object is the zero sheaf.

Natural functors Sh(X) → Sh(Y) between categories of sheaves are usually left exact allowing for the use of injective resolutions to define their derived functors.  (See  Cohomology of Sheaves   by Birger Iversen for an introduction to sheaves.)

Problem 8.  Are there naturally occurring right exact functors between categories of sheaves of k-vector spaces? If not, why are they missing?

For sheaves of modules over sheaves of rings, the tensor product functor is right exact, but that's due to the structure of categories of modules over rings.  For sheaves of vector spaces the tensor product functor is exact.  We are looking to build functors that are right exact only, not exact.

Problem 9.  Compare sheaves to comodules.  In the category of comodules over a coalgebra A, the free object A is injective, not projective.  Are there enough projective comodules, for a general coalgebra A?  My memory is that this category does not have enough projectives, in general.

Problem 10.  Is the category of sheaves of vector spaces analogous to the category of comodules, in some sense?  Apparently neither category has enough projectives.

Problem 11.  Define the notion of a cosheaf.  Why do cosheaves rarely appear in mathematics, unlike sheaves?  Look up categories of cosheaves and what people do with them.

Problem 12.  (a)  Consider presheaves and sheaves of k-vector spaces over a finite topological space.  Look at specific examples of such spaces, with up to 4 points, and understand the relation between categories of presheaves and sheaves in those concrete cases. 

Problem 12.  (b)  Are sheaves over the Cantor set interesting? Can one describe that category explicitly? 

Problem 13.  You can suggest a research problem also  (even going through a paper on topics related to algebraic curves is fine).  Contact me in regards to this.

Additional references  (optional).

1. Simon K. Donaldson,  Riemann surfaces

2. Jürgen Jost,  Compact Riemann Surfaces: An Introduction to Contemporary Mathematics

3. Jean-Benoît Bost,  Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties

4. Alexander Bobenko,  Compact Riemann Surfaces

5. Phillip Griffiths and Joseph Harris,  Principles of algebraic geometry

6. Roger Godement,  Topologie algébrique et théorie des faisceaux

7. Charles A. Weibel,  An introduction to homological algebra

8. Joseph J. Rotman,  An introduction to homological algebra

Please note that our schedule may change, depending on how much we cover each day.

Mon Aug 26:  Definition of Riemann surfaces  (Forster, HW page 8:  1.1 and 1.2)

Wed Aug 28:  Elementary properties of holomorphic mappings  and  homotopy of curves; the fundamental group  (Forster, HW page 13:  2.2, 2.3, and 2.5)

Mon Sept 2:  No Classes,  Labor Day

Wed Sept 4:  Homotopy of curves; the fundamental group  (finish the section)  and  branched and unbranched coverings  (Forster, HW page 30:  4.1, 4.3, and 4.5)

Mon Sept 9:  The universal covering and covering transformations  and  sheaves  (Forster, HW page 38:  5.1 and 5.2,  HW page 44:  6.1 and 6.3)

Wed Sept 11:  Analytic continuation  and  algebraic functions  (Forster, HW page 59:  8.2)

We will use  Algebraic Curves and Riemann Surfaces by Rick Miranda  starting today.

Mon Sept 16:  Algebraic functions (examples)  (Miranda, HW pages 23-37:  read about singularities of functions, meromorphic functions, Laurent series, harmonic functions, smooth projective curves, and come to Wednesday's class with questions, or email a list of questions to everyone in the class)

Propositions:  A holomorphic map from a compact Riemann surface to the complex plane must be constant.  If F is a nonconstant holomorphic map from a compact Riemann surface to Y, where Y is connected, then Y is compact and F is surjective.

Wed Sept 18:  Differential forms  and  operations on differential forms  (Miranda, HW page 111:  IV.1.A,  HW page 117:  IV.2.C)

Why do we need \bar{z} when describing 1-forms on Riemann surfaces?

Mon Sept 23:  Differentials as operators  (Miranda, HW page 117:  IV.2.H and I)

Lemma:  A holomorphic 1-form is d-closed.

Wed Sept 25:  Integration on a Riemann surface  (Miranda, HW page 127:  IV.3.E and F)

Prop:  let F: X → Y be a holomorphic map between Riemann surfaces.  Then F_* on paths is adjoint to F* on 1-forms.

Mon Sept 30:  Group actions on Riemann surfaces  (Miranda, HW page 83:  III.3.F)

Wed Oct 2:  Divisors  (Miranda, HW page 137:  V.1.A)

Mon Oct 7:  Boundary, ramification and branch, and intersection divisors,  partial ordering on divisors,  linear equivalence of divisors  (Miranda, HW page 145:  V.2.C and F)

Wed Oct 9:  Complete linear systems of divisors:  if X compact Riemann surface and deg(D) < 0, then L(D) = {0};  P(L(D)) ≅ |D|;  if D ~ E, then L(D) ≅ L(E);  if D ~ E, then L^{(1)}(D) ≅ L^{(1)}(E);  L(D+K) ≅ L^{(1)}(D), where D = div(ω), a canonical divisor  (Miranda, HW page 152:  V.3.D and H)

Mon Oct 14:  Divisors for Riemann sphere  and  bound on the dimension of L(D)  (Miranda, HW page 152:  V.3.I)

Wed Oct 16:  Divisors and maps to projective space  and  base points of linear systems  (Miranda, HW page 166:  V.4.A, D and E)

Mon Oct 21:  Defining holomorphic map via a linear system,  removing base points,  very ample divisors,  rational and elliptic normal curves  (Miranda, HW page 166:  V.4.H, J and K)

Wed Oct 23:  Separating points and tangents,  constructing meromorphic functions with specified Laurent tails,  transcendence degree of the function field M(X) for compact Riemann surface X  (Miranda, HW page 178:  VI.1.A, I and J)

Mon Oct 28:  Laurent tail divisors  and  Mittag-Leffler problems  (Miranda, HW page 178:  VI.1.L  and  page 185:  VI.2.E, F, and G)

Wed Oct 30:  Riemann-Roch theorem  and  Serre duality  (Miranda, HW page 193:  VI.3.B, C, D, G, and K)

Prop:  let K be a canonical divisor on an algebraic curve of genus g.  Then deg(K) = 2g - 2.

Mon Nov 4:  Proof of Serre duality  and  applications of Riemann-Roch:  if X is a compact Riemann surface satisfying Riemann-Roch for every divisor D then X is an algebraic curve  (Miranda, HW page 202:  VII.1.A and B)

Wed Nov 6:  Criterion for divisor to be very ample,  every algebraic curve can be holomorphically embedded into a projective space,  if X is an algebraic curve whose dim L(p) > 1 then X is isomorphic to the Riemann sphere,  if X is a compact Riemann surface of genus ≥ 1 then L(p) ≅ constants,  any algebraic curve of genus 0 is a Riemann sphere,  any algebraic curve of genus 1 is isomorphic to a smooth projective plane cubic curve  and  is also a complex torus,  if |K| is a canonical linear system on an algebraic curve of genus g ≥ 1 then |K| is basepoint-free,  periods of 1-forms,  Jacobians  (Miranda, HW page 264:  VIII.4.B and C)

Mon Nov 11:  Abel-Jacobi map,  Abel's theorem  and  group law on a smooth projective plane cubic  (Miranda, HW page 267:  VIII.5.B and C)

Wed Nov 13:  Short exact sequence of sheaves,  sheaf/Zariski cohomology,  and  Čech cohomology  (by Fabian Espinoza)  (Miranda, HW page 301:  IX.3.M and N  and  Miranda, HW page 312:  X.1.D and E)

• NOTES

Mon Nov 18:  Special topics  (by Haihan Wu):  Gentle introduction to Witten–Reshetikhin–Turaev (WRT) quantum invariants and webs  (HW:  write 1 paragraph summarizing WRT quantum invariants and webs)*

• NOTES

Wed Nov 20:  Special topics  (by Mikhail Khovanov):  Elliptic curves in characteristic p and applications to cryptography  (HW:  write 1 paragraph summarizing elliptic curves in char p and cryptography)

• NOTES  (pages 1 - 14)

Mon Nov 25:  No Classes,  Thanksgiving Break

Wed Nov 27:  No Classes,  Thanksgiving Break

Mon Dec 2:  Special topics  (by Fabian Espinoza):  Čech cohomology computations,  moduli spaces of algebraic curves,  moduli constructions of (stable) nodal/marked algebraic curves,  examples,  properties of these moduli constructions  (References:  Peter E. Newstead,  David Mumford, John Fogarty, Frances Kirwan,  Joe Harris)  (HW:  write 1 paragraph summarizing moduli constructions)  Fabian asked to move today's class to Thursday Dec 5.

Wed Dec 4:  Special topics  (by Mitch Majure):  Sheaves on finite spaces and the Cantor set  (HW:  write 1 paragraph summarizing today's special topics)

• NOTES  (Mitch most likely presented some original results today, so his paper will not be posted here;  instead, please talk to Mitch to get a short summary of his presentation)

Thurs Dec 5  (10:30 - 12:00 pm):  Special topics  (by Fabian Espinoza):  Čech cohomology computations,  moduli spaces of algebraic curves,  moduli constructions of (stable) nodal/marked algebraic curves,  examples,  properties of these moduli constructions  (References:  Peter E. Newstead,  David Mumford, John Fogarty, Frances Kirwan,  Joe Harris)  (HW:  write 1 paragraph summarizing moduli constructions),  ROOM:  Krieger 411

• NOTES  (Fabian will send over his notes by the end of the final exam week;  please contact Fabian to discuss what he talked about today)

* = one paragraph means just a  couple of sentences  about the mathematics, your thoughts, or any questions you have about the presentation.

I enjoyed having you all in my class.

Happy Holidays!