학과 세미나 및 콜로퀴엄
In this talk, we will introduce vector field method for the wave equation. The key step is to establish the Klainerman-Sobolev inequality developed in [1]. Using this inequality, we will provide dispersive estimates of the linear wave equation, and prove small-data global existence for some nonlinear wave equations. The main reference will be Chapter II in [2].
참고문헌:
[1]. Sergiu Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 321–332. MR 784477
[2]. Christopher D. Sogge, Lectures on Nonlinear Wave Equations, Second Edition
The Langlands program, introduced by Robert Langlands, is a set of conjectures that attempt to build bridges between two different areas: Number Theory and Representation Theory (Automorphic forms). The program is also known as a generalization of a well-known theorem called Fermat’s Last Theorem. More precisely, when Andrew Wiles proved Fermat’s Last Theorem, he proved a special case of so-called Taniyama-Shimura-Weil Conjecture, which states that every elliptic curve is modular. And as a corollary, he was able to prove Fermat’s Last Theorem since Taniyama-Shimura-Weil Conjecture implies that certain elliptic curves associated with Fermat-type equations must be modular, leading to a contradiction. Note that the Langlands program is a generalization of the Taniyama-Shimura-Weil conjecture. In the first part of the colloquium, we briefly go over the following subjects:
(1) Fermat’s Last Theorem
(2) Taniyama-Shimura-Weil conuecture
And then, in the remaining of the talk, we start to explain a bit of the Langlands program
(3) Langlands program and L-functions
(4) (If time permits) Recent progress
This colloquium will be accessible to graduate students in other fields of mathematics (and undergraduate students who are interested in Number theory) at least in the first part.
In this note, we investigate threshold conditions for global well-posedness and finite-time blow-up of solutions to the focusing cubic nonlinear Klein–Gordon equation (NLKG) on $\bbR^{1+3}$ and the focusing cubic nonlinear Schrödinger equation (NLS) on $\bbR$. Our approach is based on the Payne–Sattinger theory, which identifies invariant sets through energy functionals and conserved quantities. For NLKG, we review the Payne–Sattinger theory to establish a sharp dichotomy between global existence and blow-up. For NLS, we apply this theory with a scaling argument to construct scale-invariant thresholds, replacing the standard mass-energy conditions with a $\dot{H}^{\frac12}$-critical functional. This unified framework provides a natural derivation of global behavior thresholds for both equations.
A surface can be decomposed into a union of pairs of pants, a construction known as a pants decomposition. This fundamental observation reveals many important properties of surfaces. By forming a simplicial graph whose vertices represent pants decompositions, connecting two vertices with an edge whenever the corresponding decompositions differ by a simple move, we obtain a graph that is quasi-isometric to the Weil–Petersson metric on Teichmüller space. Meanwhile, topologists often study a structure called a rose, formed by attaching multiple circles at a single point. A rose is homotopy equivalent to a compact surface with boundary. Consequently, we can define a pants decomposition of a rose as the pants decomposition of a surface homotopy equivalent to it. In this talk, we will explore the concept of pants decompositions specifically in the context of roses.
The singular limit problem is an important issue in various forms of ODEs and PDEs, and it is particularly known as a fundamental problem in equations derived from fluid dynamics. In this presentation, I will introduce some general phenomena of the singular limit problem through several examples. Subsequently, I will examine how the solution of the Euler-Maxwell equations converges to the MHD equations under the assumption that the speed of light approaches infinity, and how the Boussinesq equations converge to the QG equations in certain regimes.
Modular forms continue to attract attention for decades with many different application areas. To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to be able to compute a large number of Fourier coefficients. In this talk, firstly, we will show that this can be achieved in level 4 for a large range of half-integral weights by making use of one of three explicit bases, the elements of which can be calculated via fast power series operations.
After having "many" Fourier coefficients, it is time to ask the following question: Can the dis- tribution of normalised Fourier coefficients of half-integral weight level 4 Hecke eigenforms with bounded indices be approximated by a distribution? We will suggest that they follow the generalised Gaussian distribution and give some numerical evidence for that. Finally, we will see that the appar- ent symmetry around zero of the data lends strong evidence to the Bruinier- Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and absolute values are distributed independently.
This is joint work with Gabor Wiese (Luxembourg), Zeynep Demirkol Ozkaya (Van) and Elif Tercan (Bilecik).
Diophantine equations involving specific number sequences have attracted considerable attention. For instance, studying when a Tribonacci number can be expressed as the product of two Fibonacci numbers is an interesting problem. In this case, the corresponding Diophantine equation has only two nontrivial integer solutions. While finding these solutions is relatively straightforward, proving that no further solutions exist requires a rigorous argument-this is where Baker’s method plays a crucial role. After conducting a comprehensive literature review on the topic, we present our recent results on Diophantine equations involving Fibonacci, Tribonacci, Jacobsthal, and Perrin numbers. Furthermore, as an application of Baker’s method, we will briefly demonstrate how linear forms in logarithms can be effectively applied to Diophantine equations involving Fibonacci-like sequences.
This is joint work with Zeynep Demirkol Ozkaya (Van), Zekiye Pinar Cihan (Bilecik) and Meltem Senadim (Bilecik).
Abstract:
We consider the initial-boundary value problem (IBVP) for the 1D isentropic Navier-Stokes equation (NS) in the half space. Unlike the whole space problem, a boundary layer may appear due to the influence of viscosity.
In this talk, we first briefly study the asymptotic behavior for the initial value problem of NS in the whole space. Afterwards, we will present the characterization of the expected asymptotics for the IBVP of NS in the half space. Here, we focus only on the inflow problem, where the fluid velocity is positive on the boundary.
Reference:
Matsumura, Akitaka. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Methods Appl. Anal. 8 (2001), no. 4, 645–666.
We study stochastic motion of objects in micrometer-scale living systems: tracer particles in living cells, pathogens in mucus, and single cells foraging for food. We use stochastic models and state space models to track objects through time and infer properties of objects and their surroundings. For example, we can calculate the distribution of first passage times for a pathogen to cross a mucus barrier, or we can spatially resolve the fluid properties of the cytoplasm in a living cell. Recently developed computational tools, particularly in the area of Markov Chain Monte Carlo, are creating new opportunities to improve multiple object tracking. The primary remaining challenge, called the data association problem, involves mapping measurement data (e.g., positions of objects in a video) to objects through time. I will discuss new developments in the field and ongoing efforts in my lab to implement them. I will motivate these techniques with specific examples that include tracking salmonella in GI mucus, genetically expressed proteins in the cell cytoplasm, active transport of nuclei in multinucleate fungal cells, and raphid diatoms in seawater surface interfaces.