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Information Center for Mathematical Science

세미나

Information Center for Mathematical Science

세미나

POSTECH IBS-CGP Seminar

KAIST Discrete Math 세미나

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational geometry. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p∈P is seen by at least one guard g∈G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃R. The class ∃R consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP⊆∃R. We prove that the art gallery problem is ∃R-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃R. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many geometric approaches to the problem. This is joint work with Mikkel Abrahamsen and Anna Adamaszek.

2018 서울대학교 수리과학부 세미나

Quiver Hecke algebra is a generalization of the affine Hecke algebra from the point of view of categorificaiton of a half of the quantum group. In this lecture, we will review some basic properties on quiver Hecke algebras. In this lecture, we will review some basic properties on quiver Hecke algebras. We will study the finite dimensional graded representations of quiver Hecke algebras. The notions of characters and convolution products will be reviewed; the categorification theorems will be presented without proof.

한국고등과학원 세미나

A mathematical treatment of TQFT, based on category theory, was initiated in the early 90's. In more modern times the mathematics of TQFT has come to use advanced techniques from derived geometry and higher algebra (? la Lurie). This series of talks is loosely aimed at explaining how some of these advanced techniques apply in familiar examples of supersymmetric field theory and its topological twists. For physics, this will lead to some surprising new structure, as well as some useful organizing principles. The first and second lectures are based on work with C. Beem, D. Ben-Zvi, M. Bullimore, and A. Neitzke. The first lecture will re-examine operator algebras in topological twists of supersymmetric field theories. In d dimension, the algebras naturally come equipped with a Lie bracket of degree 1-d, which can be realized very concretely via topological descent. We'll look at examples of this bracket, in 2d, 3d, and 4d. We'll also use topological descent to give a new interpretation of the statement that turning on on Omega background is a form of quantization. The second lecture focuses on another sort of homological/descent operation, this time in the context of SUSY quantum mechanics with G symmetry. We'll find that Hilbert spaces in (de Rham) SQM come equipped with a natural homological action of G, i.e. an action of the exterior algebra H_*(G). This action controls the process of gauging the G symmetry. A nice physical way to understand the H_*(G) action and gauging/ungauging comes from considering SUSY boundary conditions for 2d G gauge theory; this will lead us to a definition of the "Fukaya category of point/G." The mathematical structures involved come from work of Goresky, Kottwitz, and MacPherson. Given time, we'll discuss some higher-dimensional examples, and the physics of Koszul duality. The third lecture applies some of the ideas from the first two in the specific context of 3d N=4 gauge theories. Our goal will be to define the category of line operators in the A and B twists of these theories, and to understand -- in a concrete and computational way -- the algebras of local operators bound to a line. These algebras get quantized in an Omega background; and they act on modules defined by boundary conditions. In the special case of an A twist and the trivial (identity) line operator, we will recover the Braverman-Finkelberg-Nakajima construction of the Coulomb-branch chiral ring.

한국고등과학원 세미나

A mathematical treatment of TQFT, based on category theory, was initiated in the early 90's. In more modern times the mathematics of TQFT has come to use advanced techniques from derived geometry and higher algebra (? la Lurie). This series of talks is loosely aimed at explaining how some of these advanced techniques apply in familiar examples of supersymmetric field theory and its topological twists. For physics, this will lead to some surprising new structure, as well as some useful organizing principles. The first and second lectures are based on work with C. Beem, D. Ben-Zvi, M. Bullimore, and A. Neitzke. The first lecture will re-examine operator algebras in topological twists of supersymmetric field theories. In d dimension, the algebras naturally come equipped with a Lie bracket of degree 1-d, which can be realized very concretely via topological descent. We'll look at examples of this bracket, in 2d, 3d, and 4d. We'll also use topological descent to give a new interpretation of the statement that turning on on Omega background is a form of quantization. The second lecture focuses on another sort of homological/descent operation, this time in the context of SUSY quantum mechanics with G symmetry. We'll find that Hilbert spaces in (de Rham) SQM come equipped with a natural homological action of G, i.e. an action of the exterior algebra H_*(G). This action controls the process of gauging the G symmetry. A nice physical way to understand the H_*(G) action and gauging/ungauging comes from considering SUSY boundary conditions for 2d G gauge theory; this will lead us to a definition of the "Fukaya category of point/G." The mathematical structures involved come from work of Goresky, Kottwitz, and MacPherson. Given time, we'll discuss some higher-dimensional examples, and the physics of Koszul duality. The third lecture applies some of the ideas from the first two in the specific context of 3d N=4 gauge theories. Our goal will be to define the category of line operators in the A and B twists of these theories, and to understand -- in a concrete and computational way -- the algebras of local operators bound to a line. These algebras get quantized in an Omega background; and they act on modules defined by boundary conditions. In the special case of an A twist and the trivial (identity) line operator, we will recover the Braverman-Finkelberg-Nakajima construction of the Coulomb-branch chiral ring.

한국고등과학원 세미나

A mathematical treatment of TQFT, based on category theory, was initiated in the early 90's. In more modern times the mathematics of TQFT has come to use advanced techniques from derived geometry and higher algebra (? la Lurie). This series of talks is loosely aimed at explaining how some of these advanced techniques apply in familiar examples of supersymmetric field theory and its topological twists. For physics, this will lead to some surprising new structure, as well as some useful organizing principles. The first and second lectures are based on work with C. Beem, D. Ben-Zvi, M. Bullimore, and A. Neitzke. The first lecture will re-examine operator algebras in topological twists of supersymmetric field theories. In d dimension, the algebras naturally come equipped with a Lie bracket of degree 1-d, which can be realized very concretely via topological descent. We'll look at examples of this bracket, in 2d, 3d, and 4d. We'll also use topological descent to give a new interpretation of the statement that turning on on Omega background is a form of quantization. The second lecture focuses on another sort of homological/descent operation, this time in the context of SUSY quantum mechanics with G symmetry. We'll find that Hilbert spaces in (de Rham) SQM come equipped with a natural homological action of G, i.e. an action of the exterior algebra H_*(G). This action controls the process of gauging the G symmetry. A nice physical way to understand the H_*(G) action and gauging/ungauging comes from considering SUSY boundary conditions for 2d G gauge theory; this will lead us to a definition of the "Fukaya category of point/G." The mathematical structures involved come from work of Goresky, Kottwitz, and MacPherson. Given time, we'll discuss some higher-dimensional examples, and the physics of Koszul duality. The third lecture applies some of the ideas from the first two in the specific context of 3d N=4 gauge theories. Our goal will be to define the category of line operators in the A and B twists of these theories, and to understand -- in a concrete and computational way -- the algebras of local operators bound to a line. These algebras get quantized in an Omega background; and they act on modules defined by boundary conditions. In the special case of an A twist and the trivial (identity) line operator, we will recover the Braverman-Finkelberg-Nakajima construction of the Coulomb-branch chiral ring.

한국고등과학원 세미나

Compatibility between Hecke and Excursion Operators

한국고등과학원 세미나

Given $u_{1},u_{2}$ each solutions to a quasilinear equation, under some mild boundedness assumptions on the coefficients, we show a Hopf boundary point lemma for the difference $u=u_{1}-u_{2}.$ We apply this, for example, to a regularity problem in area-minimizing currents.

한국고등과학원 세미나

I will present the study of two overdetermined elliptic boundary value problems on exterior domains (the complement of a ball and the complement of a solid cylinder in R^3 respectively). The Neumann condition is non-constant and involves the mean curvature of the boundary. For each problem I show there exist infinitely many bifurcation branches of domains which are small deformations of the initial domain and which support the solution of the overdetermined boundary value problem.

한국고등과학원 세미나

A Clifford-Klein form is a quotient of a homogeneous space G/H by a discrete subgroup of G acting properly and freely on G/H. It naturally admits the structure of a manifold locally modelled on G/H. We focus on the non-Riemannian case (i.e. the case when H is noncompact). The study of Clifford-Klein forms in this case is much harder than the Riemannian case because not every discrete subgroup of G acts properly on G/H. I will explain some obstructions to the existence of compact Clifford-Klein forms, obtained by comparing relative Lie algebra cohomology and de Rham cohomology, and their reinterpretation in terms of invariant theory. Corollaries to these results includes: - every complete pseudo-Riemannian manifold of signature (p, q) with positive constant sectional curvature is noncompact if p, q > 0, q: odd. - every Clifford-Klein form of a homogeneous space G/H of reductive type is noncompact if rank G - rank K < rank H - rank K_H, where K and K_H are the maximal compact subgroups of G and H, respectively (T. Kobayashi’s rank conjecture).

한국고등과학원 세미나

한국고등과학원 세미나

A Clifford-Klein form is a quotient of a homogeneous space G/H by a discrete subgroup of G acting properly and freely on G/H. It naturally admits the structure of a manifold locally modelled on G/H. We focus on the non-Riemannian case (i.e. the case when H is noncompact). The study of Clifford-Klein forms in this case is much harder than the Riemannian case because not every discrete subgroup of G acts properly on G/H. I will explain some obstructions to the existence of compact Clifford-Klein forms, obtained by comparing relative Lie algebra cohomology and de Rham cohomology, and their reinterpretation in terms of invariant theory. Corollaries to these results includes: - every complete pseudo-Riemannian manifold of signature (p, q) with positive constant sectional curvature is noncompact if p, q > 0, q: odd. - every Clifford-Klein form of a homogeneous space G/H of reductive type is noncompact if rank G - rank K < rank H - rank K_H, where K and K_H are the maximal compact subgroups of G and H, respectively (T. Kobayashi’s rank conjecture).

한국고등과학원 세미나

Construction of Excursion Operators

한국고등과학원 세미나

한국고등과학원 세미나

Geometric Satake Equivalence

한국고등과학원 세미나

Sheaves on Moduli of Shtukas

한국고등과학원 세미나

Preverse Sheaves and Affine Grassmanian

한국고등과학원 세미나

The scenario of Type I singularity is a natural generalization of the self-similar or discretely self-similar singularities. Studying the problem of Type I singularities would be eventually helpful for future understanding of the possible singularities in the Euler system. In this talk, after short review of the studies of the self-similar solutions we survey recent works on scenario of the Type I blow-up. By applying the local analysis methods, which have been useful in the elliptic or parabolic regularity theories, we could make some progresses in our study of the Type I blow-up in the Euler equations. The project is jointly done with J. Wolf.

한국고등과학원 세미나

In this talk, we will talk about 3-dimensional immersed crosscap number of a knot, which is a non-orientable analogue of the immersed Seifert genus. We study knots with immersed crosscap number 1, and show that a knot has immersed crosscap number 1 if and only if it is a nonntrivial (2p,q)-torus or (2p, q)-cable knot. We show that unlike in the orientable case the immersed crosscap number can differ from the embedded crosscap number by arbitrarily large amounts, and that it is neither bounded below nor above by the 4-dimensional crosscap number. We then use these constructions to find an infinite family of 3-manifolds such that one of its second Z2 homology class can be represented by an immersed real projective plane but any embedded representative of it has a component with Euler characteristic strictly less than 1.

한국고등과학원 세미나

In this lecture series I will sketch some relationships between geometric Langlands correspondence and symplectic duality, by identifying their physical origin as 4d N=4 and 3d N=4 supersymmetric theories, respectively, and investigating consequences of the expected results of quantum field theory. This program is jointly conceived with Justin Hilburn, but the aim of the series is rather modest, introducing the subject of symplectic duality and providing an overview of the program with simplest examples.