Department Seminars & Colloquia
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We consider the one-dimensional cubic fractional nonlinear Schrödinger equation. Due to non-locality of the fractional Laplacian, the equation does not have any Galilean-type invariance. Despite of lack of this symmetry, we can still construct a new class of traveling soliton solutions by a rather involved variational argument.
By coarse classification theorem of tight contact structures, it is known that every closed, atoroidal 3-manifolds attains at most finite tight contact structures up to contact isotopy. However, the explicit number, even the existence, of the tight contact structures remains a mystery. In this talk we introduce basic theories for classification problem and investigate the number of tight contact structures of certain hyperbolic 3-manifolds up to contact isotopy.
A fractional matching of a graph G is a function f giving each edge a number between 0 and 1 so that 
for each 
, where 
is the set of edges incident to v. The fractional matching number of G, written 
, is the maximum of 
over all fractional matchings f. Let G be an n-vertex graph with minimum degree d, and let 
be the largest eigenvalue of G. In this talk, we prove that if k is a positive integer and
, then 
Determination of coronary physiology is critical to the diagnosis and treatment of patients with coronary artery disease. Traditionally, assessment of coronary physiology required invasive coronary angiography. Here the non-invasive assessment of coronary physiology based on the image analysis of coronary computed tomography (CT) angiography, which might replace the invasive assessment methods, would be discussed.
The notion of a cohomological invariant of analgebraic group was introduced by J-P. Serre. Cohomological invariants of an algebraic group G relate principal homogeneous spaces of G over a field extension of the base field (G-torsors) and Galois cohomology of the field. If A is an "algebraic object", then the principal homogeneous spaces for the automorphism group G = Aut(A) are in one-to-one correspondence with the twisted forms of A. In such a way many classical algebraic objects arise: simple algebras, quadratic and hermitian forms, algebras with involutions, Cayley-Dickson algebras, etc. Thus, cohomological invariants assign to algebraic objects the cohomology classes.
We will compute cohomological invariants of small degrees. Some applications will be given. In particular, unramified invariants can be used to determine non-rationality property of classifying spaces of algebraic groups.
Abstract: This 8-hour course will cover many of the big themes in combinatorial and integer optimization introduced in the last half century.
Thurs Dec 11, 10:00-12:00
Combinatorial optimization: Shortest paths and dynamic programming. Nonbipartite matching. Matroid intersection. TSP. Submodular function maximization.
We prove that on a punctured oriented surface with Eulercharacteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a closed surface.
Essential dimension of an algebraic object is the smallest number of algebraically independent parameters required to define the object. This notion was introduced by Buhler, Reichstein and Serre about 20 years ago.The relations to different parts of algebra such asalgebraic geometry, Galois cohomology and representation theory will be discussed.
The notion of a cohomological invariant of analgebraic group was introduced by J-P. Serre. Cohomological invariants of an algebraic group G relate principal homogeneous spaces of G over a field extension of the base field (G-torsors) and Galois cohomology of the field. If A is an "algebraic object", then the principal homogeneous spaces for the automorphism group G = Aut(A) are in one-to-one correspondence with the twisted forms of A. In such a way many classical algebraic objects arise: simple algebras, quadratic and hermitian forms, algebras with involutions, Cayley-Dickson algebras, etc. Thus, cohomological invariants assign to algebraic objects the cohomology classes.
We will compute cohomological invariants of small degrees. Some applications will be given. In particular, unramified invariants can be used to determine non-rationality property of classifying spaces of algebraic groups.
Abstract: This 8-hour course will cover many of the big themes in combinatorial and integer optimization introduced in the last half century.
Tues Dec 9, 10:00-12:00
IP Formulation and cuts: UFL. Big M's. Generic cutting planes: Gomory for pure; BMI, MIR, GMI for mixed. Disjunctive cuts and the CGLP. Combinatorial cuts.
The notion of a cohomological invariant of analgebraic group was introduced by J-P. Serre. Cohomological invariants of an algebraic group G relate principal homogeneous spaces of G over a field extension of the base field (G-torsors) and Galois cohomology of the field. If A is an "algebraic object", then the principal homogeneous spaces for the automorphism group G = Aut(A) are in one-to-one correspondence with the twisted forms of A. In such a way many classical algebraic objects arise: simple algebras, quadratic and hermitian forms, algebras with involutions, Cayley-Dickson algebras, etc. Thus, cohomological invariants assign to algebraic objects the cohomology classes.
We will compute cohomological invariants of small degrees. Some applications will be given. In particular, unramified invariants can be used to determine non-rationality property of classifying spaces of algebraic groups.
Abstract: This 8-hour course will cover many of the big themes in combinatorial and integer optimization introduced in the last half century.
Mon Dec 8, 14:00-16:00
Integrality for free: part 1 -total unimodularity and networks; part 2 - matroids and the greedy algorithm.
It is known that every knot bounds a singular disk whose singular set consists of only clasp singularities. Such a singular disk is called a clasp disk. The clasp number of a knot is the minimum number of clasp singularities among all clasp disks of the knot. The $Gamma$-polynomial is the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials. I will talk about a characterization of the $Gamma$-polynomials of knots with the clasp numbers at most two.
Abstract: This 8-hour course will cover many of the big themes in combinatorial and integer optimization introduced in the last half century.
Fri Dec 5, 10:00-12:00
Convexity-based algorithms: Submodular function minimization. Subgradient optimization. Ellipsoid method and its implications for combinatorial optimization.
자연과학동 E6-1 Room 1409
Discrete Math
Jon Lee (University of Michigan, Ann Arbor, USA.)
Matroid Optimization
자연과학동 E6-1 Room 1409
Discrete Math
Matroids can be seen an an abstraction for understanding the essence of algorithms for some classical graph problems: (i) optimum-weight spanning tree, and (ii) optimum-weight assignment. We will see some recent results showing how matroids have further natural roles in various combinatorial-optimization problems where some local-search and linear-programming-based algorithms find provably-good approximations. In particular, we will look at: (iii) constrained submodular-function maximization, (iv) the well-known matroid matching problem, and (v) various nonlinear-objective matroid-intersection problems.
In this talk, I will present recent progress on the following subjects: (1) Smooth transonic flow of Euler-Poisson system; (2) Transonic shock of Euler-Poisson system. This talk is based on collaboration with Ben Duan(Dalian Univ. of Technology), Chujing Xie(SJTU) and Jingjing Xiao(Chinese Univ of Hong Kong).
자연과학동 E6-1 room 1409
Discrete Math
O-Joung Kwon
Upper bounds on the size of obstructions for graphs of linear rank-width at most k
자연과학동 E6-1 room 1409
Discrete Math
Graph layout problems are a class of optimization problems whose goal is to find a linear ordering of an input graph in such a way that a certain objective function is optimized. The matrix rank function has been studied as an objective function. The linear rank-width of a graph G is the minimum integer k such that G admits a linear ordering $v_1, v_2, ldots , v_n$ satisfying that the maximum over all values [operatorname{rank}A_G[{v_1, v_2, ldots, v_t}, {v_{t+1}, ldots, v_n}]] is k, where $A_G$ is the adjacency matrix of $G$ and the rank is computed over the binary field.
In this talk, we present a result that for every graph G that is vertex-minor minimal with the property having linear rank-width larger than p, the number of vertices in G is at most doubly exponential in $mathcal{O}(p)$. The number of vertex-minor obstructions for linear rank-width at most p is of interest because the only known fixed parameter tractable algorithm to test whether linear rank-width is at most p is using the finiteness of the number of forbidden vertex-minor obstructions. Our result gives an upper bound of the complexity on this algorithm. Our basic tools are the algebraic operations on labelled graphs introduced by Kante and Rao, and we extend the notion of vertex-minors in our purpose. This is joint work with Mamadou Moustapha Kante.
The warping degree of an oriented knot diagram with a base point is the number of crossing points such that we meet the point as an under-crossing first when we travel the diagram with the orientation from the base point.
In this talk, we introduce a matrix of an oriented knot diagram which is defined from a viewpoint of the warping degree, and consider its properties.
In particular, we show that the matrix represents an oriented knot diagram on the sphere uniquely.
As an application, we will consider a sudoku-like puzzle.