Math 503

Spring 2025 Undergraduate and Graduate Research

Room:  411 Krieger,  Johns Hopkins University  

(if we are using 203 Krieger on some days, then I will let you know in advance)

Dates:  January 21, 2025  (Tuesday)  -  April 28, 2025  (Monday)

Time:  Fridays,  5:00 - 7:30 pm

Postdoc Assistant:  Haihan Wu

We will go through

• Knot homology via derived categories of coherent sheaves I, sl(2)-case

• Knot homology via derived categories of coherent sheaves II, sl(m)-case  (if we have time)

carefully each week.

I will begin the semester presenting the contents in the sl(2) paper, and eventually, a subset of you will pair off and present the material also.  We will take turns doing this.  We will go through every detail in the paper.  Asking questions and asking each other for concrete examples are always welcome and encouraged!  Sometime in the semester, we will shift towards working on a collaborative and related research project, with the aim towards publication.

Note that we will not have homework, exams, or end-of-the-semester write-ups.  The goal of this research course is to write a collaborate paper by the end of the semester.

Note that we started the semester as a learning seminar, learning a lot of basic and important ideas. As we are approaching the end of the semester, we are having a lot of intense collaboration discussions. We are developing new mathematics and new ideas during these meetings, not outside these meetings. We are not reading books after books or papers after papers.

Jan 24  (Friday):  

No meeting today  (students are having problems with registration and the building heater needs to be fixed)

• We will most likely meet additional days later in the semester.

Jan 31  (Friday,  5 - 8pm):  

Geometric background:  flag varieties  (Section 2)  [Mee Seong Im]

HW:  Read through the rest of Section 2,  discuss this in our next meeting

Feb 7  (Friday,  5 - 8pm):  

Complete Geometric background:  Springer resolution, Grothendieck-Springer resolution  (Section 2)  [Mee Seong Im]

Feb 14  (Friday, 5 - 6:30pm):  

Variational geometric invariant theory on stacks and the Hilbert scheme  [Mee Seong Im]

Feb 19  (Wednesday,  5 - 7pm):  

S^n C^2 as the Hamiltonian reduction of a certain moment map and quiver varieties  (explicit constructions following Nakajima,  Gan--Ginzburg,  GIT techniques)  [Haihan Wu]

Feb 21  (Friday,  5 - 6pm):  

Hilbert scheme of points on a complex plane as a blow up of an ideal (sheaf)  [Mee Seong Im]

Feb 24  (Monday, 5 - 7pm):  

Hilbert scheme of points on a complex plane as a blow up  [Mee Seong Im]

Feb 28  (Friday, 5 - 7pm):  

An overview of the categorical braid group action on the bounded derived category of coherent sheaves on the cotangent bundle of the flag variety  (all Sections)  [Mee Seong Im]

Mar 3  (Monday, 5 - 7pm):  

Representation theory of quantum groups and diagrammatic algebras  [Haihan Wu]

Our research problem  [Mee Seong Im]

Mar 7  (Friday, 5 - 7pm):

Geometric categorification in full detail, including Khovanov-Thomas and Huerfano-Khovanov papers  [Catie Lillja]

Our research problem,  further discussions  [Mee Seong Im]

Derived categories, a short intro/computation  [Tomas Mejia Gomez]

Mar 10  (Monday, 5 - 7pm): 

Springer fibers and webs correspondence, and their combinatorial and topological structures  [Mee Seong Im]

Mar 14  (Friday, 5 - 7:30pm):  

Certain webs  (research problem)  [Haihan Wu]

Review Monday's class  (flags in an irreducible component of a Springer fiber of type A and Delta Springer fiber)  [Mee Seong Im]

Mar 17 - 21:  Spring Break

Mar 27  (Thursday, 4 - 6pm):

Cohomology ring of a Springer fiber and Delta Springer fiber  [Nicolas Rugo]

Research updates  [everyone]

Mar 31  (Monday, 5 - 7pm):

Cohomology ring of a Delta Springer fiber  [Nicolas Rugo]

Research updates  [everyone]

April 4 - 6  (Friday - Sunday):

Meet in Washington, DC

Knots in Washington 51 at George Washington University  [I encourage each person participating to give a talk]

Apr 11  (Friday, 5 - 7pm):

Standard and semistandard Young tableaux and sl(3)-webs, and affine pavings  [Haihan Wu and everyone]

Web and link correspondence, and quantum invariants  [Mikhail Khovanov,  Zoom]

Apr 14  (Monday, 5 - 7pm):

Nonelliptic web bases and Young tableaux  [Tomas Mejia Gomez]

Braid group action on bounded derived category of coherent sheaves on the cotangent bundle of flag manifolds,  Spaltenstein varieties,  triangulated categories  [Mikhail Khovanov,  In-person]

Apr 18  (Friday, 4 - 6:15pm,  Khovanov's office):

Research updates  [Tomas Mejia Gomez]

Feedback  [Mikhail Khovanov,  Zoom]

Apr 25  (Friday,  4 - 6:30pm,  Khovanov's office):

Diagrammatics, geometry, and representation theory of hook partitions  [Mikhail Khovanov]

Certain Springer and Spaltenstein varieties and their proofs  [Tomas Mejia Gomez and Mee Seong Im]

Cohomology ring and top nonvanishing cohomology module of Springer fibers applied to our research setting  [Nicolas Rugo]

sl4 and sln-webs  [Haihan Wu]

Braid group action on bounded derived category of coherent sheaves on the cotangent bundle of flag manifolds,  Spaltenstein varieties,  triangulated categories  [Mikhail Khovanov]

Apr 25  (Friday, 6:30 - 8:30pm):

Research Group Dinner!

May 2 - 4  (Friday - Sunday):

Southeastern Lie Theory Conference at the College of Charleston

Research Group Dinner in Baltimore, MD.