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Seminar series · Math in simple words
Simple Words 2024
Simple Words aims to make mathematical research more accessible to high-school, undergraduate, and graduate students, with a particular focus on helping students from underrepresented groups become familiar with a range of mathematical fields.
Experts from fields such as mathematics, statistics, and computer science discuss their research in simple language that a first-year graduate student can follow. Unlike traditional talks focused on a single result, speakers provide a broad overview of their work while emphasizing connections to other disciplines.
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October 01Tuesday
Application of Deep Learning Models in Medical Image Analysis
Willy Rodriguez · Torus AI
Application of Deep Learning Models in Medical Image Analysis
Willy Rodriguez · Torus AI
Abstract
In recent years, the development of deep learning models has revolutionized medical imaging analysis, significantly enhancing the speed and accuracy of diagnosis. This presentation will introduce two key families of models: Variational Autoencoders (VAEs) and U-Net-based architectures. VAEs are powerful tools for unsupervised classification, capable of learning and identifying patterns in medical images without the need for labeled data. On the other hand, U-Net models excel in performing precise segmentation of Regions of Interest (ROI), crucial for detailed image analysis. As a practical example, the presentation will showcase an AI module designed for automatic measurement and analysis of the cervical spine region, demonstrating the real-world impact of these technologies in medical diagnostics.
October 15Tuesday
Harnessing the Power of AI and LLMs in Mathematics: Current Trends and Future Directions
Marly Gotti · Apple
Harnessing the Power of AI and LLMs in Mathematics: Current Trends and Future Directions
Marly Gotti · Apple
Abstract
Large Language Models (LLMs) such as GPT and their derivatives have transformed numerous fields by offering advanced capabilities in language understanding, generation, and reasoning. While these models excel in tasks involving natural language, their application in mathematics, a domain characterized by precision and formal structure, presents unique challenges and opportunities. In this talk, we explore the evolving role of LLMs in mathematics, from automating theorem proving to assisting in symbolic reasoning and mathematical research.
October 29Tuesday
Newton’s Method or: How to Solve Nonlinear Equations Fast
Matthew Dallas · University of Dallas
Newton’s Method or: How to Solve Nonlinear Equations Fast
Matthew Dallas · University of Dallas
Abstract
Numerical Analysis is the rigorous study of algorithms used to solve mathematical problems. The study of numerical algorithms dates back to antiquity; with some modern techniques, such as Newton’s method for solving nonlinear equations, originating centuries ago. Since the development of modern computing in the mid-20th century, Numerical Analysis has matured and become fundamental to numerous disciplines such as physics, chemistry, engineering, and computer graphics. This talk offers an overview of Numerical Analysis as a whole before delving into a specific subfield: the numerical solution of nonlinear equations. Nonlinear equations abound in the sciences, and solving them accurately and efficiently remains a top priority. A standard tool in this subfield is Newton’s method. We will derive Newton’s method, see it in action, and develop some basic theory to understand why it performs so well compared to other methods. Additionally, we will discuss ongoing research in nonlinear solvers, addressing instances where Newton’s method falls short and exploring techniques to enhance its performance in such scenarios.
November 12Tuesday
Solving Large-Scale Linear Systems
Anna Ma · UC Irvine
Solving Large-Scale Linear Systems
Anna Ma · UC Irvine
Abstract
In applied mathematics and data science, we leverage our ability to access large amounts of data to make better decisions. While more data can provide more information about the world around us, working with large-scale data creates new and interesting challenges. In this talk, we will discuss these challenges with respect to algorithmic approaches for a specific problem that often occurs in data science applications: solving large-scale linear systems. We will explore a range of algorithmic approaches, from classical methods like Gradient Descent to more modern techniques such as Stochastic Gradient Descent (SGD), highlighting how these methods are adapted to handle the complexities of big data efficiently. By examining both traditional and cutting-edge algorithms, we will provide insight into how numerical techniques evolve to meet the demands of contemporary data-driven applications.
November 26Thanksgiving Break
December 04Wednesday
Monogenic Number Fields by Example
Hanson Smith · California State University San Marcos
Monogenic Number Fields by Example
Hanson Smith · California State University San Marcos
Abstract
By adjoining a root of an irreducible polynomial \(f(x)\in \mathbb{Z}[x]\) to the rationals, we obtain a larger field that can be thought of as a generalization of \(\mathbb{Q}\). Such extensions are called \(\textit{number fields}\). Just as \(\mathbb{Z}\subset \mathbb{Q}\), each number field \(K\) has an analogue of the integers which we call \(\textit{the ring of integers}\) of \(K\) or simply a \(\textit{number ring}\). Studying the properties of number rings is a key aspect of algebraic number theory.
A number field is \(\textit{monogenic}\) over \(\mathbb{Q}\) if it admits “one generator” over \(\mathbb{Z}\). More formally, a number field is monogenic if the ring of integers admits a \(\mathbb{Z}\)-basis of the form \(\{1, \alpha, \dots, \alpha^{n-1}\}\). Here we call \(\alpha\) a \(\textit{monogenerator}\). This talk will explore monogenicity by looking a a variety of examples. If time permits, we will survey some current work and open questions.
December 10Tuesday
Error-Correcting Codes: The Mathematics of Communication
Nathan Kaplan · UC Irvine
Error-Correcting Codes: The Mathematics of Communication
Nathan Kaplan · UC Irvine
Abstract
Suppose we are trying to communicate over a ‘noisy channel’. I want to send you a single bit, a 1 or a 0, but there is some probability that the bit I send is not the bit you receive. We could communicate more reliably by agreeing to repeat the intended message, for example, instead of sending ‘0’ or ‘1’, I would send ‘000’ or ‘111’. But, there is a cost to this repetition. A major goal in the theory of error-correcting codes is to understand how to efficiently build redundancy into messages so that we can identify and correct errors. In this talk we will give an introduction to the ideas that go into the mathematics of communication. We will highlight some open problems and places where ideas from number theory can help to build efficient communication schemes.