We introduce the algebraic connective K-theory and discuss its relations with some other oriented cohomology theories. Then we present recent results on connective K-theory of varieties of parabolic subgroups in semisimple algebraic groups.

A well-known Ramsey-type puzzle for children is to prove that among any 6 people either there are 3 who know each other or there are 3 who do not know each other. More generally, a graph G arrows a graph H if for any coloring of the edges of G with two colors, there is a monochromatic copy of H. In these terms, the above puzzle claims that the complete 6-vertex graph K_6 arrows the complete 3-vertex graph K_3. We consider sufficient conditions on the dense host graphs G to arrow long paths and even cycles. In particular, for large n we describe all multipartite graphs that arrow paths and cycles with 2n edges. This implies a conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi from 2007 for such n. Also for large n we find which minimum degree in a (3n-1)-vertex graph G guarantees that G arrows the 2n-vertex path. This yields a more recent conjecture of Schelp. This is joint work with Jozsef Balogh, Mikhail Lavrov and Xujun Liu. (*Joint Colloquium between KAIST Mathematical Sciences and IBS Discrete Mathematics Group)

Support vector machine (SVM) is a very popular technique for classification. A key property of SVM is that its discriminant function depends only on a subset of data points called support vectors. This comes from the representation of the discriminant function as a linear combination of kernel functions associated with individual cases. Despite the direct relation between each case and the corresponding coefficient in the representation, the influence of cases and outliers on the classification rule has not been examined formally. Borrowing ideas from regression diagnostics, we define case influence measures for SVM and study how the classification rule changes as each case is perturbed. To measure case sensitivity, we introduce a weight parameter for each case and reduce the weight from one to zero to link the full data solution to the leave-one-out solution. We develop an efficient algorithm to generate case-weight adjusted solution paths for SVM. The solution paths and the resulting case influence graphs facilitate evaluation of the influence measures and allow us to examine the relation between the coefficients of individual cases in SVM and their influences comprehensively. We present numerical results to illustrate the benefit of this approach.

After the discovery of algebraic $K$-theory of schemes by Quillen, it was anticipated that algebraic cycles would play a fundamental role in computing the $K$-groups of schemes. These groups are usually very hard to compute even as they contain a lot of information about the underlying scheme. For smooth schemes, the problem of describing $K$-theory by algebraic cycles was satisfactorily settled by works of several mathematicians, spanned over many years.

However, solving this problem under the presence of singularities is still a very challenging task.

The theory of additive Chow groups and the Chow groups with modulus were invented to address this problem.

Even if it is not yet clear if these Chow groups will finally solve the problem, I shall present some positive results in this direction in my talk. In particular, I shall show that the additive Chow groups do completely describe the $K$-theory of infinitesimal neighborhoods of the origin in the affine line. This was the initial motivation of the discovery of additive Chow groups by Bloch and Esnault. This talk is based on a joint work with Rahul Gupta.

 

After the discovery of algebraic $K$-theory of schemes by Quillen, it was anticipated that algebraic cycles would play a fundamental role in computing the $K$-groups of schemes. These groups are usually very hard to compute even as they contain a lot of information about the underlying scheme. For smooth schemes, the problem of describing $K$-theory by algebraic cycles was satisfactorily settled by works of several mathematicians, spanned over many years.

However, solving this problem under the presence of singularities is still a very challenging task.

 

The theory of additive Chow groups and the Chow groups with modulus were invented to address this problem.

 

Even if it is not yet clear if these Chow groups will finally solve the problem, I shall present some positive

 

results in this direction in my talk. In particular, I shall show that the additive Chow groups do completely describe the $K$-theory of infinitesimal neighborhoods of the origin in the affine line. This was the initial motivation of the discovery of additive Chow groups by Bloch and Esnault. This talk is based on a joint work with Rahul Gupta.