Dec 20 (F), Youngjin Bae (Incheon National University), Topic: Lagrangian fillings for Legendrian links of Dynkin type
Dec 20 (F), Youngjin Bae (Incheon National University), Topic: Lagrangian fillings for Legendrian links of Dynkin type
Title: Lagrangian fillings for Legendrian links of Dynkin type
Abstract: We will introduce Legendrian links of finite or affine Dynkin type, and then argue that there are at least as many Lagrangian fillings as seeds in the corresponding cluster structure. The main ingredients are N-graphs developed by Casals-Zaslow, and cluster structures by Fomin-Zelevinsky. This is a joint work with Byung Hee An and Eunjeong Lee.
Time and Location: Dec 20 (F), 14:00-16:00, Room 1423.
Title: Skein algebra and cluster algebra.
Abstract: For any triangulable oriented surface S, there are two algebras generated by curve classes on S. One is the skein algebra, and the other is the cluster algebra. In this talk, I will explain their relationship and some implications on the structure of these algebras. This is joint work with Helen Wong.
Time and Location: Dec 19 (R), 9:00-11:00, Online.
Notes: will be uploaded after the talk.
Title: Cluster algebras and monotone Lagrangian tori.
Abstract: A monotone Lagrangian torus is a Lagrangian torus in a monotone symplectic manifold satisfying a certain balanced condition. Each monotone Lagrangian torus assigns a Laurent polynomial (called a potential function) which is an invariant of a Lagrangian up to Hamiltonian isotopy. In this talk, we will construct an infinitely many monotone Lagrangian tori in a full flag manifold and distinguish them by calculating their potential functions using the theory of cluster algebras. If time permits, I will also talk about a generalization of this result to the case of partial flag manifolds. This talk is based on joint works with Myungho Kim, Yoosik Kim, and Euiyong Park.
Time and Location: Dec 10 (T), 14:00-16:00, Room 1423.
Title: Differential cohomology and gerbes: An introduction to higher differential geometry
Abstract: Differential cohomology is a topic that has been attracting considerable interest. Many interesting applications in mathematics and physics have been known; description of WZW terms, string structures, study of conformal immersions, classifications of Ramond-Ramond fields to list a few, and it is also an interesting application of the theory of infinity categories. I will try to give an audience-friendly overview of differential cohomology and a classification of higher line bundles (a. k. a. U(1)-banded gerbes) with connection. I will start from scratch and assume only some basic differential geometry and algebraic topology so that it would be accessible to most graduate students.
Time and Location: Nov 12 (T), 11:00-12:00 and 13:30-15:30, at Room 1423.
Notes: Note.
Title: Contact invariants in bordered Floer homology
Abstract: Given a contact 3-manifold with convex boundary, we apply a result of Zarev to derive contact invariants in the bordered modules CFA and CFD. We show that these invariants satisfy a pairing theorem, which is a bordered extension of the Honda-Kazez-Matić gluing map for sutured Floer homology. We also show that there is a correspondence between A-infinity operations on bordered type-A modules and bypass attachment maps. we are also able to apply the immersed curve interpretation of Hanselman-Rasmussen-Watson to prove results involving contact surgery.
Time and Location: Oct 14 (M) 10:00 - 11:00 AM, Online (if you would like to get the zoom information, don't hesitate to get in touch with the organizer).
Title: Singularities in symplectic geometry.
Abstract: In this series of survey talks, I will explain many classical results about singularities and their symplectic analogies/enhancements.
Time and Location: Sep 30 (M), 2:00 - 3:30 PM, Oct 7 (M) 2:00 - 3:30 PM, at Room 1423.
Notes: Notes.
Title: Big Out(Fₙ) and its algebraic properties (10:30-12:00)
Abstract: The group Out(Fₙ) consists of automorphisms of the free group of rank n, modulo inner automorphisms, and is regarded as the mapping class group of finite graphs. Algom-Kfir and Bestvina introduced “Big Out(Fₙ)” as the mapping class group of (locally finite) infinite graphs. In this talk, I will give a gentle introduction to Big Out(Fₙ) and its various interesting algebraic properties. This is joint work with George Domat and Hannah Hoganson.
Title: Big Out(Fₙ) and its coarse geometry (14:00-15:30)
Abstract: As a geometric group theorist, one might be also interested in studying Big Out(Fₙ) as a metric space. One typical way for an arbitrary group is to use its generating set and define the metric using word length. It is a classical result that a finitely generated or a compactly generated group has a well-defined metric regardless of the choice of the generating set. However, Big Out(Fₙ) is not even locally compact, so fails to be in a classical context. In this talk, I will introduce Rosendal’s remedy by extending the framework beyond compactly generated groups. Then we will see which Big Out(Fₙ) admits a well-defined coarse geometry under his frame. This is joint work with George Domat and Hannah Hoganson.
Time and Location: July 23(T), 10:30 - 12:00, 14:00 - 15:30, Room 1423.
Notes: First talk, Second talk.
Title: Realisation problems in mirror symmetry
Abstract: Tropical geometry plays a key role in the modern development of mirror symmetry. Based on an idea of S. Payne, I introduced the notion of tropical Lagrangian multi-sections, which is the tropical/combinatorial replacement of Lagrangian multi-sections in the SYZ proposal. In this talk, I will introduce two realisation problems in toric equivariant mirror symmetry; A-realisation and B-realisation. Given a tropical Lagrangian multi-section, the A-realisation problem asks if there exists a Lagrangian multi-section in the cotangent bundle of a vector space that has asymptotic condition prescribed by the tropical Lagrangian, while the B-realisation problem asks if there exists a toric equivariant vector bundle whose total equivariant Chern class is prescribed by the tropical Lagrangian. When the tropical Lagrangian is a 2-fold cover over a 2-dimensional complete fan, I am going to give a complete answer on both realisation problems. This is partially a joint work with Yong-Geun Oh.
Time and Location: June 20, 10:30-12:00 at Room 1423, 14:00-15:30 at Room 1423.
Notes: To be uploaded
Title: The Application of Symplectic Homology to Powered Flyby Manoeuvres
Abstract: Powered flybys are an important manoeuvre commenly used in space missions to increase the kinetic energy of the spacecraft as effective as possible. Proving in which situations such an manoeuvre exists can therefore be helpful for improving the design of future space missions. The goal of this talk is to show how one can use the tools developed in symplectic topology to prove the existence of such powered flyby orbits. In the first part we start by discussing the basic idea behind the powered flyby manoeuvre. After a short repetition of Hamiltonian dynamics we introduce the restricted three body problem and investigate its Levi-Civita regularisation. To finish the first part we will see how to describe the powered flybys mathematically and how we can use the specific properties of the Levi-Civita regularization to gain some more control over these orbits. In the second part we then focus more on the symplectic topology, i.e. we give the definition of the Lagrangian Rabinowitz Floer homology and discuss the main ideas behind this homology. With this new tool at hand, we can then go ahead an prove the main theorem of this talk, the existence of infinitely many powered flyby orbits in the restricted three body problem for all energies below the first Lagrange point.
Time and Location: May 29, 10:30-12:00 at Room 1423, 14:00-15:30 at Room 1424.
Notes: Notes.
Title: Chern-Simons Theory for 3-Manifolds (10:30-12:00)
Abstract: We review the Chern-Simons theory for 3-manifolds. The talk will start with a gentle introduction to principal bundles and their connections, and reach the definition and properties of the Chern-Simons 3-form and invariant. We will also discuss holonomy representation and the case of hyperbolic 3-manifolds.
Title: The Renormalization of Volume and the Chern-Simons Invariant for Hyperbolic 3-Manifolds (14:00-15:30)
Abstract: For hyperbolic 3-manifolds, many interesting results support a deep relationship between volume and the Chern-Simons invariant. In this talk, we consider noncompact hyperbolic 3-manifolds having infinite volume. For these manifolds, there is a well-defined invariant called the renormalized volume which replaces classical volume. We will renormalize the Chern-Simons invariant and discover a close relationship with the renormalized volume.
Time and Location: May 21, 10:30-12:00, 14:00-15:30, at 1423.
Notes: Notes.
Beomjun will give a talk introducing persistence module theory, barcodes, and their applications in symplectic geometry.
Title: Persistence Modules in Floer theory (10:30-12:00).
Abstract: Ever since it was introduced by Shelukhin-Polterovich, persistence module has been used as a powerful tool for quantitative studies in symplectic geometry. In the first talk, we review the language of persistence module through Morse theory. Next, for persistence modules associated with filtered Floer homology, we understand their stability under Floer-theoretic norms and introduce several recent developments as applications.
Title: Floer theoretic Entropy for Reeb flows (14:00-15:30).
Abstract: Inspired by the work of Cineli-Ginzburg-Gürel, we introduce a Floer-theoretic entropy for Reeb flows. The barcode entropy of symplectic homology is defined as the exponential growth rate of the number of not-too-short bars in the barcode of the associated Floer complex. In the second talk, we will compare barcode entropy with topological entropy for Reeb flows. The talk is based on a joint work with Elijah Fender and Sangjin Lee.
Time and Location: May 2, 10:30-12:00, 14:00-15:30, Room 1423.
Sungho will introduce the theory of Rabinowitz Floer homology and also will explain his research.
Title: Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence.
Abstract: In the first talk, we explain the symplectic (co)homology of filled Liouville cobordisms and then provide several properties, computations, and applications of symplectic homology for Liouville domains.
In the second talk, we restrict our attention to symplectic homology of filled trivial cobordisms, i.e. Rabinowitz Floer homology of Liouville domains. For a prequantization bundle $Y$ over a closed monotone symplectic manifold $\Sigma$, we study the Rabinowitz Floer homology of $Y$ (and its Liouville filling) and construct a Gysin-type exact sequence. This sequence connects the Rabinowitz Floer homology and the quantum homology of $\Sigma$ where the map between the quantum homology equals the quantum product with the Euler class of $Y$ under certain conditions. The concrete description of the sequence allows one to compute the Rabinowitz Floer homology in various situations. Several other applications will also be discussed. This is based on joint work with Joonghyun Bae and Jungsoo Kang.
Time and Location: Apr 16, 10:30-12:00, 14:00-15:30, Room 1423.
Notes: The first talk and the second talk.
Kyungmo will introduce the notion of Fukaya category, and he will also explain his recent result on it.
Title: Fukaya Category of Tagged Arc System and Derived Tame Algebra.
Abstract: In the first talk, we review topological Fukaya category, gentle algebra, and relation between them. Topological Fukaya category is a topological version of partially wrapped Fukaya category of surface with marked boundary. The endomorphism algebra of a formal generators of topological Fukaya category becomes a special type of algebra called gentle algebra. Gentle algebra is a finite dimensional algebra associated with a quiver with relation, which is derived-tame. It is known that there is one-to-one correspondence between graded marked surfaces with arc system and homologically smooth and proper graded gentle algebra.
In the second talk, we generalize topological Fukaya category to $\mathbb{Z}/2\mathbb{Z}$-orbifold surface with marked boundary and tagged arcs. A tagged arc is an arc whose boundary is on a boundary of a surface or on interior marked points together with a choice of $\mathbb{Z}/2\mathbb{Z}$ on each interior boundary. The endomorphism algebra of a formal generators gives a new class of derived-tame algebras, which include skew-gentle algebras, generalization of gentle algebras. This talk is partially based on a joint work with Cheol-Hyun Cho and Kyungmin Rho.
Time and Location: Apr 2, 10:30-12:00, 14:00-15:30, Room 1423.
Notes: Notes.
Seokbong will give an introductory talk on dg manifold and Atiyah class. Detailed information will be given soon.
Title: Introduction to the Atiyah class of differential graded manifolds.
Abstract: The notion of differential graded manifolds (dg manifolds) is a generalisation of the notion of smooth manifolds, in a way that the algebra of smooth functions is enriched to include graded commutative elements — Lie algebras, foliations, and complex manifolds give a rise to examples of dg manifolds. Moreover, dg manifolds are related to the emerging field of derived differential geometry. The Atiyah class, first introduced by Atiyah as an obstruction to the existence of holomorphic connection, generalises to dg manifolds.
In this talk, I will explain the basic definitions and examples on dg manifolds and their Atiyah class. If time permits, I will introduce some theorems and applications where the Atiyah class of dg manifolds takes an important role.
Time and Location: Mar 25 (M), 11:00 - 12:00, 14:00-15:30, Room 1423.
Notes: Notes.
Jisu will give an introductory talk on Topological Data Analysis. Detailed information will be given soon.
Title: Statistical Inference on Topological Data Analysis and Application to Machine Learning.
Abstract: Topological Data Analysis (TDA) generally refers to utilizing topological features from data. Typical examples are cluster tree and persistent homology. The cluster tree gathers similar data together to make clusters. The persistent homology quantifies salient topological features that appear at different resolutions of the data. TDA provides useful information, such as delivering scientific information from data, or extracting useful features for learning.
The first part of this talk will be about the statistical inference on TDA. I will first present statistical inference on the cluster tree. Then, I will present how the randomness of the persistent homology computed from data can be statistically quantified and significant topological features can be identified.
The second part of this talk will be about the application of TDA to machine learning. This talk will specifically focus on two aspects: featurization and evaluation. I will present how the persistent homology is featurized in Euclidean space or functional space. Then, I will end this talk by presenting how TDA can be applied to evaluate data or machine learning models.
Time and location: Mar 6 (W), 14:00-16:00, Room 1423.
Notes: Note.
Sangwook will give two talks, one (Feb 27, morning time) introduces Matrix factorizations, and the other (Feb 27, afternoon) explains his research on the theory. Detailed information will be given soon.
Title: Introduction to Matrix factorization.
Abstract: We review the basics of matrix factorizations of a polynomial, with emphasis on their roles in homological mirror symmetry. Topics include: dg category of matrix factorizations, graded matrix factorizations, localized mirror functors (due to Cho-Hong-Lau), Hochschild invariants etc.
KyeongRo will give two talks introducing Teichmüller theory. The title and abstract will be updated soon.
Title: Revisit the Tits-alternative of mapping class groups
Abstract: Thurston used the concept of measured geodesic laminations to compactify the Teichmuller spaces and describe the boundary behavior under the mapping class group actions. After that, McCarthy refined the description of the boundary action of the mapping class groups to show the Tits-alternative of mapping class groups. In this talk, I will review basic notions related to measured geodesic laminations and the McCarthy’s description of the boundary action of a mapping class group in terms of measured geodesic laminations. And I will review McCarthy’s proof of the Tits-alternative of mapping class groups.
Time and Location: Fab 1(R) 14:00-16:00, Room 1423, and Fab 2(F) 14:00-16:00, Room 1423.
Notes: Notes.
Geunho will give a talk on his research in the field of geometric topology. The title and abstract will be updated soon.
Title: Universal linear bounds on rho-invariants on high-dimensional manifolds.
Abstract: Using L^2 cohomology, Cheeger and Gromov define the L^2 rho-invariant on (4k-1)-manifolds endowed with an arbitrary representation of the fundamental group, as a generalization of the Atiyah-Singer rho-invariant. With interesting applications, Cha computed explicit universal linear bounds for 3-manifolds in terms of simplicial complexity of 3-manifolds. Using the hyperbolization, there was progress for high-dimensional manifolds with a faithful representation. In this talk, we provide explicit universal linear bounds on (4k-1)-manifolds with an arbitrary representation, giving a new geometric construction of cobordism. This is a joint work with Jae Choon Cha.
Time and Location: Jan 15(M) 14:00-15:30. Location will be decided.
Notes: Notes.
Tianyu will give a series of talks introducing the notion of Higher-dimensional Heegaard Floer homology. The title and abstract will be updated soon.
Title: Introduction to higher-dimensional Heegaard Floer homology (Jan 9, T)
Abstract: In this talk, we will briefly review the foundations of higher-dimensional Heegaard Floer homology (HDHF) and its applications. As a first example, we define a link invariant analogous to Khovanov homology by performing HDHF on Type A Milnor fibers.
Title: Higher-dimensional Heegaard Floer homology and Hecke algebras (Jan 10, W)
Abstract: We show some applications of HDHF in representation theory. Given a closed oriented surface of genus greater than 0, we construct a map from the HDHF group of the cotangent fibers to the surface Hecke algebra and show it's an isomorphism. We show that the conormal bundle of the torus meridian in HDHF naturally gives the polynomial representation of double affine Hecke algebra. (The talk is based on joint works with Ko Honda, Roman Krutovski, Eilon Reisin-Tzur, and Yin Tian)
Title: Folded Morse flow trees (Jan 11, R)
Abstract: We present an approach to Morse theory on symmetric products of surfaces using the notion of folded ribbon trees. We introduce an A_\infty-category with objects defined as \kappa-tuples of Morse functions, where the differential of the tuple has no self-intersection. This construction is closely related to HDHF. As an example, we show that when the graph of the differential of the \kappa-tuple of Morse functions on T^*\mathbb{R}^2 is the wrapped \kappa disjoint cotangent fibers, its endormorphism is the finite Hecke algebra.
Time and Location: Time for the three talks are the same, 14:00-16:00, and the location is KIAS Building 1, Room 1423 for all three talks.