Saint Louis University Mathematics & Statistics Events

Title:  Fast food for thought: what can chicken nuggets tell us about linear algebra?

Abstract:  A simple question about chicken nuggets connects everything from analysis and combinatorics to probability theory and computer-aided design.  Linear algebra is a recurring theme: piecewise-polynomial interpolation, ``random norms'', trace polynomials, and operator algebras all make an appearance.  Students are invited to attend.  In fact, much of this work was done with students!

Canceled

Title: Weighted Weak-Type Inequalities in Harmonic Analysis

Abstract: A central problem in analysis is understanding the mapping properties of operators acting on function spaces defined by specific integrability conditions. Weighted norm estimates address these properties when the underlying measure is non-uniform, and weak-type estimates further refine the scope of these spaces. In this talk, I will explore the significance of weighted weak-type estimates and discuss recent generalizations that have attracted considerable attention in the field.

Title: Geometry in Finite Fields

Abstract:  Finite field models have frequently been used as a simple setting in which many problems can be studied while avoiding some technical issues (such as convergence).  Nonetheless, many problems in finite fields remain out of reach.  One such problem is the finite field distance problem, which has the benefit of being relatively new, easily accessible, and still completely unsolved.

We will start with defining the integers modulo n and discussing vector spaces over finite fields.  Once the preliminary background has been covered, we will discuss some notions of geometry in finite fields as well as a finite field analogue of both the Erdos distance problem in combinatorics and the Falconer distance problem in geometric measure theory.  We will give an overview of the known results in the area as well as proving some landmark results.

Frame Theory: The mathematical foundation for acoustics, quantum physics, numerics and machine learning.

In this overview talk we give a broad reflection of frame theory and the connection to various application areas, in particular acoustics. 

We will introduce the basic definition and concept of frames. We will link them to the concept of coherent states in quantum physics. We will talk about time-frequency analysis and its link to frame theory. As a particular form of quantization operators we will present frame multipliers - operators that can be represented as a weighted version of the frame decomposition. We will show how they are applied in signal processing as time-variant filters. We will introduce the representation of operators using frames, and show the link to numerical approaches like FEM/BEM and the applications to acoustical simulations. We will present sound signals of a particular application in acoustics: audio inpainting. Finally, we will hint at the connection of frames to deep neural networks.

Frame Theory Day is a midwestern regional mathematics conference dedicated to the study of frame theory and its applications.  The goal of this weekend conference is to promote awareness and collaboration within the strong network of frame researchers in the Midwest through a modest schedule of short talks that includes ample time for discussion.  This meeting also strives to promote the work of budding researchers by reserving a number of speaking slots for graduate students. For more details visit https://sites.google.com/view/frametheoryday24/home.

Title: Constructing Discrete Frames from Continuous Wavelet Transforms

Abstract: In the early 1950s, Duffin and Schaeffer introduced the concept of discrete frames for a Hilbert space $\mathcal H$. In pursuit of ``non-harmonic Fourier series", they defined discrete frames as countable sequences $\{e_j\}_{j=1}^\infty$ in $\mathcal H$ together with real constants $0<A\leq B < \infty$ such that

\[

  A\|f\|_{\mathcal H}^2 \leq \sum_{j=1}^\infty |\langle f,e_j \rangle|^2 \leq B \|f\|_\mathcal H^2,

\]

for all $f\in \mathcal H$. I will introduce discrete frames in two distinct settings: infinite sets for function spaces $L^2(\mathbb R^{n^2})$ and finite sets for graph signals (functions defined on the vertex set of a graph).

A systematic approach for producing frames is to discretize continuous wavelet transforms, which are produced using square integrable representations of certain locally compact groups. Most results to date regarding discrete frames for high-dimensional Euclidean domains have been limited to proving existence of discrete frames, with useful, concrete constructions proving difficult to find for $L^2(\mathbb R^n)$ when $n\geq 3$. I will discuss a generalization of one such construction from $L^2(\mathbb R^4)$ to $L^2(\mathbb R^{n^2})$ for all $n$, along with other improvements to the construction. I will describe how techniques from abstract harmonic analysis and the geometry of GL$_n(\mathbb R)$ were used in finding this construction, techniques which generalize to many other groups.

Title: Tropical algebra and geometry

Abstract: Algebraic varieties are defined as the zeroes of polynomials with coefficients in a field (usually the complex numbers). Tropical varieties are combinatorial shadows of algebraic varieties. They can be described by tropical equations, obtained from the algebraic ones by changing the coefficients and the operations. The resulting algebra and geometry are very different from the classical ones. We will discuss various ways to understand and use tropical algebra. We will draw parallels with classical algebraic geometry. 

Title: Simultaneous universal circles

Abstract: In low-dimensional topology, it is common to study spaces by studying associated geometric and dynamical structures. Three examples are codimension-1 foliations, flows, and actions of the fundamental group on the circle. I will give some background on these things before describing a construction that brings all three together, called a simultaneous universal circle. This is joint work with Minsky and Taylor.

Title: Planar Crystallographic Groups: Mapping Classes

Abstract: A planar crystallographic group π is a uniform discrete group of isometries on the complex plane. There are seventeen of these groups up to isomorphism. Via the universal covering map, the orbifold Mπ = C/π is endowed with an affine structure and with a flat structure. The mapping class group mcg(π) is the group of components of the affine self-diffeomorphism group Aff(Mπ). We will discuss mcg(π) for these groups.

Title: Spectral invariants and positive scalar curvature on 4-dimensional cobordism

Abstract: Information about sectional curvature and Ricci curvature tends to make the underlying manifold "rigid" topologically. This is not the case for scalar curvature, e.g., obstruction to existence of positive scalar curvature (psc) is often via some topological invariants. In this talk, we use the Chern-Simons-Dirac functional to define an R-filtration on monopole Floer homology HM(Y) of a rational homology 3-sphere Y. We define a numerical quantity ρ (spectral invariant) that measures the non-triviality of HM(Y). It turns out that ρ is an invariant of Y with a geometric structure. Using ρ, we give an obstruction to psc on ribbon homology cobordsim between 2 rational homology spheres.

Title: Application of convex surfaces theory to Anosov flows

Anosov flows are an important class of dynamical systems due to their ergodic and geometric properties. Even though they represent examples of chaotic dynamics, they enjoy the remarkable property of being stable under small perturbations. In this talk, I will explain how, perhaps surprisingly, Anosov flows are related to both integrable plane fields (foliations) and totally non-integrable plane fields (contact structures). The latter represents a less-studied approach that has the potential to make new connections to other branches of mathematics, such as symplectic geometry and Hamiltonian dynamics. As main application, I will show how convex surface theory introduced by Giroux in the 90s in the context of contact structures, gives a general framework for cut-and-paste techniques on Anosov flows.

Title: TBA

Title: TBA

Title: Frames Meet Neural Networks

Abstract: This talk will provide an overview of the work I have been doing in collaboration with P. Balazs, H. Eckert, M. Ehler, N. Holighaus, V. Lostanlen, and F. Perfler as part of my PhD project at the Acoustics Research Institute.

The primary objective of this work is to explore the potential of frames as a tool for analyzing and potentially enhancing the design of neural networks within the context of machine learning. The discussion is structured around two main threads.

(1) Injectivity and stability of neural network layers with ReLU activation:

We employ a general frame-theoretic perspective to characterize the injectivity of ReLU layers, motivated by phase retrieval. Our findings also inform the development of algorithmic approaches that allow us to address the still-open problem of verifying the injectivity of ReLU-layers in practice.

2) Tightness of learnable filterbanks:

We employ the theory of oversampled filterbank frames to study the stability characteristics of convolutional layers utilized for encoding audio signals (aka. filterbank learning). This includes a stability analysis of random filterbanks and the development of stabilization techniques that preserve the tightness property of a convolutional layer throughout training.

Title: Localization of operator-valued frames

Abstract: An operator-valued frame is a countable family of bounded operators on some separable Hilbert space, which satisfies a Parseval type of inequality and generalizes the notion of a frame. Operator-valued frames are well-established in the literature, but so far, not much has been done in the direction of localization of such operator-valued frames. In my talk I will introduce the concept of intrinsic localization of an operator-valued frame. In particular, we will motivate this concept by considering the toy example of a certain operator-valued frame with a Gabor structure. Generalizing this idea, we will define intrinsic localization of an operator-valued frame in terms of its Gram matrix belonging to some suitable Banach algebra of operator-valued matrices. We then explain why localization is preserved under canonical duality. Finally, we define a whole class of Banach spaces (coorbit spaces) associated to an intrinsically localized operator-valued frame and show that perfect reconstruction also remains true in these Banach spaces and not only in the underlying Hilbert space.

Title: A Greedy Version of the Frame Algorithm

Abstract: The classical frame algorithm uses a simple recursive formula to approximate an unknown vector from its frame coefficients.  One weakness of the frame algorithm is that knowledge of the frame bounds is required to achieve the optimal convergence rate.  This talk will introduce an adaptive version of the frame algorithm that does not require knowledge of the frame bounds, yet achieves the same rate of convergence as its classical counterpart.  It will also be shown that the adaptive version of the frame algorithm is robust with respect to imperfect knowledge of the frame coefficients.

Title:  DG Algebra Structures on Leavitt Path Algebras

Title: Decompositions in Low-Dimensional Topology

Abstract: In the field of low-dimensional topology, four-dimensional spaces are notoriously hard to study, which has inspired the rapid growth of the theory of “trisections” in the last decade. A trisection of a (smooth, connected, closed, oriented) 4-manifold is a decomposition into three “simple” pieces, with the complexity of the manifold relegated to the maps gluing the pieces together.

This theory is the latest in a growing body of literature tackling the general theme of decompositions in low-dimensional topology. In this introductory talk, we will explore decompositions of the following objects, one for each “low” dimension: knots (dimension 1), knotted surfaces (dimension 2), 3-manifolds (dimension 3), and 4-manifolds (dimension 4). As a byproduct we will learn about each of these aforementioned objects, how they relate to one another, and why topologists care about them. In particular, we assume no prior knowledge of topology!

The AWM Student Chapter is excited to invite you to a fun and creative Origami Making Event happening today! Whether you are a beginner or an origami enthusiast, this event will offer a great opportunity to unwind, learn something new, and connect with others.

Event Details:

• Date: September 9

• Time: 4:00 PM

• Location: Ritter Hall Lobby

Materials will be provided, so just bring yourself and your creativity!

Everyone is welcome to attend! We will have snacks and beverages. We hope to see you there for an enjoyable afternoon of folding and fun