Abstract. Bott proved the following vanishing theorem for complex projective space in his seminal paper (Annals of Mathematics, 1957) on homogeneous vector bundle: $ H^*({\mathbb P}^n, \Omega ^q_{\mathbb P^n}\otimes \mathcal{O}(k))=0, ~ 1\leq k \leq q.$ We say that a smooth projective variety $X$ satisfies the Bott vanishing property if the twisted holomorphic forms or Dolbeault cohomology H$^{p,q}(X, \mathcal{L})=$H$^q(X, \Omega^p(\mathcal{L}))=0$ for all ample line bundle $L, ~ \forall q >0 ~ \forall p >0$. Where, $\Omega ^ p $ stands for $\Lambda ^p T^*_X$ and $\Omega^p(\mathcal{L}) = \Omega ^p\otimes \mathcal{L}$. The flag varieties are smooth projective varieties. We show that the flag variety fails to hold Bott vanishing property for certain ample line bundles, and also give numerical conditions on ample line bundles to satisfy Bott vanishing property.

Abstract. We will prove $\pi$ is the minimum value of $\pi_p$ following the results in Alder and Tanton's paper MR1766159. Given there is enough time we can also discuss the area of an $l_p$ circle and cite a result from Xianfu Wang (2005).

Abstract. We consider the numerical range the product of a composition operator $C_{\phi}$ with the adjoint of a composition operator $C_{\psi}^*$, both acting on the Hardy space $H^2.$ For certain choices of $\phi$ and $\psi$ with small $L^{\infty}$ norm we show that the numerical range is a non-tangential approach region on the disc and explicitly compute the aperture at the boundary point of this region. We relate this problem to the problem of finding the spectrum of a certain class of Jacobi matrices.

Abstract. I shall talk about the notion of complex symmetric operators and their basic properties, based on Stephan Garcia's note ``What is a Complex Symmetric Operator?'' in the January 2017 issue of the Notices of the AMS. Recent results on complex symmetric composition, Toeplitz and truncated Toeplitz operators will be then discussed.

Abstract. Bergman projections and Bergman kernels are among the central objects in complex analysis. In this talk we will discuss the $L^p$ regularity of weighted Bergman projections on various domains in $\mathbb{C}^n$. Then we will show an $L^p$ irregularity of weighted Bergman projections on complete Reinhardt domains with exponentially decaying weights (joint work with Yunus Zeytuncu). Finally we establish estimates of the $L^p$ norms of Bergman projections on strongly pseudoconvex domains. The talk should be accessible to graduate students.

Abstract. I shall talk about the notion of complex symmetric operators and their basic properties, based on Stephan Garcia's note ``What is a Complex Symmetric Operator?'' in the January 2017 issue of the Notices of the AMS. Recent results on complex symmetric composition, Toeplitz and truncated Toeplitz operators will be then discussed.

Abstract. First, I will give an introduction to the geometry of bounded convex Reinhardt domain $\Omega\subset \mathbb{C}^2$ with both 'horizontal' and 'vertical' nontrivial analytic disks in the boundary. I will discuss how the existence of analytic disks in the boundary is an impediment to compactness of the Hankel operator. Then, assuming the boundary of $\Omega$ contains certain classes of analytic disks, I will show that the Hankel operator with conjugate holomorphic symbols is compact on the Bergman space of $\Omega$ if and only if the symbol is identically constant.

Abstract. In this talk, I will prove, for any $L^{\infty}(\partial\mathbb{D})$-symbol, the positivity of Berezin transform of the symbol on the disk implies the positivity of truncated Toeplitz operator on the $2$-dimensional model space. Furthermore I will show a counterexample that proves positivity of the operator does not always imply the positivity of the symbol.

Abstract. I will give some background, and then a sufficient condition for Toeplitz operators with Carleson measure symbols to be "almost" invertible on the weighted Bergman space of the disc, and a sufficient condition for a positive Borel measure to be "almost" Reverse Carleson on the Bargmann-Fock space. This type of condition shows up sometimes when trying to construct forward or reverse Carleson inequalities on function spaces.

Abstract. I will give some background, and then a sufficient condition for Toeplitz operators with Carleson measure symbols to be "almost" invertible on the weighted Bergman space of the disc, and a sufficient condition for a positive Borel measure to be "almost" Reverse Carleson on the Bargmann-Fock space. This type of condition shows up sometimes when trying to construct forward or reverse Carleson inequalities on function spaces.

Abstract. I will give some background, and then a sufficient condition for Toeplitz operators with Carleson measure symbols to be "almost" invertible on the weighted Bergman space of the disc, and a sufficient condition for a positive Borel measure to be "almost" Reverse Carleson on the Bargmann-Fock space. This type of condition shows up sometimes when trying to construct forward or reverse Carleson inequalities on function spaces.

Abstract. Let $T_f$ and $T_g$ be two Toeplitz operators on the Bergman space $A^2(\mathbb{D})$. In general, the product $T_f T_g$ is not a Toeplitz operator. A question one may ask is: can a finite rank perturbation of $T_f T_g$ be a Toeplitz operator? We offer solutions for a certain class of quasi-homogeneous symbols.

Abstract. Let $T_f$ and $T_g$ be two Toeplitz operators on the Bergman space $A^2(\mathbb{D})$. In general, the product $T_f T_g$ is not a Toeplitz operator. A question one may ask is: can a finite rank perturbation of $T_f T_g$ be a Toeplitz operator? We offer solutions for a certain class of quasi-homogeneous symbols.

Abstract. I will talk about the joint work with S. Şahutoğlu which provides an approximation result for a large class of domains in $\mathbb{C}^n$. The talk will be aimed at grad students and no prior knowledge of the subject will be assumed.

Abstract. Let $\phi$ be a linear fractional map of the unit disk $\mathbb{D}$ into itself. A fair amount of energy has been expended in examining the unital $C^{*}$-algebra generated by the composition operator $C_{\phi}$ and either the compacts or the Toeplitz operator $T_z$. The addition allows one to quotient out by the compacts in either case to obtain reasonable isomorphism results, detailed in the works of Kriete, Macluer, and Moorhouse and Quertermous. We attempt to describe the unital $C^{*}$-algebra $A$ generated by $C_{\phi}$, beginning by asking when this algebra actually contains nonzero compact operators. This is work in progress with John Clifford and Yunus Zeytuncu at the University of Michigan-Dearborn.

Abstract. I will talk about the joint work with S. Şahutoğlu which provides an approximation result for a large class of domains in $\mathbb{C}^n$. The talk will be aimed at grad students and no prior knowledge of the subject will be assumed.

Abstract. We will study how to build an integral kernel to make a solution operator for the $\bar{\partial}$-equation for $(0,1)$-forms. We shall see how one can use the Koszul complex to develop good solutions to the Cauchy-Fantappi equation. This involves to make good choices of the smooth solutions C-F equations. Further we shall see how one can handle the transition from line type to curve type.

Abstract. We will introduce the famous Diederich-Fornæss worm domains and talk about their properties.

Abstract. We will introduce the famous Diederich-Fornæss worm domains and talk about their properties.

Abstract. Let $\Omega$ be a bounded convex Reinhardt domain in $\mathbb{C}^2$ and $\phi\in C(\overline{\Omega})$. We show that the Hankel operator $H_{\phi}$ is compact if and only if $\phi$ is holomorphic along every non-trivial analytic disc in the boundary of $\Omega$. This is joint work with Sönmez Şahutoğlu.

Abstract. Let $\Omega$ be a bounded convex Reinhardt domain in $\mathbb{C}^2$ and $\phi\in C(\overline{\Omega})$. We show that the Hankel operator $H_{\phi}$ is compact if and only if $\phi$ is holomorphic along every non-trivial analytic disc in the boundary of $\Omega$. This is joint work with Sönmez Şahutoğlu.

Abstract. Let $\Omega$ be a bounded convex Reinhardt domain in $\mathbb{C}^2$ and $\phi\in C(\overline{\Omega})$. We show that the Hankel operator $H_{\phi}$ is compact if and only if $\phi$ is holomorphic along every non-trivial analytic disc in the boundary of $\Omega$. This is joint work with Sönmez Şahutoğlu.

Abstract. Let $\Omega$ be a bounded convex Reinhardt domain in $\mathbb{C}^2$ and $\phi\in C(\overline{\Omega})$. We show that the Hankel operator $H_{\phi}$ is compact if and only if $\phi$ is holomorphic along every non-trivial analytic disc in the boundary of $\Omega$. This is joint work with Sönmez Şahutoğlu.