학과 세미나 및 콜로퀴엄
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).
Serrin’s overdetermined problem is a famous result in mathematics that deals with the uniqueness and symmetry of solutions to certain boundary value problems. It is called "overdetermined" because it has more boundary conditions than usually required to determine a solution, which leads to strong restrictions on the shape of the domain. In this talk, we discuss overdetermined boundary value problems in a Riemannian manifold and discuss a Serrin-type symmetry result to the solution to an overdetermined Steklov eigenvalue problem on a domain in a Riemannian manifold with nonnegative Ricci curvature and it will be discussed about an overdetermined problems with a nonconstant Neumann boundary condition in a warped product manifold.
De novo mutations provide a powerful source of information for identifying risk genes associated with phenotypes under selection, such as autism spectrum disorder (ASD), obsessive-compulsive disorder (OCD), congenital heart disease, and schizophrenia (SCZ). However, identifying de novo variants is costly, as it requires trio-based sequencing to obtain parental genotypes. To address this limitation, we propose a method to infer inheritance class using only offspring genetic data. In our new integrated model, we evaluate variation in case and control samples, attempt to distinguish de novo mutations from inherited variation, and incorporate this information into a gene-based association framework. We validate our method through ASD gene identification, demonstrating that it provides a robust and powerful approach for identifying risk genes.
A knot bounds an oriented compact connected surface in the 3-sphere, and consequently in the 4-ball. The 4-genus of a knot is the minimal genus among all such surfaces in the 4-ball, and the 4-genus of a link is defined analogously. In this talk, I will discuss lower bounds on the 4-genus derived from Cheeger-Gromov-von Neumann rho-invariants. This is joint work with Jae Choon Cha and Min Hoon Kim.
Abstract: In this talk, we discuss the global-in-time existence of strong solutions to the one-dimensional compressible Navier-Stokes system. Classical results establish only local-in-time existence under the assumption that the initial data are smooth and the initial density remains uniformly positive. These results can be extended to global-in-time existence using the relative entropy and Bresch-Desjardins entropy under the same hypotheses. This approach allows for possibly different end states and degenerate viscosity.
Reference: A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J. Math. Anal., 39(4):1344–1365, 2007/08.
Abstract :
When a plane shock hits a wedge head on, it experiences a reflection diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. In particular, the C^{1,1}-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock where the wedge has large-angle. Also, one can obtain the C^{2,\alpha} regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region.
Reference :
Myoungjean Bae, Gui-Qiang Chen, and Mikhail Feldman. "Regularity of solutions to regular shock reflection for potential flow." (2008)
Gui-Qiang Chen and Mikhail Feldman. "Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow"