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Using elliptic regularity results, we construct for every starting point, weak solutions to SDEs in R^d with Sobolev diffusion and locally integrable drift coefficient up to their explosion times. Subsequently, we develop non-explosion criteria which allow for linear growth, singularities of the drift coefficient inside an arbitrarily large compact set, and an interplay between the drift and the diffusion coefficient. Moreover, we show strict irreducibility of the solution, which by construction is a strong Markov process with continuous sample paths on the one-point compactification of R^d. Joint work with Haesung Lee
A chordless cycle in a graph G is an induced subgraph of G which is a cycle of length at least four. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the Erdős-Pósa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either k+1 vertex-disjoint chordless cycles, or c k^2 log k vertices hitting every chordless cycle for some constant c. It immediately implies an approximation algorithm of factor O(OPT log OPT) for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least ℓ for any fixed ℓ≥ 5 do not have the Erdős-Pósa property.
We prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies a conjecture of Kohayakawa, Nagle, Rödl, and Schacht on random-like counts for small graphs in locally dense graphs. This implies an approximate version of the conjecture for graphs with bounded tree-width. We also prove an analogous result for odd cycles instead of complete multipartite graphs.
Okounkov bodies have become a very interesting and useful tool to understand the positivity of divisors. Although the Okounkov body carries rich positivity data of a divisor, it only provides information near a single point. In this talk, we introduce a new convex body of a divisor that is effective in handling the positivity theory in a multi-point setting. We study its various properties, and observe local positivity data via this convex body.
I will explain our recent progress on the construction of exceptional vector bundles on surfaces when they admit Q-Gorestein degenerations to singularities of class T_d. This is a generalization of the result of Hacking who has studied the case d=1. We give the construction of block(=completely orthogonal exceptional collection) of length d when d>1. If the underlying spaces are del Pezzo surfaces, then our construction explains the paralleism between toric degenerations and three block collections in derived categories.