2016 Pohang Mathematics Workshop
November 24–27, 2016
Novotel Ambassador Busan, Busan, South Korea
The aim of this workshop is to encourage academic exchange and promote friendly relations among researchers at mathematics institutions in the Pohang area.
Speakers
- Florin Ambro (Institute of Mathematics of the Romanian Academy)
- Jinseok Cho (Busan National University of Education)
- Sung Rak Choi (Yonsei University)
- Dmitry Doryn (Institute for Basic Science, Center for Geometry and Physics)
- Jin Woo Jang (Institute for Basic Science, Center for Geometry and Physics)
- Myeonggi Kwon (Seoul National University)
- Sangwook Lee (Korea Institute for Advanced Study)
- Byeongho Lee (National Institute for Mathematical Sciences)
- Damien Lejay (Institute for Basic Science, Center for Geometry and Physics)
- Yoshikazu Nagata (Center for Geometry and its Applications)
- Jongil Park (Seoul National University)
- Mehdi Tavakol (Institute for Basic Science, Center for Geometry and Physics)
- Youngho Yoon (Korea Institute for Advanced Study, Center for Mathematical Challenges)
Venue
Organizers
- Youngjin Bae (Institute for Basic Science, Center for Geometry and Physics)
- Hwajong Yoo (Institute for Basic Science, Center for Geometry and Physics)
Talk Schedule
| November 24 (Thu) | November 25 (Fri) | November 26 (Sat) | November 27 (Sun) | |
|---|---|---|---|---|
| 10:00 – 10:50 | Arrival and Registration | Florin Ambro | Dmitry Doryn | Jin Woo Jang |
| 11:10 – 12:00 | Byeongho Lee | Jongil Park | Yoshikazu Nagata | |
| 12:00 – 14:00 | Lunch | |||
| 14:00 – 14:50 | Damien Lejay | Free discussion Excursion |
Departure | |
| 15:00 – 15:50 | Sangwook Lee | Sung Rak Choi | ||
| 16:10 – 17:00 | Youngho Yoon | Jinseok Cho | ||
| 17:10 – 18:00 | Mehdi Tavakol | Myeonggi Kwon | ||
| 18:00 – | Dinner | |||
Title & Abstract
- Speaker
- Florin Ambro
- Title
- On normal crossing singularities
- Abstract
- I will discuss Kodaira type vanishing theorems for varieties with certain mild singularities, which generalize normal crossing singularities.
- Speaker
- Jinseok Cho
- Title
- Various aspects of the volume conjecture
- Abstract
- In this talk, I will explain my personal viewpoint of the volume conjecture. Especially, the physical, combinatorial and geometrical aspects of the volume conjecture will be discussed.
- Speaker
- Sung Rak Choi
- Title
- Introduction to potential pairs
- Abstract
- I will explain the notion of potential pairs. This is a generalization of the usual klt, lc singularities studied in the minimal model program.
Since the notion of potential pairs also captures the positivity of the divisors, we expect that the new results for potential pairs will be useful in understanding the varieties with negative Kodaira dimension.
Some ongoing project will be also discussed.
- Speaker
- Dmitry Doryn
- Title
- The $c_2$ invariant of completed Feynman graphs in $\phi^4$
- Abstract
- The $c_2$ invariant is the coefficient of $q^2$ in the $q$-expansion of the number of $\mathbb{F}_q$-rational points of the (graph) hypersurface associated to a Feynman graph. It is considered to be an arithmetic analogue of the Feynman period. I will discuss the properties of the $c_2$ invariant and the ways it can be computed. I will present the formula that can be used for calculation of this invariant for graphs without a vertex of degree less than 4. In particular, this allows to compute the $c_2$ for small 4-regular graphs in $\phi^4$ theory.
- Speaker
- Jin Woo Jang
- Title
- Global well-posedness and stability of the relativistic Boltzmann equation without angular cut-off
- Abstract
- We prove the unique existence and exponential decay of global in time classical solutions to the special relativistic Boltzmann equation without any angular cut-off assumptions with initial perturbations in some weighted Sobolev spaces. We consider perturbations of the relativistic Maxwellian equilibrium states. We work in the case of spatially periodic box. We consider the general conditions on the collision kernel from Dudynski and Ekiel-Jezewska (Commun Math Phys 115(4):607–629, 1985). Additionally, we prove sharp constructive upper and coercive lower bounds for the linearized relativistic Boltzmann collision operator in terms of a geometric fractional Sobolev norm; this shows a spectral gap exists and this behavior is similar to that of non-relativistic case as shown by Gressman and Strain(Journal of AMS 24(3), 771–847, 2011). This is the first global existence and stability result for relativistic Boltzmann equation without angular cutoff and this resolves the open question of perturbative global existence for the relativistic kinetic theory without the Grad's angular cut-off assumption.
- Speaker
- Myeonggi Kwon
- Title
- Morse-Bott spectral sequences and the links of weighted homogeneous polynomials
- Abstract
- In this talk, we introduce a version of Morse-Bott spectral sequences for (equivariant) symplectic homology. We apply this to a special kind of symplectic manifolds, named Milnor fibers, whose contact type boundary is the links of isolated hypersurface singularities. In particular, for weighted homogeneous polynomials, the links admit a periodic Reeb flow. This periodicity is useful for computing an invariant of contact structures extracted from equivariant symplectic homology, called the mean Euler characteristic. We give some applications, for example, to exotic contact structures.
- Speaker
- Sangwook Lee
- Title
- Mirror symmetry between Calabi-Yau categories
- Abstract
- We review Calabi-Yau category structures on Fukaya categories and matrix factorization categories. Then we investigate relations between them via localized mirror functors(due to Cho-Hong-Lau), boundary-bulk maps and Kodaira-Spencer maps(due to Fukaya-Oh-Ohta-Ono). This is a work-in-progress jointly with Cheol-hyun Cho and Hyung-seok Shin.
- Speaker
- Byeongho Lee
- Title
- Frobenius manifolds and topological conformal field theories
- Abstract
- Frobenius manifolds are central figures in classical mirror symmetry. For example, (big) quantum cohomology in the A model or universal unfolding in the Landau-Ginzburg B model are examples of Frobenius manifolds. Since Dubrovin introduced this notion, the main tool to investigate them has been mostly geometric and analytic. However, it was introduced as a theory of a type of conformal field theory, as can be seen from one of his earlier papers on this subject. Then a natural question is, what the relationships are with representations of the Virasoro-type algebra that is associated with this specific conformal field theory, as is usual in the literature on conformal field theories. We expect a new tool to analyze problems in Frobenius manifolds, such as orbifolding. This is a work in progress, and we will report on the current status in this approach.
- Speaker
- Damien Lejay
- Title
- Factorisation Algebras and Vertex Algebras
- Abstract
- Both factorisation algebras and vertex algebras are tools to encode the
mathematics of field theories. The former is much newer than the latter. We shall review the ideas leading to the definition of the two objects and have a look at the known links between them.
- Speaker
- Yoshikazu Nagata
- Title
- On Lie group structure of automorphism groups
- Abstract
- We give a sufficient condition for complex manifolds that the automorphism groups become Lie groups. As an application we see that the automorphism group of any strictly pseudoconvex domain has a Lie group structure.
- Speaker
- Jongil Park
- Title
- Symplectic fillings and rational blowdowns
- Abstract
- In this talk, I'd like to explain that any minimal symplectic filling of the link of a quotient surface singularity equipped with the canonical contact structure can be obtained by a sequence of rational blowdowns and blowing-ups from the minimal resolution the corresponding quotient surface singularity.
- Speaker
- Mehdi Tavakol
- Title
- Differential models for B-type open-closed topological Landau-Ginzburg theories
- Abstract
- I will describe a family of differential models for B-type open-closed topological Landau-Ginzburg theories defined by a pair (X,W), where X is a non-compact Calabi-Yau manifold and W is a holomorphic complex-valued function defined on X whose critical set is compact. For the particular case of Stein manifolds there is more explicit description of the differential model. This is based on recent joint works with Mirela Babalic, Dmitry Doryn and Calin Iuliu Lazaroiu.
- Speaker
- Youngho Yoon
- Title
- Combinatorics of hyperplane arrangements
- Abstract
- Hyperplane arrangements connect many areas of mathematics. An important question is the following: how many algebraic/geometric/topological properties and invariants are determined combinatorially? I will introduce some of known results and open problems.
Accommodation — Novotel Ambassador Busan, Busan
Contact
- Speaker
- Florin Ambro
- Title
- On normal crossing singularities
- Abstract
- I will discuss Kodaira type vanishing theorems for varieties with certain mild singularities, which generalize normal crossing singularities.
- Speaker
- Jinseok Cho
- Title
- Various aspects of the volume conjecture
- Abstract
- In this talk, I will explain my personal viewpoint of the volume conjecture. Especially, the physical, combinatorial and geometrical aspects of the volume conjecture will be discussed.
- Speaker
- Sung Rak Choi
- Title
- Introduction to potential pairs
- Abstract
- I will explain the notion of potential pairs. This is a generalization of the usual klt, lc singularities studied in the minimal model program. Since the notion of potential pairs also captures the positivity of the divisors, we expect that the new results for potential pairs will be useful in understanding the varieties with negative Kodaira dimension. Some ongoing project will be also discussed.
- Speaker
- Dmitry Doryn
- Title
- The $c_2$ invariant of completed Feynman graphs in $\phi^4$
- Abstract
- The $c_2$ invariant is the coefficient of $q^2$ in the $q$-expansion of the number of $\mathbb{F}_q$-rational points of the (graph) hypersurface associated to a Feynman graph. It is considered to be an arithmetic analogue of the Feynman period. I will discuss the properties of the $c_2$ invariant and the ways it can be computed. I will present the formula that can be used for calculation of this invariant for graphs without a vertex of degree less than 4. In particular, this allows to compute the $c_2$ for small 4-regular graphs in $\phi^4$ theory.
- Speaker
- Jin Woo Jang
- Title
- Global well-posedness and stability of the relativistic Boltzmann equation without angular cut-off
- Abstract
- We prove the unique existence and exponential decay of global in time classical solutions to the special relativistic Boltzmann equation without any angular cut-off assumptions with initial perturbations in some weighted Sobolev spaces. We consider perturbations of the relativistic Maxwellian equilibrium states. We work in the case of spatially periodic box. We consider the general conditions on the collision kernel from Dudynski and Ekiel-Jezewska (Commun Math Phys 115(4):607–629, 1985). Additionally, we prove sharp constructive upper and coercive lower bounds for the linearized relativistic Boltzmann collision operator in terms of a geometric fractional Sobolev norm; this shows a spectral gap exists and this behavior is similar to that of non-relativistic case as shown by Gressman and Strain(Journal of AMS 24(3), 771–847, 2011). This is the first global existence and stability result for relativistic Boltzmann equation without angular cutoff and this resolves the open question of perturbative global existence for the relativistic kinetic theory without the Grad's angular cut-off assumption.
- Speaker
- Myeonggi Kwon
- Title
- Morse-Bott spectral sequences and the links of weighted homogeneous polynomials
- Abstract
- In this talk, we introduce a version of Morse-Bott spectral sequences for (equivariant) symplectic homology. We apply this to a special kind of symplectic manifolds, named Milnor fibers, whose contact type boundary is the links of isolated hypersurface singularities. In particular, for weighted homogeneous polynomials, the links admit a periodic Reeb flow. This periodicity is useful for computing an invariant of contact structures extracted from equivariant symplectic homology, called the mean Euler characteristic. We give some applications, for example, to exotic contact structures.
- Speaker
- Sangwook Lee
- Title
- Mirror symmetry between Calabi-Yau categories
- Abstract
- We review Calabi-Yau category structures on Fukaya categories and matrix factorization categories. Then we investigate relations between them via localized mirror functors(due to Cho-Hong-Lau), boundary-bulk maps and Kodaira-Spencer maps(due to Fukaya-Oh-Ohta-Ono). This is a work-in-progress jointly with Cheol-hyun Cho and Hyung-seok Shin.
- Speaker
- Byeongho Lee
- Title
- Frobenius manifolds and topological conformal field theories
- Abstract
- Frobenius manifolds are central figures in classical mirror symmetry. For example, (big) quantum cohomology in the A model or universal unfolding in the Landau-Ginzburg B model are examples of Frobenius manifolds. Since Dubrovin introduced this notion, the main tool to investigate them has been mostly geometric and analytic. However, it was introduced as a theory of a type of conformal field theory, as can be seen from one of his earlier papers on this subject. Then a natural question is, what the relationships are with representations of the Virasoro-type algebra that is associated with this specific conformal field theory, as is usual in the literature on conformal field theories. We expect a new tool to analyze problems in Frobenius manifolds, such as orbifolding. This is a work in progress, and we will report on the current status in this approach.
- Speaker
- Damien Lejay
- Title
- Factorisation Algebras and Vertex Algebras
- Abstract
- Both factorisation algebras and vertex algebras are tools to encode the mathematics of field theories. The former is much newer than the latter. We shall review the ideas leading to the definition of the two objects and have a look at the known links between them.
- Speaker
- Yoshikazu Nagata
- Title
- On Lie group structure of automorphism groups
- Abstract
- We give a sufficient condition for complex manifolds that the automorphism groups become Lie groups. As an application we see that the automorphism group of any strictly pseudoconvex domain has a Lie group structure.
- Speaker
- Jongil Park
- Title
- Symplectic fillings and rational blowdowns
- Abstract
- In this talk, I'd like to explain that any minimal symplectic filling of the link of a quotient surface singularity equipped with the canonical contact structure can be obtained by a sequence of rational blowdowns and blowing-ups from the minimal resolution the corresponding quotient surface singularity.
- Speaker
- Mehdi Tavakol
- Title
- Differential models for B-type open-closed topological Landau-Ginzburg theories
- Abstract
- I will describe a family of differential models for B-type open-closed topological Landau-Ginzburg theories defined by a pair (X,W), where X is a non-compact Calabi-Yau manifold and W is a holomorphic complex-valued function defined on X whose critical set is compact. For the particular case of Stein manifolds there is more explicit description of the differential model. This is based on recent joint works with Mirela Babalic, Dmitry Doryn and Calin Iuliu Lazaroiu.
- Speaker
- Youngho Yoon
- Title
- Combinatorics of hyperplane arrangements
- Abstract
- Hyperplane arrangements connect many areas of mathematics. An important question is the following: how many algebraic/geometric/topological properties and invariants are determined combinatorially? I will introduce some of known results and open problems.
eunyoung.jang@ibs.re.kr
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