The class of stabilized sequential quadratic programming (SQP) methods for nonlinearly constrained optimization solve a sequence of related quadratic programming (QP) subproblems formed from a two-norm penalized quadratic model of the Lagrangian function subject to shifted, linearized constraints. While these methods have been shown to exhibit superlinear local convergence even when the constraint Jacobian is rank deficient at the solution, they generally have no global convergence theory. To address this, primal-dual SQP methods (pdSQP) employ a certain primal-dual augmented Lagrangian merit function and solve a subproblem that consists of bound-constrained minimization of a quadratic model of the merit function. The model of the merit function is constructed in a particular way so that the resulting primal-dual subproblem is equivalent to the stabilized SQP subproblem. Together with a flexible line-search, the use of the merit function guarantees convergence from any starting point, while the connection with the stabilized subproblem allows pdSQP to retain the superlinear local convergence that is characteristic of stabilized SQP methods. 

A new dynamic convexification framework is developed that is applicable for nonconvex general standard form, stabilized, and primal-dual bound-constrained QP subproblems. Dynamic convexification involves three distinct stages: pre-convexification, concurrent convexification and post-convexification. New techniques are derived and analyzed for the implicit modification of symmetric indefinite factorizations and for the imposition of temporary artificial constraints, both of which are suitable for pre-convexification. Concurrent convexification works synchronously with the active-set method solving the subproblem, and computes minimal modifications needed to ensure the QP iterates are uniformly bounded. Finally, post-convexification defines an implicit modification that ensures the solution of the subproblem yields a descent direction for the merit function.

A new exact second-derivative primal-dual SQP method (dcpdSQP) is formulated for large-scale nonconvex optimization. Convergence analysis is presented that demonstrates guaranteed global convergence. Extensive numerical testing indicates that the performance of the proposed method is comparable or better than conventional full convexification while significantly reducing the number of factorizations required.

In harmonic analysis, the Muckenhoupt $A_p$ condition characterizes weighted spaces on which classical operators are bounded. An analogue $B_p$ condition for the Bergman projection on the unit ball was given by Bekolle and Bonami. As the development of the dyadic harmonic analysis techniques, people have made progress on weighted norm estimates of the Bergman projection for various settings. In this talk, I will discuss some of these results and outline the main ideas behind the proof. I will also mention the application of these results in analyzing the $L^p$ boundedness of the projection. This talk is based on joint work with Nathan Wagner and Brett Wick.
 

We consider a generalization of the maximum degree in random graphs. Given a rooted tree $T$, let $X_v$ denote the number of copies of T rooted at v in the binomial random graph $G_{n,p}$. We ask the question where the maximum $M_n = max \{X_1, ..., X_n\}$ of $X_v$ over all vertices is concentrated. For edge-probabilities $p=p(n)$ tending to zero not too fast, the maximum is asymptotically attained by the vertex of maximum degree. However, for smaller edge probabilities $p=p(n)$, the behavior is more complicated: our large deviation type optimization arguments reveal that the behavior of $M_n$ changes as we vary $p=p(n)$, due to different mechanisms that can make the maximum large.

Based on joint work with Pedro Araújo, Simon Griffiths and Matas Šileikis; see arXiv:2310.11661 

I will present the Poincare polynomial, the Chow ring, and some tautological relations on the moduli space of elliptic K3 surfaces.

In this talk we discuss a problem in Combinatorial Probability, that concerns some finer details of the so-called 'giant component' phase transition in random graphs. More precisely, it is well-known that the size $L_1(G_{n,p})$ of the largest component of the binomial random graph $G_{n,p}$ has a scaling limit for $p=c/n$, i.e., that $L_1(G_{n,p})/n$ converges in probability to some limiting function $\rho(c)$. It is of interest to understand finer details of this limiting function, in particular if $\rho(c)$ is well-behaved for some range of $c$, say analytic. Analyticity can be shown directly for the binomial random graph $G_{n,p}$, since explicit descriptions and formulas for $\rho(c)$ are available. In this talk I will outline a somewhat more robust approach, that also works in models where explicit formulas are not available. Our approach combines tools from random graph theory (multi-round exposure arguments), stochastic processes (differential equation approximation), generating functions, and partial differential equations (Cauchy-Kovalevskaya Theorem).

The goal of the talk is to explain some geometric results on quasi-projective manifolds from the perspective of Carathéodory metrics and distances.  We will study some conjectures of Lang on manifolds which satisfied some Carath\'eodory conditions. The results are also used to study hyperbolicity of suitable compactifications of the non-compact manifolds involved.  As applications, we also prove some statements to the effect of non-existence of level structures on manifolds such as abelian varieties over function fields, as well as the  so-called volume estimates for mapping of curves into the manifolds involved. Most of the results to be presented are joint work with Kwok-Kin Wong.

In this talk we will introduce the level set flow. Then we will talk about the relation between the rate of curvature blow-up near a singularity of the flow and the distribution of surrounding singular points.

In theory, Chabauty-Coleman provides an explicit method to obtain rational points on any curve, so long as its genus exceeds its Mordell-Weil rank. In practice, when applied to modular curves, we often encounter difficulties in finding a suitable plane model, which only worsens as the genus increases. In this talk we describe how to skip this step and instead work directly with the coarse moduli space. This is joint work with Steve Huang and Jun Bo Lau.

The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First every polynomial with odd degree and real coefficients has a real root. Second every nonnegative real number has a square root. It is proved in characteristic zero that the assumption about odd-degree polynomials is stronger than necessary any field of characteristic zero in which polynomials of prime degree have roots is algebraically closed. In this talk we show that this result is the case for all fields regardless of their characteristics.

The bidomain model is the standard model describing electrical activity of the heart. We discuss the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen‐Cahn equation) in two spatial dimensions. In the bidomain Allen‐Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen‐Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen‐Cahn equation in striking contrast to the classical or anisotropic Allen‐Cahn equations. We identify two types of instabilities, one with respect to long‐wavelength perturbations, the other with respect to medium‐wavelength perturbations. Interestingly, whether the front is stable or unstable under long‐wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate‐wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate‐wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions. Time permitting, I will also discuss properties of the bidomain FitzHugh Nagumo equations. This is joint work with Hiroshi Matano, Mitsunori Nara and Koya Sakakibara.

Hassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when we reduce weights. I will discuss some joint work with Ziquan Zhuang which constructs similar morphisms between moduli of stable pairs in higher dimension.