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Supported by the Global Basic Research Lab (LFANT) at Chonnam National University and the National Research Foundation of Korea
Speaker: Jaime Hernandez Palacios (Chonnam National University)
Title: Gaps between zeros of zeta and L-functions of high degree
Abstract: There is a great deal of evidence, both theoretical and experimental, that the distribution of zeros of zeta and L-functions can be modeled using statistics of eigenvalues of random matrices from classical compact groups. In particular, we expect that there are arbitrarily large and small normalized gaps between the ordinates of (high) zeros zeta and L-functions. Previous results are known for zeta and L-functions of degrees 1 and 2. We discuss some new results for higher degrees, including Dedekind zeta-functions associated to Galois extensions of the rationals and principal automorphic L-functions.
Speaker: Ayan Maiti (Chonnam National University)
Title: Weyl's law for arbitrary Archimedean type
Abstract: Let $M$ be a compact Riemannian manifold. It was proved by Weyl that number of Laplacian eigenvalues less than $T$, is asymptotic to $C(M)T^{dim(M )/2}$, where $C(M)$ is the product of the volume of $M$, volume of the unit ball and $(2π)^{dim(M)}$. Let $Γ$ be an Arithmetic subgroup of $SL_2(Z)$, $H^2$ be an upper-half plane. When $M = Γ\H^2$, Weyl’s asymptotic holds true for the discrete spectrum of Laplacian. It was proved by Selberg, who used his celebrated trace formula.
Let $G$ be a semisimple algebraic group of Adjoint and split type over $Q$. Let $G(R)$ be the set of $R$-points of $G$. For simplicity of this exposition let us assume that $Γ ⊂G(R)$ be an torsion free arithmetic subgroup. Let $K_{\infty}$ be the maximal compact subgroup. Let $L^2(Γ\G(R))$ be space of square integrable $Γ$ invariant functions on $G(R)$. Let $L^2_{cusp}(Γ\G(R))$ be the cuspidal subspace. Let $M = Γ\G(R)/K_{\infty}$ be a locally symmetric space. Suppose $d= dim(Γ\G/K_{\infty})$. Then it was proved by Lindenstrauss and Venkatesh, that number of spherical, i.e. bi-$K_{\infty}$ invariant cuspidal Laplacian eigenfunctions, whose eigenvalues are less than $T$ is asymptotic to $C(M)T^{dim(M )/2}$, where $C(M)$ is the same constant as above.
We are going to prove a similar Weyl’s asymptotic estimates for $K_{\infty}$-finite cusp forms for the above space. We are going to closely follow the pre-trace formula and the test functions used in Lindenstrauss and Venkatesh.
Speaker: Héctor del Castillo Gordillo (Chonnam National University)
Title: Langlands functoriality and L-functions
Abstract: We will discuss the Langlands functoriality principle and explore its relation with the L-functions.
Speaker: Wonwoong Lee (Chonnam National University)
Title: Classification of rectangular representations
Abstract: In this talk, we will discuss the complete classification of faithful rectangular representations of a complex semisimple Lie algebra. We will also discuss its application to $\lambda$-independence on the algebraic monodromy groups of compatible systems of $\lambda$-adic Galois representations of number fields. This is a joint work with Chun-Yin Hui.
Speaker: Andrei Seymour-Howell (Chonnam National University)
Title: Numerical computations of Maass cusp forms via the trace formula
Abstract: In the 1950s Selberg derived his well celebrated trace formula to prove the existence and infinitude of Maass cusp forms. Explicit examples of such forms have been rare, only occurring from examples that Maass originally constructed and those arising from Galois representations. From this, we mainly rely on numerical approximations to study specific cusp forms. In this talk I will give an overview and history of numerical computations of Maass cusp forms and furthermore, I will discuss personal progress in this area in using the trace formula for numerical computations.
Speaker: Chan-Ho Kim (Jeonbuk National University)
Title: TBA
Abstract: TBA
Speaker: Asif Zaman (University of Toronto)
Title: TBA
Abstract: TBA