We will start this talk by discussing general results about the fundamental group of the link of a singularity. We will continue by studying the singularities of the Minimal Model Program. We will start by discussing the state of the art for log terminal singularities. Lastly, we will study restrictions for fundamental groups of log canonical singularities in low dimensions, and construct log canonical singularities with certain prescribed fundamental groups. This is based on joint work with Joaquín Moraga

We prove the existence of a class of orbitally stable bound state solutions to nonlinear Schrodinger equations with super-quadratic confinement in two and three spatial dimensions. These solutions are given by time-dependent rotations of a non-radially symmetric spatial profile which in itself is obtained via a doubly constrained energy minimization. One of the two constraints imposed is the total mass, while the other is given by the expectation value of the angular momentum around the z-axis. Our approach also allows for a new description of the set of minimizers subject to only a single mass constraint.

Groups acting linearly on projective spaces are examples of holomorphic dynamical systems which exhibit a variety of different behaviours. We introduce the notion of proximal stability which measures a form of dynamical stability for the action of a holomorphic family of group representations and we will explain how this property can be detected using a bifurcation current on the parameter space of the family. This bifurcation current measures the pluriharmonicity of the top Lyapunov exponent of the family of representations, defined using a random walk on the group.

Given an elliptic curve over Q and a rational point P on E, Rafe Jones and Jeremy Rouse (and others) have considered the problem of determining the density of primes p such that the order of P modulo p is odd. The main tool for studying such a question is the arboreal Galois representation. We’ll discuss Galois representations of elliptic curves generally and how they relate to solving the odd order reduction problem, as well as an extension for abelian surfaces.

A common way to picture a singular point of an algebraic variety is by analyzing its link, the intersection of a small embedded sphere with the variety. Heuristically, more complex links correspond to worse singularities. Over the past few years, certain numerical invariants have been used to (roughly) give an upper bound on the size of the fundamental groups of such links for some important classes of singularities in a number of different settings. In positive characteristic, one can use the F-signature -- an invariant giving a quantitative measure of F-regularity and closely related to Kawamata Log Terminal (KLT) singularities in characteristic zero. In this talk, I aim to motivate and discuss a new mixed characteristic analogue of the F-signature defined using the perfectoidization functor of Bhatt-Scholze. As an application, we are able to bound the size of the étale fundamental group of the regular locus of BCM-regular singularities. This is joint work with Hanlin Cai, Seungsu Lee, Linquan Ma and Karl Schwede.

Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view, singular perturbations generate thin layers near the boundary of a domain, called boundary layers, where many important physical phenomena occur. In this talk, we discuss some recent results on the viscous boundary layer analysis and their applications in implementing effective numerical schemes including the Physics Informed Neural Networks (PINNs).

The tree property is an uncountable analogue of Konig's infinity lemma. At a cardinal $\kappa$, it states that every tree of height $\kappa$ and levels of size less than $\kappa$ has a cofinal branch. At $\aleph_1$ the tree property fails, and at $\aleph_2$ the tree property is equiconsistent with a weakly compact cardinal (Mitchell; Silver, 1972). Going further, in 1983 Abraham showed the tree property can hold simultaneously at $\aleph_2$ and $\aleph_3$, using a much stronger large cardinal hypothesis -- a supercompact cardinal. Since then, it has been a long standing project in set theory to obtain the tree property simultaneously at large intervals of regular cardinals. The ultimate goal being to force the tree property at every regular cardinal greater than $\aleph_1$. By a theorem of Specker a positive answer would require many failures of SCH. We show that from large cardinals, we can force the tree property at every regular cardinal in the interval $[\aleph_2, \aleph_{\omega^2}+2]$ with $\aleph_{\omega^2}$ a strong limit. This is joint work with Cummings, Hayut, Magidor, Neeman, and Unger.

We study the problem of distribution-free dependence detection and modeling through the new framework of binary expansion statistics (BEStat). The binary expansion testing (BET) avoids the problem of non-uniform consistency and improves upon a wide class of commonly used methods (a) by achieving the minimax rate in sample size requirement for reliable power and (b) by providing clear interpretations of global relationships upon rejection of independence. The binary expansion approach also connects the symmetry statistics with the current computing system to facilitate efficient bitwise implementation. Modeling with the binary expansion linear effect (BELIEF) is motivated by the fact that wo linearly uncorrelated binary variables must be also independent. Inferences from BELIEF are easily interpretable because they describe the association of binary variables in the language of linear models, yielding convenient theoretical insight and striking parallels with the Gaussian world. With BELIEF, one may study generalized linear models (GLM) through transparent linear models, providing insight into how modeling is affected by the choice of link. We explore these phenomena and provide a host of related theoretical results. This is joint work with Benjamin Brown and Xiao-Li Meng.

Solutions of many nonlinear PDE systems reveal a multiscale character; thus, their numerical resolution presents some major difficulties. Such problems are typically characterized by a small parameter representing, say, a low Mach or Fraude number. In the limiting regimes, the propagation speeds are very low, and therefore the use of explicit numerical methods would require very restrictive time and space discretization steps due to the CFL condition and restrictions on the smallness of numerical diffusion. This becomes rapidly too costly from a practical point of view, and consequently, numerical solutions for small parameter values may be out of reach. Moreover, standard implicit schemes, which will be uniformly stable, may be inconsistent with the limiting problem and may provide a wrong solution in the zero limits. Thus, designing robust numerical algorithms whose accuracy and efficiency are independent of the values of the small parameter is an important and challenging task. A widely used numerical approach applicable in all-speed regimes is based on asymptotic preserving (AP) numerical methods. AP methods guarantee that for a fixed mesh size and time step, the numerical scheme should automatically transform into a consistent and stable discretization of the limiting system. In this talk, we will present several AP schemes for Navier–Stokes–Korteweg equation, rotational shallow water equations with Coriolis, and, if time permits, kinetic equations with singular limits.

In this talk, I will give a brief introduction to a trivariate spline collocation method for solving the Dirichlet problem of the 3D elliptic Monge-Ampère equation. Specifically, I will explain the spline collocation method introduced in [SIAM J. Numerical Analysis, 2405-2434,2022] to numerically solve iterative Poisson equations, and incorporate an averaged algorithm to ensure convergence of the iterations. Furthermore, I present various computational results, including testing known convex and non-convex solutions over convex and non-convex domains to demonstrate the efficiency and effectiveness of the method.

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Liquid crystal polymeric networks are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy. The derivation is similar to derivations in Ozenda, Sonnet, and Virga (2020) and Cirak et. al. (2014). We characterize the zero energy deformations and prove that the energy lacks certain convexity properties. We propose a finite element method to discretize the problem. To address the lack of convexity of the membrane energy, we regularize with a term that mimics a higher order bending energy. We prove that minimizers of the discrete energy converge to minimizers of the continuous energy. For minimizing the discrete problem, we employ a nonlinear gradient flow scheme, which is energy stable. Additionally, we present computations showing the geometric effects that arise from liquid crystal defects. Computations of configurations from nonisometric origami are also presented.