Why, What, hoW?

Korea Institute for Advanced Study

Mondays, 5:10–6:10 pm, Room 1423/24

This seminar is inspired by Idiot's Guide seminar and the New Member Seminar that used to be held at KIAS. Research talks often focus on *what* and *how*, but don't address the question of *why* the speaker finds the topic interesting. By encouraging speakers to avoid technical details and think about *why*, we hope to create a friendly seminar where:

- All members of KIAS and visitors feel welcome.
- Members can learn about areas of mathematics from a down-to-earth perspective.
- Speakers can share the topics they are interested in with a wide audience.
- People from all research areas can socialize.

We invite anyone interested in giving a talk to contact one of the organizers.

This seminar is oriented not to be a usual research seminar. We request that speakers avoid technical details and make the topic accessible to broader audience. Speakers may find it helpful to keep in mind the three questions suggested by the title of the seminar:

*"Why is it interesting to study this topic?"**"What are the main ideas in this topic?"**"How does one study this topic?"*

Speakers may present their own theorems but it is neither necessary nor recommended. As we encourage stupid questions and a lively discussion, speakers should expect less material than in a usual research seminar.

Feel free to contact us at threeW.kias [at] gmail.com

This page uses MathJax. This may cause math processing errors in older browsers such as Internet Explorer 8. If there is a problem, please visit the official announcement of seminars from KIAS.

Based on the guideline of KIAS, the seminar is canceled and will not be held for a month (March 2–April 6).

**
February 24:
Yeping Zhang (KIAS-Math)
**
17:10–18:10 #1424

*
Determinant line, Reidemeister metric and Ray–Singer metric
*

The determinant line of a finite dimensional vector space is its top exterior power. In this talk, I will give several applications of this simple construction.

- Reidemeister metric.

It is a topological invariant which may distinguish certain mutually homotopic topological spaces. - Ray–Singer metric.

It could be viewed as an analytic analogue of the Reidemeister metric. - Cheeger–Müller–Bismut–Zhang Theorem.

This theorem clarifies the relation between the two metrics mentioned above.

**
February 17:
Yoshihiko Matsumoto (Osaka University)
**
17:10–18:10 #1423

*
On Cartan connections
*

Riemannian manifolds are infinitesimally Euclidean. Likewise, one may think about geometries in which spaces are infinitesimally modeled on homogeneous spaces––Cartan connections are what describe them. In this talk, I’m going to discuss the naturalness of the idea of Cartan connections, putting some emphasis on the notion of development of curves, and explain what the Cartan connections describing Riemannian and conformal geometry are like. It is worth noting that the Weyl curvature tensor, whose vanishing characterizes the local conformal flatness, is best understood along this line of thought. I'm also planning to touch upon the relation of the conformal Cartan connection to the Fefferman–Graham “ambient metric” construction and to Poincaré-Einstein manifolds.

**
February 10:
Kevin Morand (Sogang University-Physics)
**
17:10–18:10 #1423

*
Kontsevich's graph complex and deformation quantization
*

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a graph version of the deformation theory for the Schouten algebra of polyvector fields on a manifold. In this talk, we will review Kontsevich's graph construction and the role it plays in deformation quantization. If time allows, we will discuss possible generalisations of Kontsevich's construction to higher structures.

February 3: It was canceled.

January 27: It was skipped because it is a public holiday (Seollal Holiday)

**
January 20:
Zhihao Duan (KIAS-Physics)
**
17:10–18:10 #1423

*
A glimpse of topological string theory
*

Topological string theory lies at the crossroads between physics and mathematics, and this talk tries to give a glimpse of it. On the physics side, I will review its physical definition and introduce several important developments. On the mathematics side, I will talk about a conjecture known as the Gromov-Witten/Donaldson-Thomas correspondence, formulated by Maulik, Nekrasov, Okounkov and Pandharipande. Finally, we will treat $C^3$ as a toy example and see how combinatorial pictures naturally arise.

**
January 13:
DongSeon Hwang (Ajou University)
**
17:10–18:10 #1423

*
A glimpse of Manin's conjecture
*

Diophantine problems have fascinated many mathematicians throughout the ages. Manin's conjecture provides a precise prediction on some types of Diophantine problems. More precisely, it predicts the asymptotic behavior on the number of rational points of bounded anticanonical height on Fano varieties. In this talk, I will try to explain the historical development of the conjecture and how the geometry governs the arithmetic.

January 6:
It was skipped because of the conference:
*Workshop on Atiyah classes and related topics*.

December 23–30: Winter Break

**
December 16:
Yuto Yamamoto (IBS-CGP)
**
17:10–18:10 #1423

*
Introduction to tropical geomery
*

Tropical geometry is algebraic geometry over the max-plus algebra. There is a certain procedure called tropicalization, to associate a tropical algebraic variety with a classical one. It connects tropical geometry with combinatorial aspects of classical algebraic geometry. In this talk, I will explain basic ideas of this field by using a couple of elementary results and simple examples.

December 9: It was skipped.

**
December 2:
Joachim Koenig (KAIST)
**
17:10–18:10 #1423

**
Polynomials and their monodromy groups,
**
Abstract

- Given an irreducible integer polynomial $f(t,X)$ in two variables, what can we say about the set of $t_0\in\mathbb{Z}$ such that $f(t_0,X)$ is reducible?
- Given two polynomials $f$ and $g$, can the two-variable polynomial $f(X)-g(Y)$ factor in any non-trivial way?
- What do the value sets $\{f(t) \text{ mod } p \mid t\in \mathbb{Z}\}$ (with $p$ a prime) tell about the polynomial $f$ itself?
- How ``far" can a polynomial map $x\mapsto f(x)$ on the rational numbers be from being injective?

All of these are (more or less) classical arithmetic-geometric problems. Their common feature is that they lead to interesting questions in group theory. I will motivate this connection by introducing monodromy groups of polynomials and explaining how they can be used to attack the above problems. Along the way, I will try to show how group theory can explain some astonishing examples occurring around the above questions.

**
November 25:
Kang-Ju Lee (Seoul National University)
**
17:10–18:10 #1423

*
A relationship between commutative algebra and topological combinatorics
*

In this talk, we explain a relationship between combinatorial commutative algebra and topological combinatorics, and present formulas for the betti numbers arising from ideals defined by graphs and those of simplicial complexes associated with these ideals.

**
November 18:
Jaigyoung Choe (KIAS-Math)
**
17:10–18:10 #1424

*
Unknottedness of minimal surfaces in $S^3$
*

There are two parallels on $R^2$. But there are no parallels on $S^2$. This is mainly why we have no knotted minimal surface in $S^3$.

**
November 11:
WonTae Hwang (KIAS-Math)
**
17:10–18:10 #1423

*
Brief Story of Elliptic Curves and Fermat's Last Theorem
*

In the history of number theory, the Fermat's Last Theorem, proven by Andrew Wiles and Richard Taylor, is one of the deep and beautiful results, in which, at least two seemingly unrelated mathematical areas were used in its proof. Roughly speaking, the theorem was proved by relating the existence of integer solutions of certain Diophantine equations to a special property, called modularity, of the corresponding elliptic curves. This kind of phenomenon occurs many times in various fields of math to reveal the beauty of mathematics. In this talk, we briefly review the story of the FLT (as non-experts) with the aid of the book: Rational Points on Elliptic Curves.

**
November 4:
Daehwan Kim (KIAS-Math)
**
17:10–18:10 #1423

*
Solitons for the mean curvature flow and weighted minimal surfaces
*

The mean curvature flow (MCF) arises from the study of crystal growth, grain growth, image processing and other scientific fields. The reason for these study is that MCF is the negative gradient flow of the area functional and it is a well-known property that any closed (hyper)surface flows in the direction of steepest descent for the area and occurs singularities in finite time under MCF. There are two types of singularities that are type 1 and type 2 represented by the self-similar solution and translating soliton, respectively. In terms of weighted minimal surfaces, these two models are regarded as critical points of the area functional with density $e^{-f}$, which are $f$-minimal surfaces, where $f$ is linear or quadratic function, respectively. In this talk, we observe the relation between solitons for MCF and $f$-minimal surfaces and provide several examples.

**
October 28:
Richard Eager (KIAS-Physics)
**
17:10–18:10 #1423

*
What is a BPS state?
*

BPS states are particles or extended objects in supersymmetric quantum field theories that transform in a short representation of the supersymmetry algebra. One of the main motivations for studying and counting BPS states is to give a microscopic understanding of the Beckenstein--Hawking entropy of supersymmetric black holes. BPS state counting is also closely related to enumerative geometry and Donaldson--Thomas invariants in algebraic geometry. In addition to reviewing the role of BPS states in Seiberg--Witten theory, we give a simple example of BPS state counting for a local Calabi--Yau threefold using exponential networks.

October 21:
It was skipped because of the conference:
*Progress in Several Complex Variables*.

**
October 14:
Aeryeong Seo (Kyungpook National Univesity)
**
17:10–18:10 #1423

*
Geometry of domains in several complex variables
*

In this talk, I will introduce some domains(open and connected sets) in several complex variables including domains of holomorphy, pseudoconvex domains, bounded symmetric domains and bounded homogeneous domains. I will also give a brief introduction to some invariant metrics/distances, automorphism groups, scaling methods and their applications.

October 7: Skipped.

**
September 30:
Nessim Sibony (Université Paris-Sud / KIAS)
**
17:10–18:10 #1423

*
Entropy
*

How to measure the complexity of a dynamical system?

A possibility is to count the number of orbits one can distinguish
at a certain scale before time $n$.
Then look at the growth of that function as a function of time and scale.
Amazingly it's possible to end up with a number.

I will discuss notions of entropy and give examples from complex geometry.

**
September 23:
Clifford Blakestad (Postech)
**
17:10–18:10 #1423

*
Mathematics from the p-adic perspective
*

The real numbers are only one of infinitely many completions of the rational numbers. For each prime integer $p$, we define a new metric on the rational numbers by declaring $p$ to be "small," yielding a new completion called the $p$-adic numbers. We will discuss ways to think about these new completions and explore notions such as $p$-adic calculus and $p$-adic manifolds.

**
September 16:
Hsuan-Yi Liao (KIAS-Math)
**
17:10–18:10 #1423

**
Deformation of algebra and Hochschild cochains**, Notes

I'll explain a problem about deformation of algebra and its relationship with the differential graded Lie algebra of Hochschild cochains. If time permitted, I'll talk about a coalgebra point of view and deformation quantization.

**
September 9:
Matthieu Sarkis (KIAS-Physics)
**
17:10–18:10 #1423

*
A connection between algebraic surfaces and Mathieu moonshine
*

After recalling some background motivational results on moonshine, we will try and motivate in which sense Mathieu moonshine should have to do with an infinite family of algebraic surfaces, hence pointing towards a more global geometric understanding of moonshine phenomena.

**
September 2:
Ryo Suzuki (KIAS-Physics)
**
17:10–18:10 #1424

*
Gauge theory correlators from quiver calculus
*

In Gauge/String Duality, non-planar corrections to the correlators of gauge theory correspond to higher-genus corrections to the string worldsheet. I talk on how to study such quantities by finite-group theory and quiver calculus, whose ultimate goal would be to construct "Riemann surfaces" from permutations.

**
August 26:
Jeonggyu Huh (KIAS-CS)
**
17:10–18:10 #1423

**
Introduction to Deep Learning**, Slides

I will have a brief talk about deep learning for researchers studying pure mathematics or pure physics. The participants do not have to know about deep learning at all because I will shortly review basic concepts of deep learning (artificial neural network, error back propagation etc). I will also explain how the new method is different from traditional statistics and how it is changing the world.

August 5–19: Summer Break

**
July 29:
Geunho Lim (Indiana University in Bloomington)
**
17:10–18:10

*
Cheeger-Gromov $L^2$ rho-invariants of 3-manifolds*

Atiyah, Patodi and Singer defined rho-invariants to generalize Hirzebruch signature theorem for Riemannian manifolds with boundary. Cheeger-Gromov extended rho-invariants to $L^2$ rho-invariants. As Chang-Weinberger tried first, $L^2$ method allows us to topologically approach rho-invariants. Cha showed the existence of universal bounds of $L^2$ rho-invariants by using geometric topology. In this talk, we first recall the definition and properties of Cheeger-Gromov $L^2$ rho-invariants including the existence of universal bounds. The goal of this talk is to share geometric topological idea which is used to find explicit bounds. If time permits, I will discuss recent progress on elliptic 3-manifolds.

**
July 22:
Sreedhar Bhamidi (KIAS-Math)
**
17:10–18:10

*
What are motives?
*

Motives refer to a 'universal cohomology theory' for algebraic varieties proposed by Grothendieck in the 1960's. Till date the construction of a category of motives is still conjectural. Regardless, the idea has led to far reaching consequences in Algebraic Geometry and has been a guiding light. The goal of this lecture is to motivate and discuss some conjectures and their implications. The lecture would be elementary and we hope by avoiding technical details we shall be able to get across the underlying ideas surrounding this beautiful theory.

**
July 15:
Sanghyeon Lee (KIAS-Math)
**
17:10–18:10 #1423

*
Splitting of genus 1 Gromov-Witten invariants
*

Gromov-Witten invariant, which can be considered as a number counting curves in a variety, appears in both symplectic/algebraic geometry. When a variety is a complete intersection in a projective space, A. Zinger developed a way to split genus 1 GW invariant (counting genus 1 curves) to reduced invariant and genus 0 invariants. This reduced invariant can be expressed by an Euler class of some vector bundle on some smooth space. This splitting is related to the geometry of moduli space of curves in a variety, which is called stable map space. In this talk, we will talk about the geometry of the genus 1 stable map space and how this give us a way of splitting genus 1 invariants.

**
July 8:
Thomas Goller (KIAS-Math)
**
17:10–18:10 #1423

*
Hilbert schemes of points
*

A famous mathematician's scoring rubric? No no no, that would be "Hilbert's scheme of points". Consider a natural problem: given a geometric space X, construct a new geometric space that parametrizes all subsets of X that contain exactly k points. This is not difficult to do. But now suppose that we want to compactify our new space. Then we need a plan for dealing with the issue of several points "colliding". I will describe a nice way of solving this problem in algebraic geometry and discuss some remarkable features of the resulting spaces. The most technical detail will be the grammatical remark made at the beginning of this paragraph.

**
July 1:
Min Hoon Kim (KIAS-Math)
**
17:10–18:10 #1423

*
Spheres
*

Spheres, the simplest examples of closed manifolds, are central objects in geometric topology. Initiated by Poincare, topologists have considered homotopy spheres (manifolds which are homotopy equivalent to spheres). In this introductory talk, I will give a survey on homotopy spheres. For the first part, I will discuss celebrated results of Milnor and Smale on homotopy spheres without giving too much details. For the second part of talk, I will discuss well-known examples of homotopy 4-spheres including Cappell-Shaneson spheres and Gluck twists. If time permits, I will discuss recent progress on Cappell-Shaneson spheres which is joint work with Hakho Choi and Motoo Tange.

**
June 24:
Jae Min Lee (KTH Royal Institute of Technology)
**
17:10–18:10

*
Geometric investigation of PDEs in hydrodynamics
*

The goal of this talk is to present a geometric approach on Euler equation for ideal fluid pioneered by V. Arnold, and its applications. Following Arnold, we will first review the motion of a 3-dimensional rigid body and extract the abstract mathematical structure, i.e., geodesic equations on a Lie group endowed with one-sided invariant metrics. We will briefly discuss some technical difficulties arising when we extend this idea to the infinite dimensional setting. Finally, we will highlight some applications to problems in PDEs from fluid mechanics.

**
June 17:
Sungmin Yoo (KIAS-Math)
**
17:10–18:10 #1423

*
Generalization of the Poincare metric
*

As generalization of the Poincare metric, I will introduce two biholmorphically invariant Kahler metrics: Bergman metric and Kahler-Einstein metric. These two metrics play important roles in study of complex manifolds. In this talk, we will survey their properties and relation.

**
June 10:
Christoph Saulder (KIAS-Physics)
**
17:10–18:10 #1503

*
Why do we care about early-type galaxies?
*

We will discuss what early-type galaxies are and how one can identify them in large-scale surveys. Additionally, I will illustrate how distances are measured in the universe and how early-type galaxies and their fundamental plane can be used to do so. I will also explain why this is important for modern cosmology.

**
June 3:
Xiaoxiang Chai (KIAS-Math)
**
17:10–18:10 #1423

*
Constructing a minimal surface in a sphere with an arbitrary metric
*

Given a simple Jordan curve, there is a minimal surface spanning this Jordan curve as its boundary. This result known as Plateau problem is due to Douglas and Rado in the 30s. Douglas was the first Fields medalist. I will start from such basic aspects of minimal surface theory as definition, first variation formula, second variation formula. Later, the talk would be focused on finding minimal surfaces using Almgren-Pitts min-max theory. This part is mostly after Jon Pitts, Gromov, Guth, F. Marques and A. Neves. We present the theory in parallel to the theory of eigenfunctions and eigenvalues so that it would be easier to follow the general idea. In particular, there exists at least one minimal hypersurface in every Riemannian manifold. Recently, it is proved that there exists infinitely many of them. If time permits, we will explain some of these results for generic metrics. Also, if time permits, for people with a physics background, I will mention a quick proof of Schoen-Yau positive mass theorem using minimal surface theory would be sketched. For people with a PDE background, the connection between semilinear PDE and minimal surface would be briefly discussed. These two aspects are related to research of my own.

References:

1. Fernando C. Marques, Andr'e Neves, Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, https://arxiv.org/abs/1311.6501

2. Richard Schoen and Shing Tung Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., Volume 65, Number 1 (1979), 45-76.

3. Pedro Gaspar, Marco A. M. Guaraco, The Allen-Cahn equation on closed manifolds, https://arxiv.org/abs/1608.06575

**
May 27:
Hojoo Lee (Seoul National University)
**
17:10–18:10 #1423

*
Flat structures on soap films
*

The modern theory of minimal surfaces offers spectacular applications to, for instance, the three dimensional topology and geometry, positive mass conjecture in mathematical relativity, and the Ricci flow proof of Poincare conjecture. In this talk, we shall present explicit examples of negatively curved minimal surfaces, which can be characterized by flat structures introduced by Chern and Ricci.

[1] Hojoo Lee, Mysteries of Minimal Surfaces, YouTube Video

[2] Joaquin Perez, A new golden age of minimal surfaces, Notices Amer. Math.Soc. 64 (2017), no. 4, 347-358.

[3] Jeremy Gray, Mario Micallef, About the cover: the work of Jesse Douglas on minimal surfaces, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 2, 293-302.

**
May 20:
Tomoyuki Hisamoto (Nagoya University)
**
17:10–18:10 #1424

*
Geometric flow and the limit
*

We first look at the plane curve shortening flow in order to get the feeling of general geometric flow. The main topic is the Kähler-Ricci flow, which is the evolution of a metric on the complex manifold. I will explain the relation with the uniformization theorem for complex curve and what happens in higher dimension. This talk may serve as a prelude to my research talk on Wednesday and Thursday.

**
May 13:
Jaepil Lee (Yonsei University)
**
17:10–18:10 #1423

**
Bordered Heegaard Floer homology and knot complement**,
Slides

Lipshitz, Ozsvath and Thurston have developed the bordered Heegaard Floer homology, which is a variant of the Heegaard Floer homology designed for a 3-manifold with boundaries. In this talk I will focus on the bordered Floer chain complex of torus boundary case, especially on a knot complement in 3-sphere and its relation to the knot Floer homology. I will also explain a 2-link complement and knot Floer homology of knots in integral homology sphere.

May 6: It was skipped because it is a public holiday (Children's Day observed).

**
April 29:
Seong-Mi Seo (KIAS-Math)
**
17:10–18:10 #1423

*
Universality for random normal matrix ensembles
*

In the random matrix theory, one of the most important phenomenons is that the local eigenvalue statistics of large random matrices have a universal behavior. I will introduce some universality results for the random normal matrix model and discuss the universality class depending on the location of a point where the local eigenvalue statistics are considered.

**
April 22:
Kyunghwan Song (Ewha Womans University)
**
17:10–18:10 #1423

**
The inverses of tails of the Riemann zeta function for some natural numbers
and real numbers in critical strip and related topics**,
Slides

The Riemann zeta function appears in many contexts in mathematics each of which is very interesting. It is also an interesting object to study in other areas such as physics. I would like to introduce the relationship between the Riemann zeta function and the Stefan-Boltzmann law, photon density, and also explain some results regarding the Riemann zeta function and its variations. We begin the talk by introducing the Riemann zeta function, its generalized functions, and their properties. Next, we will see why a reciprocal sum related to the Riemann zeta function is interesting by seeing explicit examples when $s = 2, 3, 4$, and $5$. Afterwards, we present a new result on the reciprocal sum related to the Riemann zeta function at $s = 6$, which is a joint work with WonTae Hwang. Also, we give some bounds of the inverses of tails of the Riemann zeta function on $0 < s < 1$ and compute the integer parts of the inverses of tails of the Riemann zeta function for $s = 1/2, 1/3$, and $1/4$, which is a joint work with Donggyun Kim. Furthermore, we present additional results for $s = 1/d, 2/d$ where $d \ge 2$ is any integer, which is a joint work with WonTae Hwang.

April 15:
It was skipped because of the conference:
*Hyperbolic Geometry, Spectral Geometry and related topics*.

**
April 8:
Plinio Murillo (KIAS-Math)
**
17:10–18:10 #1423

*
Can one hear the shape of a drum?
*

Mathematically, a drum can be considered as a bounded region $D$
in the Euclidean plane. The vibration of the drum can be
modelled by the wave equation, and the vibrating frequencies are
codified by the eigenvalues of the Laplacian operator on $D$.
In 1966, Marc Kac popularized the question which has the title of this talk,
and which translates to ask whether the eigenvalues of the Laplacian
determine the geometry of $D$.
In this occasion, we will talk about the history of the problem,
and how its developments have touched different areas of mathematics
such as Analysis, Geometry and Number Theory.

This talk will serve as a prelude to the workshop
"*Hyperbolic Geometry, Spectral Geometry and related topics*"
to be held at KIAS from April 15th to 19th.

**
April 1:
Jean-Emile Bourgine (KIAS-QUC)
**
17:10–18:10 #1424

*
Integrable systems, quantum groups and string theory.
*

Quantum integrable systems are a special class of quantum systems in which exact calculations are possible. They play an essential role in physics, from descriptions of magnetism and cold atoms to (supersymmetric) particle physics and string theory. Exact methods have been developed for explointing the rich underlying mathematical structure of symmetries called "quantum group". In this talk, I will explain what a quantum group is, how it appears in quantum systems, and briefly mention an application to a particular problem in string theory.

**
March 25:
Chul-hee Lee (KIAS-Math)
**
17:10–18:10 #1424

*
Dilogarithm function and applications
*

The dilogarithm function is a special function of a single variable. It appears in several different contexts of mathematics, such as algebraic K-theory, hyperbolic geometry, cluster algebras and mathematical physics. I will explain the basic properties of the dilogarithm function, with a focus on the functional identities satisfied by it, and its applications to computing volumes of hyperbolic 3-manifolds.

**
March 18:
Makoto Miura (KIAS-Math)
**
5pm #1424

*
Grassmann tensors in algebraic vision
*

Algebraic vision is a new research area that aims to look at geometric problems in computer vision using algebro-geometric methods. In this talk, I will give an overview on this area with particular emphasis on the Grassmann tensors, which are one of the main ideas for the reconstruction problems and are also used in realistic algorithms.

**
March 11:
Carlos Scarinci (KIAS-CMC)
**
5pm #1423

*
3-dimensional anti-de Sitter geometry and Teichmüller theory
*

In this introductory talk I will give an overview on anti-de Sitter geometry (AdS) in dimension 3 and its relation to Teichmüller theory. The idea is to focus on the basic definitions and on some core ideas on the use of 3d geometry in Teichmüller theory.

Sreedhar Bhamidi, Jean-Emile Bourgine, Sanghyeon Lee, Makoto Miura, Matthieu Sarkis, Carlos Scarinci

- Plinio Murillo
- Thomas Goller
- Byungdo Park