Idiot's Guide

Upcoming Seminars

Upcoming Seminars

2016

2016

 February 29: Pierrick Bousseau (Imperial College)   4pm Rm 1423

   DT invariants of quivers

I will give a general introduction with geometric motivations to quivers and their representations. Then I will try to present an elementary definition of DT invariants of quivers without oriented loops. This talk is intended as preparation for my algebraic geometry seminar talk on Wednesday.

 

 February 29: Pablo Solis (Caltech)   5pm Rm 1423

   A stacky view of vector bundles on curves

I will introduce global quotient stacks X/G and look at some simple examples. In fact I will focus mainly on X = pt and G = GL(V). Describing a morphism from a curve C to pt/GL(V) is the same as giving a vector bundle on C. We will look at some low genus curves and consider what happens if the curve is nodal. If time permits I will mention how this leads to considering compactifications of groups.

 

 March 17: Otto van Koert (Seoul National University)   Thursday 5pm Rm 1423

   The 3-body problem: Results and History

The three-body problem is an old problem in mathematics first considered in its current form by Newton. We describe some of the history of this problem since Newton, and some of the work that it has inspired from the mathematical side. If time permits, we will also go into more recent developments, showing that the mathematics of this problem is still very much an active area of research.

 

 March 21: Jinhyung Park (KIAS)    Rm 1423

   Volumes in algebraic geometry and Okounkov bodies

In algebraic geometry, the volume of a divisor (or a line bundle) on a projective manifold is an invariant measuring asymptotic growth of global sections. When a divisor is ample, this notion coincides with the volume of a projective manifold with respect to the Kaehler metric up to constant. On the other hand, the Okounkov body of a divisor is a convex subset of some Euclidean space, and its Euclidean volume is the volume of a given divisor up to constant. In this talk, I first review basic properties of volumes and Okounkov bodies of big divisors, and then I introduce natural ways to extend these results to pseudoeffective divisors.

 

 April 4: Frederic Campana (University of Nancy and KIAS Scholar)   Rm 8101

   Rational connectedness and negativity of the cotangent bundle

Initiated by S. Mori in 1980, many fundamental results have been obtained relating the distribution of rational curves on complex projective manifolds to the negativity of their canonical or cotangent bundles. We shall survey some of them, which play an important role in the birational classification.

 

 May 2: Ionut Ciocan-Fontanine (University of Minnesota and KIAS Scholar)    Rm 8101

   Quasimap invariants, Gromov-Witten invariants, and mirror maps

Algebraic Gromov-Witten invariants are (virtual) counts of holomorphic maps from compact Riemann surfaces to complex quasiprojective manifolds. In the early 1990's, physicists used Mirror Symmetry to make stunning predictions about these counts in all genera for many Calabi-Yau threefolds. Namely, given such a threefold X and its "mirror" Calabi-Yau threefold Y, the generating function of genus g Gromov-Witten invariants of X is obtained from an (often explicit) "B-model partition function of Y" via a change of variables known as the mirror map. Some of these predictions have been since proved, many are still conjectural. Quasimap theory is a more recent alternative way of map counting, and a natural question to ask is how it is related to Gromov-Witten theory. The answer turns out to be that for Calabi-Yau threefolds the generating functions of quasimap invariants and of Gromov-Witten invariants are related precisely by the mirror map. Put it differently, the quasimap theory of $X$ is conjecturally _equal_ to the physicists' B-model partition function of the mirror threefold Y. This is joint work with Bumsig Kim.

 

 May 9: Cheol-Hyun Cho (Seoul National University)    Rm 8101

   What is homological mirror symmetry?

Homological mirror symmetry is a relation between two different branches of mathematics, symplectic geometry and complex (algebraic) geometry. It can be already observed from the geometry of surfaces, such as sphere or torus. We will explain main ideas of homological mirror symmetry via this surface examples, which involves quantum cohomology, Jacobian ring, Fukaya category, coherent sheaves, and matrix factorizations.

 

 May 23: David Favero (University of Alberta)    Rm 1423

   Unifying Mirror Symmetry Conjectures

While, given a Calabi-Yau manifold, string theory predicts the existence of a "mirror" Calabi-Yau manifold, the various constructions of the mirror proposed by physicists and mathematicians do not always agree. So, who has the right construction? As it turns out, everyone does! I will give an introduction to mirror symmetry and discuss some “multiple mirror” phenomena particularly in the Berglund-Hubsch-Krawitz construction of mirror pairs. Then, I will survey some conjectures and results which unify these mirrors through birational geometry and homological algebra. Slides will be available at http://www.ualberta.ca/~favero if you would like to follow along on your laptop during the talk.

 

 June 13: Renzo Cavalieri (Colorado State University)    Rm 8101

   A few points of view on GW theory

Gromov-Witten theory is a virtual curve counting theory that exhibits remarkable structure. Gromov-Witten invariants may be used to give an associative deformation of the cohomology ring of the target, to construct a vector bundle with a flat connection. In this talk I will introduce some of the points of view in GW theory that will be most useful for the seminar talk. Time permitting I will also introduce the notion of open GW theory, which is a theory that wishes to count maps from Riemann Surfaces with boundaries to a pair (X,L), with X a CY manifold and L a Lagrangian submanifold.

 

 July 28 (Thursday): Jinwon Choi (Sookmyung Women's University)    Rm 8101

   Genus zero wall crossing for quasimaps

There are series of wall crossing for the moduli space of quasimaps, which are called $\epsilon$ and $\delta$ wall crossing. I will discuss explicit and concrete examples of $\epsilon$ and $\delta$ wall crossing for genus zero quasimaps. When $\epsilon$ is infinity, we obtain the moduli space of stable maps and when $\delta$ is zero, the moduli space is a projective bundle over the moduli space of stable curves. I will describe the wall crossing with the example when $d=1$. When $d=1$, each $\epsilon$ and $\delta$ wall crossing is a blowup. When the target space is $\mathbb{P}^1$, this gives a new construction of the Fulton-MacPherson configuration space which is isomorphic to the the moduli space of stable maps. This is joint work with Young-Hoon Kiem.

 

 Aug 16 (Tuesday): Carlo Madonna (Autonoma University of Madrid)    Rm 8101

   The Tower of Hanoi - Myths and Maths

In this talk I will recall the well known game for children called "Hanoi Tower with 3 bars" and I will review the recent bibliography related to the Hanoi Tower problem with four or more bars and to some open problems.

 

 Sept 5, 5:30pm: Insong Choe (Konkuk University)    Rm 8101

   Rational parameterization of plane curves

The unit circle has the famous rational parameterization, which was probably known to Babylonians. Some higher degree curves have rational parameterizations, whose formulas are sometimes very long. After reviewing some basics on this issue, I'll explain the rational parameterization of plane quartic curves with three nodes. Everything in this talk will be quite elementary and classical.

 

 Sept 12, 5:30pm: Donghoon Hyeon (SNU)   Rm 8101

   A stratification of Hilbert schemes via generic initial ideals

I will describe the decomposition of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratication into locally closed subschemes along which the generic initial ideals remain the same. I will give two applications: First, I will give a completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Secondly, I will prove that when a Hilbert scheme is embedded by the Grothendieck-Pl ucker embedding of a high enough degree, one of its components must be degenerate.

 

 Sept 26, 5:30pm: Wanmin Liu (IBS-CGP)    Rm 8101

   Bridgeland stability conditions and wall-crossing

Bridgeland Stability Conditions (BSC) originate from Douglas's work on Π-stability for B-branes. It associates a triangulated category D a natural topological space, called stability manifolds Stab(D). If D is the bounded derived category of a CY threefold, Stab(D) up to the natural autoequivalence group action, is expected to be the mathematical way of stringy Kaehler moduli space. BSC also extend our understanding of moduli space of Gieseker semistable sheaves via wall-crossing. In this talk, I will give a basic introduction to BSC via examples, which involves tilting, t-structure, heart of categories. I will also mention some open problems.

 

 Sept 29, 4pm: Michel van Garrel (KIAS)    Rm 1423

   Gromov-Witten theory for physicists - examples and calculations

This talk is meant as an introduction to the curve counting theories that have had such an important impact in algebraic geometry. I will introduce the problems, clarify the various approaches and provide calculations in many examples. The goal of this talk is to get to Kontsevich's computation concerning counts of rational curves in the projective plane. Precise definitions and the setup of the theory will be emphasized in a later talk.

 

 Oct 10, 5:30pm: Michel van Garrel (KIAS)    Rm 8101

   Curve counting theories for physicists - moduli spaces and why we need virtual fundamental classes

This is the continuation of the series of talks for physicists on curve counting theories in algebraic geometry. I will start the talk by outlining the general strategy for defining enumerative invariants in algebraic geometry. Then I will motivate, by example, the need to consider virtual fundamental classes.

 

 Oct 17, 5:30pm: Michel van Garrel (KIAS)     Rm 8101

   Curve counting theories for physicists - basic definitions and correspondences

In this talk, I will outline the definition for the various curve counting theories, namely Gromov-Witten and Donaldson-Thomas type theories, as well as BPS numbers. I will elaborate on the correspondences that relate these invariants.

 

 Oct 31, 5:30pm: Michel van Garrel (KIAS)     Rm 8101

   Curve counting theories for physicists - the moduli space of stable maps

The goal of this talk is to give an in depth treatment of the moduli space of stable maps and its perfect obstruction theory. This will then lead to an understanding of what Gromov-Witten invariants are.

 

 Nov 3, 4pm: Michel van Garrel (KIAS)     Rm 8101

   DT-type theories for physicists

In this talk, I will give an overview of DT theory. I will explain what the moduli space is, outline how it differs from the moduli space of stable maps, and elaborate on what the correct perfect obstruction theory is.

 

 Nov 7, 5:30pm: Jinwon Choi (Sookmyung Women's University)    Rm 8101

   The KKV method in mathematics

I will talk about the Katz-Klemm-Vafa method which is a computing algorithm for the BPS invariants. I will also give a mathematical interpretation.

 

 Nov 14, 5:30pm: Youngju Kim (Konkuk University)     Rm 8101

   The geometry of complex hyperbolic 2-space

Complex hyperbolic 2-space can be seen as the unit ball in C^2 (the 2-dimensional complex vector space) with the Bergman metric. This is a complex counterpart of the Poincare disk model of real hyperbolic plane. In this talk, I will introduce the geometry of complex hyperbolic 2-space. We will start with basic definitions such as isometries and the boundary at infinity along with other geometric notions. Then we will move to a research survey regarding the action of discrete groups of isometries.

 

 Nov 17, 4pm: Jeongseok Oh (KAIST)     Rm 8101

   GLSM in mathematics

In this talk, I will introduce the Landau-Ginzburg and Calabi-Yau correspondence for specific examples (more precisely, for the quintic Calabi-Yau). This talk is based on the result by Chang-Li-Li-Liu.

 

 Nov 21, 5:30pm: Sanghoon Kwon (KIAS)     Rm 8101

   Number theoretic aspects of ergodic theory and measure rigidity

Over the past three decades, the ergodic theory for flows on homogeneous spaces has produced many beautiful applications in number theory, geometry and mathematical physics. We will discuss some of the rigidity phenomena for Lie group actions on homogeneous spaces and will present how to use those to solve problems in number theory. We start with exploring the actions on circle and its connection with Diophantine approximations. Then we study some of the dynamical results for Lie groups such as Ratner’s equidistribution theorem and the theorem about measure rigidity of diagonalizable actions. We also provide some of the ideas in the proofs, and a few of the important applications including the proof of Oppenheim’s conjecture and Littlewood’s conjecture under certain assumption.

 

 Nov 28, 5:30pm: Carlos Scarinci (KIAS)    Rm 8101

   3D gravity and Teichmuller theory

In this talk, I will describe some relations between General Relativity on 3-manifolds and Teichmuller theory. After reviewing some basic concepts of each theory, I'll explain results of G. Mess and followers characterising globally hyperbolic Einstein 3-manifolds in terms of analytic/hyperbolic geometry data on 2-dimensional surfaces. I'll then explain the generalisations need to include the treatment of 3-dimensional multi-black hole spacetimes. If time permits, I'll also mention some aspects related to quantisation.

 

 Dec 12, 4:00pm: Elena Mantovan (Caltech)     Rm 8101

   An introduction to p-adic automorphic forms

The notion of a p-adic modular form (a p-adic analogue of classical modular forms) was first introduced by Serre in 1973 via the q-expansion principle. Soon afterwards, Katz gave a new definition via the geometry of modular curves. The p-adic theory provides the appropriate framework for the study of congruences among classical forms, and p-adic interpolation (i.e. the construction of families of classical forms which converge p-adically) is a crucial tool behind many important arithmetic results on classical modular forms. Automorphic forms are a vast generalization of modular forms. Yet, in recent years, many arithmetic properties of modular forms were extended to automorphic forms, with stunning consequences. In this talk we will introduce the notions of classical and p-adic automorphic forms. We will mostly focus on aspects of the p-adic theory related to Hida's Igusa tower, the q-expansion principle and their application to the construction of p-adic families of automorphic forms.

 

 Dec 12, 5:30pm: Tom Graber (Caltech)     Rm 8101

   The virtual fundamental class

I'll discuss the ideas and motivation for the virtual fundamental class in Gromov-Witten theory (and other curve counting theories) as well as some simple examples that illustrate the nature of the construction.

 

 Dec 15, 5pm: Yang-Hui He (Oxford University)     Rm 8309

   Yang-Mills Theory and the ABC Conjecture

We establish a correspondence between the ABC Conjecture and N=4 super-Yang-Mills theory. This is achieved by combining three ingredients: (i) Elkies' method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings; (ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi-Scharaschkin; and (iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d'enfant in the sense of Grothendieck. We explore this correspondence for the highest quality ABC-triples as well as large samples of random triples. The Conjecture itself is mapped to a statement about the fundamental domain of the toroidal compactification of the string realization of N=4 SYM.

 

2015

2015

 February 2: Jinseok Cho (KIAS)   Rm 1423

   Combinatorial approach to the hyperbolic volumes of knots

Basic facts on the knot theory will be discussed briefly. Especially, I will explain the definition of knot, fundamental group, Wirtinger presentation, parabolic subgroup of SL(2,C), boundary-parabolic representation, quandle of knot, hyperbolic structure of knot complement, hyperbolic volume of boundary-parabolic representation, volume conjecture and optimistic limit. The final goal is to explain combinatorial method of calculating the hyperbolic volume of a boundary-parabolic representation. The calculation is so simple that (smart) high school students can follow it. Therefore, anyone with undergraduate level of knowledge can understand this talk. (Reference - arXiv:1410.0525)

 

 February 11 (Wednesday): Joonhyung Kim (Hannam University)   Rm 1423

   On complex hyperbolic Kleinian groups

In this talk, I will introduce my recent work(with Sungwoon Kim) on complex hyperbolic Kleinian groups with real trace. To do this, I will explain complex hyperbolic geometry, Zariski topology and some Lie group theories.

 

 February 23: Roland Abuaf (Imperial College)   Rm 1423

   Invariants of algebraic varieties

One of the main goal in complex algebraic geometry is to classify projective varieties up to some equivalence relation (e.g. isomorphism, birational isomorphism etc...). To achieve this classification, one needs to find invariants which are rich enough to distinguish between two different varieties. In this introductory talk, I will discuss many of these invariants on some down-to-earth examples. I will also explain why derived categories appear naturally as one of the richest invariants of algebraic varieties.

 

 March 2: Matt Young (Hong Kong University)   Rm 1423

   Hall algebras in combinatorics, representation theory and geometry

I will explain what a Hall algebra is, in what situations you may come across it and how it leads to interesting links between various areas of mathematics.

 

 April 17, 17h30: Jinwon Choi (Sookmyung Women's University)    Rm 1423

   Torus localization

We present some applications of the localization theorem by torus action in classical enumerative geometry.

 

 May 11: Youngju Kim (KIAS)   Rm 1423

   An introduction to Lorentzian 3-manifolds

We will provide a background of Lorentzian geometry. In particular, we will discuss isometries of Minkowski space, the Margulis invariant, group actions by isometries, the Margulis opposite sign lemma etc.

 

 May 18: Jeongseok Oh (KAIST)   Rm 1423

   Quasi-maps and mirror symmetry

Quasi-maps are one of the most important recent developments of counting theories in algebraic geometry. I will introduce them and explain how they shine new light on mirror symmetry.

 

 June 15: Yoonsuk Hyun (KIAS)   Rm 1423

   Understanding divisors on toric varieties

Divisors are a quite important object to understand for a given variety, and there are several basic definitions and properties related with these. As we begin to study algebraic geometry, some concepts often seem too abstract to understand easily. One method to get familiar with it in a more comprehensible way is using toric varieties, which give us a more concrete and clear visualization by using combinatorial arguments. In this talk, we will try to understand some properties of divisors mostly through toric examples.

 

 October 2 (Friday), 5pm: Hoil Kim (Kyungpook National University)   Rm 1423

   Supersymmetry and Morse theory

I want to review the seminal paper "Supersymmetry and Morse theory" (J. of Diff. Geom. 1982) written by E. Witten. Motivated by Morse theory, fixed point theory, index theory (heat kernel proof), he interpreted many important mathematical concepts with the ideas of supersymmetry. This opened the way to several results such as Floer cohomology, localization and supersymmetric proof of the index theorem. In this lecture we want to discuss his ideas and interpretations along these lines.

 

 October 19: Xia Liao (KIAS)    Rm 1423

   An introduction to Chern classes of vector bundles

In this talk I wish to give an introduction to Chern classes of vector bundles. To get an idea of what Chern class of vector bundles are, one essentially only has to know what the first Chern class of vector bundles is, and the latter is closely related to the divisor class of line bundles. With the help of the Chern class of vector bundles, one can define intersection of cycle classes and talk about intersection multiplicities. Another interesting application might be Riemann-Roch transformation depending on the interest of the audience. I learned these topics from Fulton’s book 'Intersection Theory'. What I plan to do is to cover some basic constructions and examples from the first 6 chapters of that book. If time permits, I will spend a little effort to explain the formulation of Grothedieck-Riemann-Roch.

 

 November 2: Binbin Xu (KIAS)    Rm 1423

   Introduction to McShane's Identity

In his thesis, McShane proved a remarkable identity for the lengths of simple closed geodesics on a hyperbolic once-punctured torus. Later this identity was generalized to a variety of different settings by many people (McShane, Mirzakhani, Bowditch, Akiyoshi-Miyachi-Sakuma, Tan-Wong-Zhang, McShane-Labourie, Kim-Kim-Tan and others). Some interesting applications were found as well. In particular, Mirzakhani used her generalized McShane's identity to get the recursive formula of the Weil-Petersson volume of moduli space of bordered hyperbolic surface, and gave a new proof of the Witten conjecture which was first proved by Kontsevich in 1992. In this talk, we will concentrate on the McShane's original work. I'll first recall the necessary background in hyperbolic geometry. Then I'll give the statement of the McShane's identity for once-punctured torus and explain its geometric meaning.

 

 November 16: Woocheol Choi (KIAS)    Rm 1423

   Shape of solutions of some nonlinear elliptic equations

Nonlinear elliptic equations appear in various areas of physics and mathematics. An important question is to know the explicit shape of solutions of such equations. In this talk, I will explain about this question for simple examples of nonlinear elliptic equations.

 

 December 7 (11am-12pm): Hyejin Park (Ohio State University)    Rm 1423

    Introduction to Asymptotics and Borel summability

While most equations cannot be solved explicitly, we can often obtain formal solutions such as power series in an algorithmic way. Sometimes the series do not converge, but we can still use them to derive some information about the solutions. I will introduce asymptotic analysis methods and their applications to the solutions of ODEs and PDEs through examples. Especially even when the formal solution is nowhere convergent, by introducing Borel summation, I will show how to regularize this divergent formal solution to get an actual one, and see what this divergence means in Borel plane.

Relevant references for the I-seminar talk ‘Introduction to Asymptotics and Borel summabiliby’ Books: -Asymptotics and Borel summability, by Ovidiu Costin -Asymptotic Expansions for Ordinary differential Equations, by Wolfgang Wasow Articles (available online): -The Devil’s Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series, by John P. Boyd -Asymptotics beyond all orders: Applications to Free surface flows, by Christopher Lustri

 

 December 21: Joohee Lee (Chung-Ang University)    Rm 1423

   Nodal flows: How cilia activity yields laterality?

Experimental work in developmental biology has recently shown that fluid flow driven by rotating cilia in the node, which is a structure present in the early stage of growth of mouse embryos, is responsible for determining the normal development of the left-right symmetry breaking, with the heart on the left of the body, the liver on the right, and so on. The role of physics, in particular, of fluid dynamics in this process is one of the important questions that remain to be answered. We apply a computational technique based on the immersed boundary method to the 3-D nodal flow problem and investigate the fluid dynamics of the nodal flow in the developing embryo. Numerical simulations allow us to find the optimal condition for the maximal leftward flow in the node, which is qualitatively comparable to the experimental and theoretical findings. In addition, we treat the rigidity of the rotating cilia, the rotational frequency, and the fluid viscosity as parameters to explore how the leftward flow is affected by the change of these parameter values.

 

2014

2014

 February 3: Hyosang Kang (KIAS)

   Constructing non-arithmetic lattices

A natural consequence of the Marguils's arithmeticity theorem is the study on the existence of a non-arithmetic lattice in a simple Lie group. In this talk, I will deliver well-known ideas which partially resolved the difficulty of constructing such lattices.

 

 February 17: Daniele Zuddas (KIAS)

   Idiot's guide to open book decompositions of 4-manifolds   Notes

I will introduce the 2-dimensional open book decompositions of 4-manifolds, as an analogous to the classical open book decompositions of 3-manifolds. Our goal is to understand a theorem of Etnyre and Fuller about the existence of these open books, as well as some interesting facts about their bindings.

 

 March 3: Junyan Cao (KIAS)

   Idiot's guide to the differences between complex projective manifolds and non projective Kähler manifolds

In this talk, I would like first to present some very basic differences between complex projective manifolds and non projective Kähler manifolds. I will also explain the counterexample of Claire Voisin about the Kodaira conjecture.

 

 March 17: Mario Chan (KIAS)

   An idiot's guide to what ∂u = f has done to complex geometry

This is an (ambitious) 1-hour tour which guides the audience from the Cousin problems to the concept of sheaf-valued cohomology, and the role of positivity in the vanishing of cohomology groups. The spirit behind the scene is to solve the ∂-equations.

 

 April 7: Kenji Hashimoto (KIAS)

   Idiot's guide to K3 surfaces

A K3 surface (over the complex numbers) is a compact complex surface which is simply connected and has a nowhere vanishing holomorphic 2-form. We will review basic results on K3 surfaces including the global Torelli theorem for K3 surfaces.

 

 April 21: Youngwoo Koh (KIAS)

   Idiot's guide to Besicovitch sets

In Rn, a Besicovitch set (or a Kakeya set) is a bounded set which contains a unit line segment in every direction. In this talk, we study the basic property of Besicovitch set and its dimension.

 

 June 2: Michel van Garrel (KIAS)

   Counting rational curves in the complex projective plane

The theory of counting curves reached its first apogee in the early 20th century. Since the late 20th century, however, with the nascent of Gromov-Witten theory, it enjoyed a renaissance. I will provide an introduction to some of these intriguing modern results in the case of the complex projective plane.

 

 July 7: Hanchul Park (KIAS)

   Introduction to Toric Topology

Toric geometry connects algebraic geometry and combinatorics in the sense that every toric variety is in a one-to-one correspondence to a fan. One can develop an analogue of this theory to topology, called toric topology. In this talk, we will give a brief introduction to toric topology focusing on its relation with combinatoric and discrete geometry and review some recent results.

 

 July 21: Hojoo Lee (KIAS)

   What makes minimal submanifolds beautiful?

Many of the techniques developed in the theory of minimal submanifolds have played key roles in calculus of variations, differential geometry, elliptic partial differential equations, geometric measure theory, and mathematical relativity. Our aim is to exploit complex analysis to highlight several beautiful properties of minimal submanifolds.

 

 Oct 6: Youngmi Hur (Yonsei University)

   Basics of multidimensional wavelet constructions

Wavelets were introduced as an alternative to the classical Fourier analysis in late 80s, and since then, they have been used in various applications including signal and image processing. In this talk, I will review wavelets and discuss some of the basics of wavelet constructions, especially in multidimensional setting.

 

 Oct 20: Elisha Falbel (l’Universit e Paris VI)

   The SL(3,C)-character variety of the figure eight knot

I will present a description of the space of representations of the fundamental group of the figure eight knot into SL(3,C).

 

 Nov 3: Jaesoon Ha (Kyunggi University)

   Geometric Permutations of Non-Overlapping Unit Balls

Let F be a family of pairwise disjoint convex sets in the d-dimensional Euclidean space. A line intersecting every member of F is called a line transversal to F. A single line transversal with its two directions induces two linear orders on F that are the reverse of each other, so the two linear orders are equivalent. The equivalence classes are called geometric permutations of F. In 2005, Cheong et al. proved that for any family F of n pairwise non-overlapping d-dimensional unit balls, there are at most two geometric permutations if n>8, and at most three if 2<n<9. In this talk, we present a new short proof for the upper bounds on the number of geometric permutations of F.

 

 Nov 17: Kyungbae Park (KIAS)

   Some recent advances in the study of the knot concordance

It is a well-known fact that a knot must be the unknot if it bounds a disk in 3-sphere. A natural question arises how a knot bounds a disk in the 4-dimesional ball. Initially people believed every knot can bound a disk in the dimension four, however it was not until the early 1960’s that a knot, which cannot bound a disk in the 4-ball, was discovered. Roughly speaking, the knot concordance studies this phenomenon, and has explored deeply for decades. Recently, the developments of two powerful homological invariants for knots, knot Floer homology and Khovanov homology, have allowed us to understand more about the knot concordance. In the seminar we introduce the subject, recall the history of the study, and present some recent results especially from knot Floer theory.

 

 Dec 1: Dongmin Gang (IPMU)

   Chern-Simons theory : Bridge between physics and mathematics

Chern-Simons (CS) theory is one of most famous topological quantum field theories (TQFTs). Since the Witten's seminal work by E. Witten on Jones polynomial, CS theory has played important roles in connecting physics and mathematics. In this talk, I will briefly review interesting mathematical aspects of the TQFT. I will start with explaining basic aspects of general quantum field theory (QFT) : path-integral (functional integral) and several techniques for performing the integral (saddle point approximation and geometric quantization). After illustrating these techniques using the simplest QFT (quantum harmonic oscillator) as an example, I will apply the techniques to the CS theory and explain how some mathematics (Volume conjecture, AJ conjecture, 3d Hyperbolic geometry and quantum Teichmuller theory) naturally appears in CS theory.

 

 December 22: I-day

    Ji Young Kim (Seoul National University)

       Sums of nonvanishing integral squares in real quadratic fields

By the Lagrange's theorem, it is clear that, for $k \geq 4$, every nonnegative integer $n$ is representable as a sum of $k$ squares. Indeed, one can always write $n = x_1^2 + x_2^2 + x_3^2 + x_4^2 + 0^2 + \dots + 0^2 $ with an arbitrary number of zeros. Compare to this, the more interesting problem is to consider the representation of integers by exactly $k$ nonvanishing integral squares. In 1911, Dubouis seems to have determined first that all positive integers which cannot be represented by a sum of nonvanishing $k$ squares, when $k \geq 4$. Note that there are infinitely many integers that do not have representation by a sum of exactly $k$ nonvanishing squares $k \leq 4$. In this talk, we determine all totally positive algebraic integers in real quadratic fields that are represented by exactly $k$ nonvanishing integral squares for $k \geq 5$ generally, provided their norm is large enough. General features of results is quite similar to that of Dubouis.

    Byungchan Kim (Seoul National University of Science and Technology)

       Positivity questions in q-series

We introduce three ways to establish the positivity of the coefficients for q-expansions. These three methods have different flavors from algebraic to analytic, and have their own merits. From the typical examples of q-series, we discuss their roles and advantages in the theory.

    Hwa Jeong Lee (Korea Advanced Institute  of  Science and Technology)

       On the arc index of knots

A knot in knot theory is a one-dimensional circle embedded in three-dimensional space. A fundamental problem in the theory is to distinguish knots that are not same. Research on knot invariants is not only motivated by the problem but also to understand basic properties of knots. In this talk, we present a small survey of known results on the arc index of knots and also introduce some recent results.

    Jang Soo Kim (SungKyunKwan University)

       Multiplex juggling probabilities

Imagine that a juggler is juggling balls. Let's assume that he catches and throws at most one ball at each discrete beat. At each beat, we can write a 0-1 sequence for which i-th bit is 1 if a ball is going to be landed after i beats, and 0 otherwise. Such a sequence is called a juggling state. Suppose that the juggling can throw a ball at most height h. In other words, when a ball is thrown, it will be landed after at most h beats. Suppose that the juggler throws each ball uniformly randomly such that we still have at most one ball landed at each beat. In 2005, Warrington showed that this random process is a Markov chain with steady state probability and found a formula for the probability. In this talk we will review Warrington's idea and consider multiplex juggling which allows to have more than one ball landed at each beat.

    Sukmoon Huh (SungKyunKwan University)

       Vector bundles on threefolds

Globally generated vector bundles on projective varieties play an important role in classical algebraic geometry, mainly because they give embeddings of the varieties into Grassmannians. Due to the Hartshorne-Serre correspondence the study on such bundles can be done in terms of the classification of smooth curves with proper numeric invariants. In this talk we will take a glipse of features of Hartshorne-Serre correspondence to deal with some simple vector bundles.

 

 

2013

2013

 February 18: Jaigyoung Choe (KIAS)

   The idiot's guide to the isopermetric inequality

We will introduce the isoperimetric inequality. And we'll give its two proofs (one easy, the other challenging for the idiot). Then we will consider the same inequality for the soap films (minimal surfaces).

 

 March 4: Fabian Ziltener (KIAS)

   The idiot's guide to Symplectic geometry

I will explain how symplectic geometry originated from classical mechanics, formulate some important questions in this field, and mention a few highlights such as Gromov's nonsqueezing result.

 

 March 18: Mihai Paun (KIAS)

   The idiot's guide to the Kaehler cone of compact complex manifolds

In this talk, we intend to present a few very basic facts concerning the Kaehler cone associated to a compact complex manifold. Some links with the algebraic geometry will be equally discussed.

 

 April 1: Jun Ho Lee (KIAS) Room 1424

   The idiot's guide to the class number problem of number fields

I will introduce the class number problem of number fields. In particular, I will focus on the class number one problem of quardratic fields and cubic fields.

 

 April 15: Hyosang Kang (KIAS)

   Tales of three groups: arithmetic groups, mapping class groups, and outer automorphisms

The study of arithmetic groups and mapping class groups started from different motivations, yet there are very close analogies between two groups. Recently, we found that groups of outer automorphisms of free groups also share some properties of the above groups. In this talk, we will observe what are these groups, how they are related, and why such similarity arise.

 

 May 6: Tarig Abdelgadir (KIAS)

   The idiot's guide to the McKay correspondence

Take a finite subgroup G of SL(2,C) and consider its natural action on the complex plane, the associated orbit space, call it X, is a variety with an isolated singularity. The classical McKay correspondence relates the geometry of the minimal resolution (a smooth approximation) of X to the representation theory of G in SL(2,C). I plan to give a gentle introduction to this correspondence and discuss some reformulations and generalizations.

 

 May 20: Thilo Kuessner (KIAS)

   The idiot's guide to hyperbolic surfaces and 3-manifolds

We will start with discussing some elementary approaches to the topology of surfaces and why Hyperbolic Geometry is useful in 2-dimensional Topology, e.g. for the classification of diffeomorphisms up to homotopy. Then we will move to 3-manifolds and discuss in some breadth the example of the figure knot complement, whose Hyperbolic Metric can be seen in several ways: from an arithmetic representation of its fundamental group, from an explicit triangulation or from its fibering over the circle. If time permits, we will also discuss more general 3-manifolds and topological invariants that can be computed from hyperbolic metrics.

 

 June 3: Leobardo Rosales (KIAS)

   So what is Geometric Measure Theory?

Many mathematicians have wondered exactly what is geometric measure theory. Amongst such mathematicians are geometric measure theorists. I will give a philisophical introduction to GMT, including why it is useful. The goal will be to introduce Allard's Regularity Theorem, a now basic albeit at once sophisticated result.

 

 June 17: Todd Drumm (Howard University)

   Lorentzian Geometry and the deformations of surfaces

We will provide an introduction to three-dimensional Lorentzian geometry. We will be especially interested in the Margulis invariant and its application to groups which act properly on Lorentzian space. Also, the connection between Lorentzian geometry and the deformations of noncompact hyperbolic surfaces will be highlighted.

 

 July 15: Woojin Jeon (KIAS)

   An idiot's guide to 'Primitive stability'

This talk would be mainly about the action of Aut(Fn) on Hom(Fn, PSL(2,C)). Starting from a brief introduction to geometric group theory, we will discuss a dynamical decomposition of Hom(Fn, PSL(2,C)) and will also discuss some open problems.

 

 August 5: Jinmyoung Seok (KIAS)

   Idiot's guide to critical points theories and elliptic PDEs

In this introductary seminar, I will discuss about the basic critical points theory, which is a very powerful tool to study superlinear elliptic PDEs. I would like to deal with simple model cases, not technical ones to show the essential ideas much clearly.

 

 September 2: Otto van Koert (Seoul National University)

   The idiot's guide to contact topology

I will describe some of the basics of contact topology, mention some popular problems in the field, and discuss some methods to tackle these problems.

 

 September 16: HwanChul Yoo (KIAS)

   Root hyperplane arrangements and graph colourings

We will introduce some enumerative problems regarding hyperplane arrangements. In particular we will see how the characteristic polynomials factor when the root hyperplane arrangements are ''smooth''. This can be seen in the connection to certain graph colouring problem.

 

 October 7: HyunKyu Kim (KIAS)

   (The idiot's guide to) quantization of Teichmüller spaces

I'll give a gentle introduction to quantum Teichmüller theory, which is established in late 1990's. I'll first describe some coordinate systems of the Teichmüller space of a Riemann surface: Penner's lambda length coordinates, Penner-Thurston's shear coordinates, and Kashaev's ratio coordinates. Mapping class group acts on the Teichmüller space, and the action will be described in terms of the coordinate change maps for the above mentioned coordinates. Then we'll discuss what it means to quantize a symplectic manifold in general, with a simple example. In the case of the quantization of Teichmüller space, the crucial point is the compatibility with the mapping class group action. I will present how this quantization is done with the help of a special function called "quantum dilogarithm". One of the main results of the quantum Teichmüller theory is a class of projective representations of the mapping class group. If (and only if) time permits, I'll also say a few words on some research topics in this area. The talk will be made as elementary as possible.

 

 November 4: Hyunsuk Kang (KIAS)

   Introduction to extrinsic curvature flows

This is a survey talk on some of well-known extrinsic curvature flows of codimension one in Euclidean space. Basic definitions and notion will be explained and some of important historical reults will be mentioned. If time permits, recent results by Huisken and Brendle will be discussed.

 

 December 2: Hanjin Lee (Handong Global University) Room 7323

   Introduction to analysis on Alexandrov spaces

Alexandrov space is an example of singular space most systematically studied. Lecture will be a short introduction to Alexandrov space and issues on how to do analysis there. Ricci curvature bounded below condition and related application to analysis will be discussed.

 

 December 16: Kwangho Choi (Seoul National University)

   On existence of the heat flow for harmonic maps between compact Riemannian manifolds

We discuss short/long-time existence of solutions of the heat flow for harmonic maps. Basic identities and analytic background for the proof of short/long-time existence of solutions are reviewed. If time permits, we present the weak formulation of the harmonic map heat flow and survey the results on global existence of weak solutions of the flow.

 

 December 23: I-day Room 7323   Program download

    11:00 Taechang Byun (University of Seoul)

   The geometry of  SO(n)\SO₀(n,1)

The Lie group SO0(n,1) has the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left and right. The geometry of the quotient space of the homogenous submersion SO0(n,1) → SO(n)\SO0(n,1) is investigated. It is a beautiful space with many symmetries. The space is expressed as a warped product. Its group of isometries, geodesics, and sectional curvatrues are calculated.

    14:00 Ji Young Kim (Seoul National University)

   The strictly regular quaternary quadratic Z-lattices

A positive definite quadratic Z-lattice is said to be strictly regular if it primitively represents all positive integers that are primitively represented by its genus. In this talk, it will be shown that there exist only finitely many isometry classes of primitive integral positive definite quaternary quadratic Z-lattices that are strictly regular.

    14:50 Jaewoong Kim (Seoul National University)

   Introduction to the invariant subspace problem

I will talk about the invariant subspace problem, which is the most famous problem in operator theory, and show partial results on that.

    16:00 Joonhyung Kim (Konkuk University)

   On complex and quaternionic hyperbolic Fuchsian groups

In the beginning of my talk, I will introduce the complex/quaternionic hyperbolic geometry. And then, I'll give the characterization of Fuchsian groups acting on complex/quaternionic hyperbolic spaces.

    16:50 Hwayoung Lee (Seoul National University)

   Local Donaldson Thomas theory on G-Hilb(C²)&timesP¹

I will start from discussing what is a DT theory and then introduce our work as follows. The quotient space of complex plane C2 with G-action where G is a finite subgroup of SL2(C) has a mild singularity at (0,0). It is known that this quotient space C2/G has a resolution G-Hilb(C2) (which is a closed subscheme of Hilbert scheme of |G|-points Hilb{|G|}(C2)). We use McKay correspondence by BKR to make a connection between stable pairs on the threefold G-Hilb(C2)&timesP1 and stable (ADE) quiver bundles on P1. This is a joint work with Bumsig Kim and Timothy Logvinenko.