The classical Moser-Trudinger inequality is a borderline case of Sobolev inequalities and plays an important role in geometric analysis and PDEs in general. Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space H1, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without mass center constraints. Efforts have also been made to show similar inequalities in higher dimensions. We have improved Beckner’s inequality, the higher dimensional counterpart of Onofri’s inequality, for axially symmetric functions when the dimension n = 4, 6, 8. Numerical computations are exploited to provide rigorous proof. I will also present some new results on higher dimensional counterpart of Huber’s isoperimetric inequalities.

A central goal of scientific computing is to develop accurate and efficient solvers for scientific problems, and this goal is often pursued through sophisticated numerical methods. In modern machine learning, by contrast, the basic optimization procedure is often comparatively simple, typically gradient descent and its variants, while much of the complexity is shifted to larger models. This first talk introduces the main motivation of the talk series: to examine advanced iterative methods in scientific computing from a viewpoint inspired by this contrast. We begin with basic examples and preliminary concepts, including classical iterative methods, convergence of stationary iterations, and elementary schemes such as Jacobi, Gauss--Seidel, and Richardson iterations. These examples provide the foundation for the auxiliary-space perspective developed in the subsequent talks.

This second talk develops the theoretical framework behind the interpretation of advanced iterative methods as elementary iterations on larger spaces. The main tool is an auxiliary-space framework, which recasts an iterative method for the original system as an equivalent, but more elementary, method for a suitably enlarged auxiliary system. Within this framework, many methods that appear sophisticated on the original problem can be understood as simple iterations applied to auxiliary variables. In particular, multigrid and domain decomposition methods can be viewed as Jacobi- or Gauss--Seidel-type iterations for appropriate expanded systems. This provides a rigorous way to connect advanced solvers with elementary iterative principles and clarifies the algebraic structure underlying these methods.

This final talk illustrates the scope and utility of the auxiliary-space viewpoint through several important classes of numerical methods. The framework applies to a broad range of advanced iterative methods, including subspace correction methods, Hiptmair--Xu preconditioners, saddle point solvers, and iterative substructuring methods. Through these applications, we show how apparently different methods can be understood within a common structure: each may be interpreted as an elementary iteration on a suitably enlarged space. This perspective clarifies relationships among existing algorithms and suggests new ways to design efficient solvers. We conclude by discussing how this viewpoint may inform the development of numerical methods for problems arising in machine learning, where complexity is often shifted from the optimization procedure to the underlying representation.

(This is a reading seminar presented by two graduate students.) In this reading seminar, we will present an introduction to étale cohomology and the Weil conjectures based on Milne's lecture notes. Beginning with the étale topology and the theory of sheaves on étale sites, we will develop the basic constructions and properties of étale cohomology. We will then explain several fundamental results of the theory including the purity theorem, the base change theorems, and the comparison theorem. Finally, we will discuss how these results culminate in the proof of the Weil conjectures.

A celebrated theorem of Mark Green from the 1980s describes the syzygies of a curve embedded by a line bundle of sufficiently large degree: if $L$ is a line bundle on a smooth projective curve $C$ of genus $g$ with $\deg(L)\geq 2g+1+p$, then $L$ satisfies property $N_p$; equivalently, the minimal free resolution of its homogeneous ideal is linear for the first $p$ steps. Since then, many efforts have been made to find analogues of Green’s theorem in broader settings. Two major directions are the study of linear syzygies of adjoint line bundles on higher-dimensional varieties, as predicted by syzygetic Fujita conjectures, and the study of higher-weight syzygies, encoded by properties $N_{d,p}$. In joint work with Wenbo Niu, we address both directions by studying the syzygies of tautological line bundle on symmetric products of curves. Let $C$ be a smooth projective curve of genus $g$, let $L$ be a line bundle on $C$, and let $T_{k+1,L}$ be the tautological line bundle on the symmetric product $C_{k+1}$, obtained by descent from $L^{\boxtimes (k+1)}$ on $C^{k+1}$. We prove that for each integer $d$ with $0\leq d\leq k$, if $\deg(L)\geq dg+2g+1+p$, then $T_{k+1,L}$ satisfies property $N_{k+2-d,p}$. This yields a family of sharp results on higher syzygies of higher weight for symmetric products of curves in arbitrary dimension, and in particular recovers Green’s classical theorem for curves. We also prove a sharp numerical criterion for the nefness of certain divisors on $C_{k+1}$. Taken together, these results provide strong evidence for syzygetic Fujita conjecture on property $N_p$ adjoint linear series and may be viewed as a natural higher-dimensional analogue of Green’s theorem.

(This is a reading seminar presented by two graduate students.) In this reading seminar, we will present an introduction to étale cohomology and the Weil conjectures based on Milne's lecture notes. Beginning with the étale topology and the theory of sheaves on étale sites, we will develop the basic constructions and properties of étale cohomology. We will then explain several fundamental results of the theory including the purity theorem, the base change theorems, and the comparison theorem. Finally, we will discuss how these results culminate in the proof of the Weil conjectures.

(This is a reading seminar presented by two graduate students.) In this reading seminar, we will present an introduction to étale cohomology and the Weil conjectures based on Milne's lecture notes. Beginning with the étale topology and the theory of sheaves on étale sites, we will develop the basic constructions and properties of étale cohomology. We will then explain several fundamental results of the theory including the purity theorem, the base change theorems, and the comparison theorem. Finally, we will discuss how these results culminate in the proof of the Weil conjectures.

(This is a reading seminar presented by two graduate students.) In this reading seminar, we will present an introduction to étale cohomology and the Weil conjectures based on Milne's lecture notes. Beginning with the étale topology and the theory of sheaves on étale sites, we will develop the basic constructions and properties of étale cohomology. We will then explain several fundamental results of the theory including the purity theorem, the base change theorems, and the comparison theorem. Finally, we will discuss how these results culminate in the proof of the Weil conjectures.

Mirror Symmetry predicts that each Calabi-Yau variety X admits a mirror-dual Calabi-Yau variety Y, so that the symplectic Gromov-Witten (GW) invariants of X are computed from period integrals on Y. Mirror Symmetry inspired calculations of GW invariants have been achieved in several large families of cases yielding lots of data, yet we are still lacking a general approach. Comes in Intrinsic Mirror Symmetry (Gross-Siebert, 2022), which provides a general construction of Y from X. I will talk about an ongoing joint project with Siebert, where we show that the generating function of GW invariants of X equals the expansion of a canonical period integral on the Intrinsic Mirror Y of X. The project heavily relies on AI-assisted proofs, using the existing databasis of mirror-symmetric GW calculations.