Share your mathematical interests with others! This is very informal seminar: no subject restriction, no time limitation, no fixed format. Just bring your topics that you have enjoyed and introduce them to others in highly interactive atmosphere.
You do not have to be an expert on your topic so please do not hesitate to give a talk!

**We usually meet at 8pm at the common room(토론실) located on the 1st floor of the math department building(E6-1), but times and places are subject to change. ** Note that there might be some alcohols and finger foods.

We are happy to have a speaker from other departments such as physics, computer science and so on.

*Inquiry: Hongtaek Jung*

TBA

Borel density theorem, is a theorem which is simply phrased as "all lattices are generally Zariski dense." In this talk we present a proof of the theorem by Furstenburg which shows some scent of dynamical ideas, and pick up some examples that, although the theorem is designed for real Lie groups only, apply the theorem for showing Zariski denseness in various situations. We will introduce some extra examples in addition, that might supplement the idea of guessing Zariski closures.

Brief Note

I will focus on the semilinear Schrödinger equation of the form $i \partial_{t} u + \Delta u = \mu |u|^{p} u$ whose initial datum lies in some $L^2$-based Sobolev space. At first, I will briefly explain the linear Schrodinger equation and its dispersive phenomenon, and then quantify it in terms of Lebesgue norms to obtain the Strichartz estimates. With the contraction mapping principle, this allows us to have the local well-posedness of the equation for a certain range of $p$. On the other hand, as a Hamiltonian equation, symmetries of the energy functional give rise to some conserved quantities (via Noether's theorem). Using these, one can upgrade the local well-posedness to the global well-posedness. Finally, if time permits, I will briefly introduce the scattering theory.

Before the seminar, I recommend you to review Young's inequality, Minkowski's integral inequality, Risez-Thorin interpolation, Hardy-Littlewood-Sobolev inequality, Sobolev embeddings and contraction mapping principle. You can find these in Wikipedia.

Lecture note

We briefly introduce old questions asked by thermodynamics and answered by equilibrium statistical mechanics. We introduce Brownian motion and see how could it be a evidence of equilibrium statistical mechanics. We introduce Loschmidt’s paradox-how the entropy could always increase even if the motions of all molecules have time reversal symmetry, which is answered by the fluctuation theorem. Lastly, recent study topics on non-equilibrium statistical mechanics are introduced.

In this talk, we give $L^p$-theory of existence and uniqueness results of second-order parabolic and elliptic equations with variable coefficients in Sobolev spaces. First, we give $L^2$-theory of existence and uniqueness result of paraboilc and elliptic PDEs of second-order in whole domain. Second, we give a scheme on $L^p$-theory and give $L^p$ results. Third, we study the difference between 2-dimension and $n$-dimension in PDEs. If time permits, we study some recent results on this topics.

Reference: https://arxiv.org/abs/1605.03322

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1, ..., x_n\}$ of integers where $x_1 < \cdots < x_n$, let the gap sequence of this set be the nondecreasing sequence $d_1,\dots, d_{n-1}$ where $\{d_1,\dots,d_{n-1}\}$ equals $\{x_{i+1}-x_i: i\in[n-1]\}$ as a multiset.

This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers $p$ and $q$ , there is a positive integer $r_0$ such that for all $r\geq r_0$, the set of integers can be partitioned into $4$-sets with gap sequence $p,q,r$.

Mathematicians have long been interested in the following problem.

"Let $K/k$ be a field extension, $V$ be an algebraic object over $k$ and $U$ be an algebraic object over $K$. Here, algebraic object means something like $k$-module, $k$-algebra, $k$-variety. Let’s write $V_K$ as a extension of $V$. If $V_K$ is isomorphic to $U$, then can we classify such $V$’s?"

A Galois cohomology is a group cohomology of Galois module. We can show that Galois cohomology classify such $V$’s. So, above classification problem becomes a problem of calculating galois cohomology. I will introduce basic theory of Galois cohomology and Galois descent.

Hyperreal numbers are the very fundamental concept in non-standard analysis. Although its existance is first guessed via the compactness theorem in formal logic, one can construct hyperreal number system $*\mathbb{R}$ over $\mathbb{R}$ using concepts of ultrafilters. In this talk we'll follow this construction in concrete terms, and see concepts of halos, galaxies and shadows which will help to imagine the structure of $*\mathbb{R}$. Finally, some set-theoretical comments are added without details.

Isoperimetric inequality is one of the most oldest problem in geometric analysis. In Euclidean space, the isoperimetric inequality has proved in various ways. However, when we generalize Euclidean space by simply connected Riemannian manifold, whose sectional curvatures are non-positive, it is known only for dimension 2,3,4. In this talk, we introduce an approach to prove isoperimetric inequality and a sketch of Kleiner’s proof for dimension 3.

Abstract Here

부드러운 3차 곡면에 들어있는 직선들을 세어본다. 이를 위해 degree, linear system, blow-up과 같은 사영기하학의 기초적인 개념들을 소개하고 surface의 intersection theory를 통해 특별한 3차 곡면의 직선들을 센다.

다음으로 모든 부드러운 3차 곡면이 앞선 예의 방법으로 구성됨을 보이며, 마지막으로 blow-up에 등장하는 contractible curve가 3차원 이상의 birational geometry에서 어떤 의미를 갖는지 간단히 설명한다.

I will first introduce the notion of geometric structures on a given manifold and their deformation spaces in general. Turns out that they are closely related to representation theory and I will briefly explore the connection between them. This correspondence gives local information of the deformation space, in particular under quite broad assumptions, it allows us to compute the dimension of the deformation space.

Secondly, as a particular example, I will talk about Teichmuller space which is the deformation space of (finite volume) hyperbolic structures on a surface.

Finally, I will briefly investigate deformations of hyperbolic 3-manifold and celebrating rigidity result by Mostow.

The statement of Ratner's theorem is explained, in rather topological sense, with few examples and nonexamples of how orbit of an 1-parameter subgroup can be in a homogeneous space. Suggesting an analogy measure theory, we prove the Ratner's theorem on unipotent flows, under a special condition that our unipotent flows generate a group isomorphic to $SL(2,\mathbb{R})$. Then we further investigate what was the original motivation of the theorem introduced, via Oppenheimer's conjecture on irrational quadratic forms.