Experimental designs for a generalized linear model (GLM) often depend on the specification of the model, including the link function, the predictors, and unknown parameters, such as the regression coefficients. To deal with the uncertainties of these model specifications, it is important to construct optimal designs with high efficiency under such uncertainties. Existing methods such as Bayesian experimental designs often use prior distributions of model specifications to incorporate model uncertainties into the design criterion. Alternatively, one can obtain the design by optimizing the worst-case design efficiency with respect to the uncertainties of model specifications. In this work, we propose a new Maximin Φp- Efficient (or Mm-Φp for short) design which aims at maximizing the minimum Φp-efficiency under model uncertainties. Based on the theoretical properties of the proposed criterion, we develop an efficient algorithm with sound convergence properties to construct the Mm-Φp design. The performance of the proposed Mm-Φp design is assessed through several numerical examples.

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A rational map on a (compact) complex curve is always holomorphic. In two dimensions or more, it is likely that the map has 'indeterminate points' where the map is not continuous; furthermore, subvarieties may get contracted to those of lower dimension. Nevertheless, it still makes sense to iterate rational maps, even though the behaviour of orbits near such singularities is more difficult to understand. In this talk I will begin by providing a perspective for the analysis and dynamics of curves on surfaces. In the second half, I will expand on a fruitful transformation of the two-dimensional complex rational map into a piecewise-linear map on a non-Archimedean (Berkovich) space. For the applications I will discuss, the latter dynamics mirror those of a one-variable rational map, and return information about the original surface mapping.

High-dimensional data have become prevalent in all fields that need statistical modeling and data analysis. I introduce my recent research in Bayesian ultrahigh dimensional variable selection, low-rank matrix regression and classification, and robust sufficient dimension reduction (SDR). We develop a Bayesian framework for mixed-type multivariate regression with continuous shrinkage priors that enables joint analysis of mixed continuous and discrete outcomes, allowing variable selection from a large number of covariates (p). We investigate the conditions for posterior contraction, especially when the number of covariates (p) grows exponentially relative to the sample size (n) and develop a two-step approach for variable selection with theorems of a sure screening property and posterior contraction and applications with simulation studies and applications to real datasets. To address challenges in analyzing regression coefficient estimation affected by high-dimensional matrix-valued covariates, we propose a framework for matrix-covariate regression and classification models with a low-rank constraint and additional regularization for structured signals, considering continuous and binary responses, introduce an efficient Riemannian-steepest-descent algorithm for regression coefficient estimation, and prove the consistency of the proposed estimator, showing improvement over existing work in cases where the rank is small with applications through simulations and real datasets of shape images, brain signals, and microscopic leucorrhea images. We propose a novel SDR method robust against outliers using α-distance covariance that effectively estimates the central subspace under mild conditions on predictors without estimating a link function, based on the projection on the Stiefel manifold. We establish convergence properties of the proposed estimation under certain regularity conditions and compare the method's performance with existing SDR methods through simulations and real data analysis, highlighting improved computational efficiency and effectiveness.

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I will discuss deformation theory of complex manifolds endowed with a meromorphic symplectic form, which has only simple poles. Such deformation theory is encoded by certain graphs with decorations, called smoothing diagrams, which were introduced in a joint work with Pym and Schedler. I will discuss how combinatorics of these diagrams predicts the geometric properties of the deformed log symplectic Poisson brackets. In particular, I will present a novel method for constructing elliptic Poisson brackets.

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, I will present some new numerical evidence that the 3D Navier-Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of $10^7$. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations. Unlike the Hou-Luo blowup scenario, the potential singularity of the 3D Euler and Navier-Stokes equations occurs at the origin. We have applied several blowup criteria to study the potentially singular behavior of the Navier-Stokes equations. We also study a variant of the axisymmetric Navier--Stokes equations with two different viscosity coefficients with smooth initial data. We show that the solution develops a stable nearly self-similar blowup solution with maximum vorticity increased by a factor of $10^{32}$.

We investigate the components determining bigness of the cotangent bundle $\Omega^1_X$ of smooth models $X$ in the birational class $\mathcal {Y}$ of an orbifold surface of general type $Y$, with a focus on the contribution given by the singularities of $Y$. A criterion for bigness of $\Omega_X^1$ is given involving only topological and singularity data on $Y$. We single out a special case, the Canonical Model Singularities (CMS) criterion, when $Y$ is the canonical model of $\mathcal Y$. We study the singularity invariants appearing in the criterion and determine them for $A_n$ singularities. Knowledge of these invariants for $A_n$ singularities allows one to evaluate the $(c_2,c^2_1)-$geographical range of the CMS criterion and compare it to other criteria. We obtain new examples of surfaces with big cotangent bundle. (Joint work with Y. Asega and M.Weiss)

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The study of magnetic monopoles in gauge theory and the connection with algebraic geometry celebrates its 50th anniversary in 2024, and in 1982 Hitchin provided the criteria for certain algebraic curves to correspond to monopole solutions. These constraints are hard to solve, being transcendental in nature, and in the ensuing 40 years only a limited number of spectral curves have been discovered, typically requiring large symmetry groups. I will describe how a combination of the representation theory of Nahm's equations, computer algebra software, and Riemann surface theory allows us to construct the first new explicit monopole spectral curves for 25 years.

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