Department Seminars & Colloquia
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A finite simple graph is called thin-chordal if it has neither a square nor a path with 4 vertices as induced subgraphs.
We show that a RAAG is thin-chordal if and only if its every finite index subgroup is a RAAG. We classfies RAAGs in the class of RAAGs whose defining graphs are thin chordal up to quasi-isometry (and commesurability).
The classical Schubert calculus arose from classical enumerative geometric problems, and is concerned with the study of the cohomology ring of complex Grassmannians, or generally, of homogeneous varieties G/P of general Lie types. One of the most central problems in this subject is to find a manifestly positive formula of the structure constants for the cup product of Schubert cohomology classes. On the other hand, a Gelfand-Cetlin polytope is a special convex polytope, occurring in many subjects such as representation theory, toric geometry, mathematical physics, and combinatorics. It turns out to be also closely related with Schubert calculus, as was studied by Kogan and Kiritchenko-Smirnov-Timorin. In these lectures, I will talk about the relationship between both sides. In Lecture I, I will give an introduction to the classical Schubert calculus. In Lecture II, I will review some basic properties of Gelfand-Cetlin polytopes and systems, as well as some relationships between them and Schubert calculus. In Lecutre III, IV, I will talk about toric degeneration of flag varieties, together with the transversal intersection of certain Schubert varieties, based on my joint work with DongSeon Hwang, Hwayoung Lee and Jae-Hyouk Lee.
The classical Schubert calculus arose from classical enumerative geometric problems, and is concerned with the study of the cohomology ring of complex Grassmannians, or generally, of homogeneous varieties G/P of general Lie types. One of the most central problems in this subject is to find a manifestly positive formula of the structure constants for the cup product of Schubert cohomology classes. On the other hand, a Gelfand-Cetlin polytope is a special convex polytope, occurring in many subjects such as representation theory, toric geometry, mathematical physics, and combinatorics. It turns out to be also closely related with Schubert calculus, as was studied by Kogan and Kiritchenko-Smirnov-Timorin. In these lectures, I will talk about the relationship between both sides. In Lecture I, I will give an introduction to the classical Schubert calculus. In Lecture II, I will review some basic properties of Gelfand-Cetlin polytopes and systems, as well as some relationships between them and Schubert calculus. In Lecutre III, IV, I will talk about toric degeneration of flag varieties, together with the transversal intersection of certain Schubert varieties, based on my joint work with DongSeon Hwang, Hwayoung Lee and Jae-Hyouk Lee.
Kempf-Laksov's resolution is a resolution of singularities
of Schubert varieties in Grassmannian, used by Kempf-Laksov to obtain
a determinant formula of Schubert classes that is equivalent to the
Jacobi-Trudi formula of Schur polynomials. In 2015,
Hudson-Ikeda-M.-Naruse used it to obtain the corresponding formula in
K-theory. In a general cohomology theory beyond K-theory, we know that
there is no well-defined notation of fundamental classes of Schubert
varieties, and therefore one uses the classes of resolutions as
replacements of Schubert classes. Bott-Samelson resolutions are such
candidates, while we focus on Kempf-Laksov resolutions.
In these two lectures, I will try to explain (1) the notion of
oriented cohomology theories and algebraic cobordism and (2) how to
compute Kempf-Laksov Schubert classes. In (1), the goals are to set up
the framework where we can do computations with enough axioms and to
introduce the (relative) Segre classes that would be key ingredients
to describe ``Schubert classes''. In (2), the main is the construction
of Kempf-Laksov's resolutions of Schubert varieties through the tower
of projective bundles. I will also try to explain a certain algebraic
technique due to Kazarian and Hudson-Ikeda-M.-Naruse to describe the
pushforward of Chern classes along the tower systematically.
The classical Schubert calculus arose from classical enumerative geometric problems, and is concerned with the study of the cohomology ring of complex Grassmannians, or generally, of homogeneous varieties G/P of general Lie types. One of the most central problems in this subject is to find a manifestly positive formula of the structure constants for the cup product of Schubert cohomology classes. On the other hand, a Gelfand-Cetlin polytope is a special convex polytope, occurring in many subjects such as representation theory, toric geometry, mathematical physics, and combinatorics. It turns out to be also closely related with Schubert calculus, as was studied by Kogan and Kiritchenko-Smirnov-Timorin. In these lectures, I will talk about the relationship between both sides. In Lecture I, I will give an introduction to the classical Schubert calculus. In Lecture II, I will review some basic properties of Gelfand-Cetlin polytopes and systems, as well as some relationships between them and Schubert calculus. In Lecutre III, IV, I will talk about toric degeneration of flag varieties, together with the transversal intersection of certain Schubert varieties, based on my joint work with DongSeon Hwang, Hwayoung Lee and Jae-Hyouk Lee.
The classical Schubert calculus arose from classical enumerative geometric problems, and is concerned with the study of the cohomology ring of complex Grassmannians, or generally, of homogeneous varieties G/P of general Lie types. One of the most central problems in this subject is to find a manifestly positive formula of the structure constants for the cup product of Schubert cohomology classes. On the other hand, a Gelfand-Cetlin polytope is a special convex polytope, occurring in many subjects such as representation theory, toric geometry, mathematical physics, and combinatorics. It turns out to be also closely related with Schubert calculus, as was studied by Kogan and Kiritchenko-Smirnov-Timorin. In these lectures, I will talk about the relationship between both sides. In Lecture I, I will give an introduction to the classical Schubert calculus. In Lecture II, I will review some basic properties of Gelfand-Cetlin polytopes and systems, as well as some relationships between them and Schubert calculus. In Lecutre III, IV, I will talk about toric degeneration of flag varieties, together with the transversal intersection of certain Schubert varieties, based on my joint work with DongSeon Hwang, Hwayoung Lee and Jae-Hyouk Lee.
Kempf-Laksov's resolution is a resolution of singularities
of Schubert varieties in Grassmannian, used by Kempf-Laksov to obtain
a determinant formula of Schubert classes that is equivalent to the
Jacobi-Trudi formula of Schur polynomials. In 2015,
Hudson-Ikeda-M.-Naruse used it to obtain the corresponding formula in
K-theory. In a general cohomology theory beyond K-theory, we know that
there is no well-defined notation of fundamental classes of Schubert
varieties, and therefore one uses the classes of resolutions as
replacements of Schubert classes. Bott-Samelson resolutions are such
candidates, while we focus on Kempf-Laksov resolutions.
In these two lectures, I will try to explain (1) the notion of
oriented cohomology theories and algebraic cobordism and (2) how to
compute Kempf-Laksov Schubert classes. In (1), the goals are to set up
the framework where we can do computations with enough axioms and to
introduce the (relative) Segre classes that would be key ingredients
to describe ``Schubert classes''. In (2), the main is the construction
of Kempf-Laksov's resolutions of Schubert varieties through the tower
of projective bundles. I will also try to explain a certain algebraic
technique due to Kazarian and Hudson-Ikeda-M.-Naruse to describe the
pushforward of Chern classes along the tower systematically.