09/05/2024 – Prof. Daren Wang: Nonparametric Density Estimation via Variance-Reduced Sketching
Abstract: Nonparametric models are of great interest in various scientific and engineering disciplines. Classical kernel methods, while numerically robust and statistically sound in low-dimensional settings, become inadequate in higher-dimensional settings due to the curse of dimensionality.
In this talk, we introduce a new framework called Variance-Reduced Sketching (VRS), specifically designed to estimate density functions in higher dimensions with a reduced curse of dimensionality. Our framework conceptualizes multivariable functions as infinite-size matrices/tensors, facilitating a new matrix/tensor-based bias-variance tradeoff in various nonparametric contexts.
We demonstrate the robust numerical performance of VRS through a series of simulated experiments and real-world data applications. Notably, VRS shows remarkable improvement over existing neural network estimators and classical kernel methods in numerous density models. Additionally, we will discuss theoretical guarantees for VRS to support its ability to deliver density estimation with a reduced curse of dimensionality.
09/12/2024 – Chase Bender: Julia Set on the Projective Line
Abstract: The field of complex dynamics was born from the work of Gaston Julia and Pierre Fatou centered around studying the limiting behavior of the sequence of iterates of a rational function of one complex variable. In this talk I will give a rough outline of the classical theory of Julia sets on one variable, and then transition to a more modern perspective which yields sharper results and is more applicable to the several variable setting.
09/19/2024 – Yuyan He: Pseudo-finite Dimension and Simple Algebraic Groups
Abstract: The talk is based on “On Pseudo-Finite Dimensions” by Hrushovski and “Approximate Subgroups of Linear Groups” by Breuillard, Green and Tao. For an ultraproduct of first order structures, given a convex subset of the nonstandard real numbers, a pseudo-finite dimension can be defined for each of the pseudo-finite subset of the ultraproduct. In the first part of the talk, I will discuss the definition and properties of pseudo-finite dimensions. The second part of the talk will be focused on application of this concept to the study of approximate subgroups of simple algebraic groups.
09/26/2024 – Cory Gillette: Introduction to Operads
Abstract: We will attempt to explain what operads are, where they came from, and more importantly, what they are used for. Roughly, an operad can be thought of as a “gadget” that parameterizes a class of algebraic structures, such as associative unital algebras or Lie algebras..
10/03/2024 – Luis Atzin: Classifying non-negatively curved 4-manifolds
Abstract: Manifolds with positive or nonnegative sectional curvature are important and interesting mathematical objects. Some fundamental results, like Gromov’s Betti number Theorem, Bonnet–Myers or Synge’s Theorem have helped understand the topology of these manifolds. However, we are far from being able to classify all of them. A classification of these manifolds in lower dimensions has been proven successful in the case when they have a large group of isometries. In this talk we will give a sketch of the proof of the classification of all closed, simply-connected, nonnegatively curved 4-manifolds with circle symmetry. This will lead us to a crash course in Alexandrov geometry, a pivotal tool essential for the proof. We will also see that the classification eventually boils down to showing that some curve in the 3-sphere is the unknotted.
10/10/2024 – Chen-Kuan Lee: Hartogs’s extension theorem
Abstract: In this presentation, we will show the difference between one and several complex variables by introducing the Hartogs’s extension theorem. In one variable, the isolated singularities are classified into removable, pole and essential; however, Hartogs said that in several complex variables, every isolated singularity is removable.
10/17/2024 and 10/31/2024 – Atticus Stonestrom: The Peter–Weyl Theorem
Abstract: I will prove the Peter-Weyl theorem on compact groups, and, if time permits, I will discuss some applications of it in model theory and in additive combinatorics.