Department Seminars & Colloquia
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The higher Chow group is introduced by S. Bloch, which satisfies localization long exact sequence extending the classical Chow group. It is also related to arithmetic questions such as the special value of L-functions. It is interesting question to find varieties with big higher Chow group. In this talk, we construct surfaces over the formal Laurent series field over C, with big higher Chow group. We use the etale cycle map c(X) and the monodromy weight spectral sequence to compute the lower bound of dim(Im(c(X))).
Computational social choice is a growing discipline at the interface of social choice theory and computer science. It is concerned with the application of computational techniques to the study of social choice mechanisms, and with the integration of social choice paradigms into computing. Several problems have been investigated in the framework of parameterized complexity. This talk will describe the election scheme of Charles Dodgson. In 1876 the mathematician Charles Dodgson (better known as Lewis Carroll) formulated a rule that defines the winner of an election even if there is no Condorcet winner. A candidate who beats every other candidate in pairwise comparison is said to be a Condorcet winner. Unfortunately, an election may have a cyclic structure (candidate a beats b, candidate b beats c and c beats a), and therefore no candidate who beats all others in pairwise comparison. The Dodgson Score measures how close a candidate is to being a Condorcet winner. Candidates with a minimum Dodgson Score are the election winners. As are many election methods, Dodgson Score is NP-hard. This talk discusses the complexity of Dodgson Score in the parameterized framework. We give a reduction from Multi-Colored Clique to show that Dodgson Score, parameterized by the number of votes, is W[1]-hard. When parameterized by the number of swaps, Dodgson Score is FPT, but we show by polynomial parameter transformation that it has no polynomial kernel.
We now have rather satisfactory answer to many questions about orbifolds CC^n/G, where n = 2 or 3 and G is a finite subgroup of SL(n,CC). For example, the G-Hilbert scheme provides a standard crepant resolution of singularities, every projective crepant resolution represents an appropriate moduli functor involving G-equivariant sheaves on CC^n, and the derived category, K theory, homology or cohomology of a crepant resolution can be treated in terms of G-equivariant structures on CC^n. However, not much is known about n >= 4, or about finite subgroups of GL(3,CC). The talk will describe some ongoing work on the case of the terminal 3-fold points 1/r(1,a,r-a), mainly due to JUNG Seung-Jo (Warwick), and on some cases in dimension >= 4 that are almost tractable.
We study the convex hull of the symmetric moment curve Uk(t)=(cost, sint, cos3t, sin3t, …., cos(2k-1)t, sin(2k-1)t) in R2k and provide deterministic constructions of centrally symmetric polytopes with a record high number faces. In particular, we prove the local neighborliness of the symmetric moment curve, meaning that as long as k distinct points t1, …, tk lie in an arc of a certain length φk > π/2, the points Ut1, …, Utk span a face of the convex hull of Uk(t). In this talk, I will use the local neighborliness of the symmetric moment curve to construct d-dimensional centrally symmetric 2-neighborly polytopes with approximately 3d/2 vertices.
This is joint work with Alexander Barvinok and Isabella Novik.
In the theory of elliptic P.D.E.'s, an overdetermined problem is one where both the Dirichlet and Neumann boundary values are prescribed. This puts strong geometric constraints on the domain. A famous result of J. Serrin asserts that if is a bounded domain D in R^n which admits a function u solution of ꠑ Delta u =-1 in D with zero Dirichlet boundary value and constant Neumann boundary values, then D is a ball. The boundary of D is then a sphere, a constant mean curvature. I will present other similar results and in particular the existence of a 1-to-1 correspondence between harmonic functions which solve an overdetermined problem and a certain type of minimal surfaces.
A braid is a structure formed by intertwining a number of strands, such as textiles or human hairs. As a mathematical object, a set of braids forms a group, called a braid group which was firstly introduced by E. Artin in 1920’s, and generalized to any topological space via configuration spaces. Nevertheless, the braid theory has been researched only on manifolds until the late 1990’s when Ghrist published some results about the braid group on graphs. After Ghrist, many people studied braid groups on graphs. However for general CW (or simplicial) complexes of dimension greater than 1, the braid theory is still an unexplored field.
In this talk, we focus on the braid group on a finite regular CW complex of dimension 2 and we explain how a decomposition of given space is related to a decomposition of its braid group and how to build up the braid group from the simple ones. As an application, we figure out the hierarchy structure that the braid group admits and the relations between group theoretical properties of the braid group and geometrical properties of a given CW complex, such as, embeddability into a manifold or planarity.
A braid is a structure formed by intertwining a number of strands, such as textiles or human hairs. As a mathematical object, a set of braids forms a group, called a braid group which was firstly introduced by E. Artin in 1920’s, and generalized to any topological space via configuration spaces. Nevertheless, the braid theory has been researched only on manifolds until the late 1990’s when Ghrist published some results about the braid group on graphs. After Ghrist, many people studied braid groups on graphs. However for general CW (or simplicial) complexes of dimension greater than 1, the braid theory is still an unexplored field.
In this talk, we focus on the braid group on a finite regular CW complex of dimension 2 and we explain how a decomposition of given space is related to a decomposition of its braid group and how to build up the braid group from the simple ones. As an application, we figure out the hierarchy structure that the braid group admits and the relations between group theoretical properties of the braid group and geometrical properties of a given CW complex, such as, embeddability into a manifold or planarity.
The $¥Gamma$-polynomial is an invariant of an oriented link in the 3-sphere, which is contained in both the HOMFLYPT and Kauffman polynomials as their common zeroth coefficient polynomial. As applications of the $¥Gamma$-polynomial, I will talk about the following
three topics:
(1) On the arc index of cable knots (joint with Hwa Jeong Lee, KAIST)
(2) On the braid index of Kanenobu knots
(3) On the arc index of Kanenobu knots (joint with Hwa Jeong Lee, KAIST)