Lectures Schedule

Time	April 25 (Tue)	April 26 (Wed)	April 27 (Thu)
08:45 – 09:00	Welcome and Registration
09:00 – 10:15	Lecture Series I (Part 1) Gwénaël Massuyeau	(9:15 – 10:15) Talk C Hongtaek Jung	Lecture Series I (Part 2) Gwénaël Massuyeau
10:15 – 10:45	Coffee/Tea break
10:45 – 12:00	Lecture Series II (Part 1) Yuka Kotorii	Lecture Series II (Part 2) Yuka Kotorii	(10:45 – 11:45) Talk F Byeorhi Kim
12:00 – 14:15	Lunch
14:15 – 15:15	Talk A Minkyoung Song	Talk D Seonhwa Kim	Talk G Arthur Soulié
15:15 – 15:45	Coffee/Tea break
15:45 – 16:45	Talk B Seonmi Choi	Talk E Hyungryul Baik	Talk H Sungkyung Kang
16:45 – 18:00	Free discussion		(16:45 – 17:00) closing remarks
18:00 – 19:30		Dinner

* The schedule may vary.

Abstracts:

Lecture Series I (Gwénaël Massuyeau)
Mapping class groups of surfaces: subgroups and infinitesimal representations
By classical results of Dehn and Nielsen, the mapping class group $M(\Sigma)$ of a surface $\Sigma$ can be studied through its action on the fundamental group $\pi_1(\Sigma)$. In the first talk, we will review all the necessary material on mapping class groups (including their generation by Dehn twists), and we will explain how the Dehn-Nielsen representation of $M(\Sigma)$ can be expanded diagrammatically by considering the action of $M(\Sigma)$ on the Malcev Lie algebra of $\pi_1(\Sigma)$. In the second talk, we will mention a few applications of this “infinitesimal” version of the Dehn-Nielsen representation for certain subgroups of $M(\Sigma)$. In particular, we shall use it to reformulate and extend previous works of Dimca-Hain-Papadima, Morita-Sakasai-Suzuki and Nozaki-Sato-Suzuki on the abelianization of the “Johnson kernel” (which is the subgroup of $M(\Sigma)$ generated by Dehn twists along separating curves). The latter part is joint work with Quentin Faes.

Lecture Series II (Yuka Kotorii)
On ribbon Yetter-Drinfeld modules and tangle invariants
Reshetikhin and Turaev introduced the notion of ribbon Hopf algebra and showed that the category of finite-dimensional modules over a ribbon Hopf algebra has a ribbon category structure. Since the category of framed, oriented tangles is a free ribbon category generated by one object, a ribbon category yields a functor from the tangle category to the category of finite-dimensional vector spaces, and thus gives a functorial invariants of tangles. In this talk, we define notions of ribbon objects in a monoidal category. These constructions give ribbon categories from a monoidal category. We apply this construction to the braided monoidal category of Yetter-Drinfeld modules over a Hopf algebra. This gives rise to the notion of ribbon Yetter-Drinfeld modules over a Hopf algebra, which form ribbon categories. This gives an invariant of framed tangles. This research is joint work with Kazuo Habiro.

Talk A (Minkyoung Song)
Homology cylinder, as generalization of both string link and mapping class group
The homology cobordism group of 3-dimensional homology cylinders can be considered as an enlargement of both the mapping class group of a surface and the concordance group of string links. In this talk, I introduce history and notion of homology cylinders and their homology cobordism group. Also, we consider invariants related to lower central series of a free group: Johnson homomorphisms and Morita homomorphisms of a mapping class group, Milnor invariants and Orr invariants of (string) links. The invariants give rise to filtrations. We extend those invariants and filtrations to homology cylinders and compare them. We get relations of the filtrations to automorphism groups of free nilpotent groups, and free Lie algebras.

Talk B (Seonmi Choi)
Marked graph mosaics
Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. Kuriya and Shehab proved Lomonaco-Kauffman conjecture which means that knot mosaic type is a complete invariant of tame knots. The mosaic number of a knot K is the smallest integer n for which K can be represented on an n × n mosaic board. In this talk, we consider the notion of mosaic diagrams for surface-links using marked graph diagrams. We establish bounds, in some cases tight, on the mosaic numbers for the surface-links with ch-index up to 10. As an application, we use mosaic diagrams to enhance the kei counting invariant for unoriented surface-links as well as classical knots and links. This is joint work with Sam Nelson.

Talk C (Hongtaek Jung)
Groups acting on the circle with invariant veering pairs
The fundamental groups of many 3-manifolds can act on a circle. Examples include closed orientable 3-manifolds with taut foliations, closed orientable 3-manifolds with pseudo-Anosov flows, hyperbolic 3-manifolds with quasi-geodesic flows and so on. A fascinating feature is that all of these actions leave some circle laminations invariant. In this talk, I will present the inverse problem, asking whether a given group is the fundamental group of a 3-manifold if it acts on a circle preserving circle laminations. The answer to this problem very depends on properties of the invariant laminations. I will introduce veering pair of circle laminations, which is motivated by recent work of Schleimer and Segerman on veering triangulations, and show that a group acting on a circle with an invariant veering pair must be the fundamental group of an irreducible 3-orbifold. This is joint work with Hyungryul Baik and KyeongRo Kim.

Talk D (Seonhwa Kim)
On the rigidity of three-dimensional Polyhedra
We demonstrate that a three-dimensional polyhedron can be uniquely determined by its dihedral angles and edge lengths, regardless of whether it is non-convex or self-intersecting. We achieve this under three plausible sufficient conditions: (1) the polyhedron is composed solely of convex faces, (2) there are no partially-flat vertices, and (3) any triple of vertices is not collinear. Our method is universally valid for Euclidean, spherical, and hyperbolic geometry. Notably, our approach is entirely different from the argument of the Cauchy rigidity theorem. We provide various counterexamples that arise when our conditions are violated, as well as several interesting corollaries, and pose further questions and conjectures.

Talk E (Hyungryul Baik)
Normal generators of mapping class groups
We will discuss how to show a given mapping class is and is not a normal generator of the mapping class group, and then discuss related open and closed questions.

Talk F (Byeorhi Kim)
On a smoothing technique of topological surfaces in 4-manifolds
In this talk, I will talk about a new smoothing technique for topologically embedded surfaces or disks in smooth 4-manifolds that provides topological isotopies to smooth surfaces. This result is motivated from recent David Gabai's Light bulb theorem. As an application, we can get some results which leading us to "topological = smooth" in dimension 4 for isotopy classifications of certain disks and spheres. This is a joint work with J. C. Cha.

Talk G (Arthur Soulié)
On homological representations for braid groups and mapping class groups
I will describe a general construction of homological representations for families of motion groups or mapping class groups, including the families of braid groups, surface braid groups and loop braid groups. This recovers the well-known constructions of Lawrence-Bigelow, and in this sense it unifies these constructions. I will also discuss indecomposability and irreducibility of these representations. The construction is moreover “global” in the sense that, for each dimension d, it is a functor on a category whose automorphism groups are all d-dimensional motion groups and mapping class groups, and which also carries a richer structure. Using this richer structure, I will discuss polynomiality of these families of representations, and use this to prove twisted homological stability for the braid groups with coefficients in any one of the Lawrence-Bigelow representations. All this represents a joint work with Martin Palmer.

Talk H (Sungkyung Kang)
One stabilization is not enough for exotic contractible 4-manifolds
We construct the first example of an exotic pair of contractible 4-manifolds which remain exotic after one stabilization.

List of Participants

Yong-Geun Oh (IBS-CGP, POSTECH)

Anderson Vera (IBS-CGP)

Yuka Kotorii (Hiroshima University)

Gwénaël Massuyeau (University of Burgundy)

Hyungryul Baik (KAIST)

Seonmi Choi (Kyungpook National University)

Hongtaek Jung (Seoul National University)

Sungkyung Kang (IBS-CGP)

Byeorhi Kim (POSTECH)

Seonhwa Kim (University of Seoul)

Minkyoung Song (Korea Science Academy of KAIST)

Arthur Soulié (IBS-CGP)

BoGwang Jeon (POSTECH)

Akash Kumar Heera (Visva-Bharati University)

Myeong Sang Cho (POSTECH)

Eric Dolores Cuenca (Yonsei University)

Javier de la Nuez Gonzalez (KIAS)

Suhyoung Choi (KAIST)

Dmitriy Voloshyn (IBS-CGP)

Donggyun Seo (Seoul National University)

Wonyong Jang (KAIST)

Junseok Kim (KAIST)

Juhun Baik (KAIST)

XIAOXIANG CHAI (POSTECH)

Yuto Yamamoto (IBS-CGP)

Seul Bee Lee (IBS-CGP)

Hyun Seok Do (POSTECH)

Seungyeol Park (KAIST)

Accommodation

We regret to say that we cannot support your travel and local expenses unless the workshop promised to pay. For booking accommodation in Pohang, please contact the hotel directly referring the list below.

There is usually a limited number of rooms available, so please make a reservation as soon as you can.

Room Type	POSCO Int'l Center	Hotel Yeongildae	Apple Tree Hotel	Tour de Pohang (Woman's Safety Sohotel)
Double Room	88,000 KRW	110,000 KRW	45,000 KRW	43,000 ~ 48,000 KRW (1 person)
Twin Room	88,000 KRW	132,000 KRW	-	58,000 ~ 63,000 KRW (2 persons)
Breakfast (1 person)	13,200 KRW	Free	Free	Free
Distance from Venue	5 min. walk	15 min. drive	15 min. drive	15 min. drive
Contact	+82-54-279-8500	+82-54-280-8900	+82-54-241-1234	+82-0507-1397-1234

* The above rate is as of August, 2022 (VAT included).

** The rate and condition may vary.

*** If you would like to stay at POSCO International Center, please inform the hotel the title of the workshop for the reservation as the hotel is not open to the public.

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Contact

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