No abstract provided
No abstract provided
The area of a circle with radius \(r\) is \(\pi r^{2}\), where \(\pi\) is the ratio between any circle's circumference to its diameter. Could you prove that this is so? If yes, how? Could you determine a rational approximation of $\pi$? How? The methods that you may think of likely involve tools that Archimedes did not have at his disposal. This talk will present Archimedes approaches to the above problems and will comment on the value of thinking deeply.
Applying for grad school can be a rather confusing process to navigate! If you're considering grad school, come to this session to learn more and help demystify the whole process.
Slides (PDF)
Rather than just explain the "what" of elliptic curve cryptography (which I'll certainly do) or the "how" (implementation details are certainly fascinating and important), I'll focus on the "why": Why do we mount the discrete log problem in groups other than the units of a finite field? Why did we settle on the group of points on an elliptic curve, such as Bernstein's Curve25519, rather than some other group? And why that elliptic curve?
In this talk, I will discuss several natural quantum problems and, in particular, how the problems change as the quantum resources change. I will show how to take an economics perspective to assign a "shadow price" to each quantum resource. To do this, I will use optimization theory and show that shadow prices are often given "for free" if you know where to look for them. No knowledge about economics, optimization theory, or quantum theory is needed for this talk. This is joint work with Gary Au (University of Saskatchewan).
This meeting will also be held on Zoom (Meeting ID: 81574260774)
The indefinite integral \(\displaystyle\int\sqrt{1+x^3}\,dx\) looks innocuous enough, but it turns out that we cannot evaluate this integral in terms of a finite number of algebraic operations of elementary functions (elementary functions are the ones that we usually deal with in elementary calculus, such as \(f(x)=e^x\), \(f(x)=\sin(x)\), \(f(x)=x^2\), etc.). In this talk, we will use two theorems from Liouville's theory of finite integration to give a clear explanation of why \(\displaystyle\int\sqrt{1+x^3}\,dx\) cannot be computed.
*This is in McBryde 455 at 5:15pm
When and how do differentiable or integrable functions appear in nature or in human affairs? How does calculus inform our thinking even when we don't use it explicitly?
Let's all meet each other and eat pizza! Tell us a bit about your mathematical background. What math topics interest you? What things might you want to learn about in Math Club?
How can undergrads get involved in research? What is an REU? Dr. Ufferman will talk to us about many of the opportunites undergraduates have at Virginia Tech (and beyond) to further develop their interests.
What are dynamic systems and why are they a big deal in modeling? What is control theory? Dr. Abaid will introduce us to her research in multi-agent systems and monitoring their collective behavior and what this means for some of the biggest problems in real world.
Zero-knowledge proofs are a kind of cryptographic protocol which can be used to prove that a statement is true, or prove that somebody knows a piece of secret information, without revealing anything else. These protocols are both interesting from a mathematical perspective, and for their applications to constructing other cryptographic protocols, such as digital signatures and many protocols with advanced functionalities. I’ll give an accessible introduction to zero-knowledge proofs in the context of a “real world” protocol, then in some cryptographic contexts. Time permitting, I’ll explain what kind of advanced functionalities you can realize using zero-knowledge proofs as a building block.
How do I apply to grad school? When should I apply? What will I get out of grad school? Is grad school really as terrible as they say it is? Applying for grad school can be a rather confusing process to navigate! If you're considering grad school, come to this session to learn more and help demystify the whole process.
As well, we'll also have an informal panel with currenty (or recently-graduated) VT graduate students who can share their thoughts and experiences while in grad school.Informational Slides (PDF)
Maximally even sets, which arose as part of Clough and Douthett’s study of musical scales and pitch classes, manifest in musical traditions across cultures in the form of interesting scales and rhythms. In a remarkable turn of events, Clough and Douthett’s research in music theory has found applications in mathematics, computer science, mathematical physics and even in the design of particle accelerators. In this talk, we will use the concept of electric potential energy from physics together with some ideas from graph theory to study maximally even sets in contexts which were not previously possible. We will go beyond the well-known one-dimensional maximally even sets into higher dimensional and more geometrically complex territory.
Giving a talk on technical matters like your research (or even expository material) is hard, and even those who are experienced lecturers may not get it right all the time. While experience is a very big factor in the accessibility and calibre of a good math talk, there are things you can do to make sure you are conveying the information in an interesting and engaging way. In this talk I'd like to present some of my ideas and past examples, and even discuss some of the unwritten rules involved in seminar and conference settings.
For those interested in other takes on a successful math lecture, you might enjoy reading this newsletter article by P.R. Halmos (PDF) .
What is coding theory? What is a code? How is it used in the world today? Dr. Valvo will introduce us to the basics of coding theory and build up to what's known as Reed-Solomon Codes.
What topics do you want to cover this semester? What activities do we want to plan? This meeting will mainly be a meet and greet with fellow club memebers so we can discuss out plans for the rest of the semester.
In 2011, Dan Shechtman was awarded the Nobel Prize in Chemistry for his discovery of quasicrystals, novel materials with properties somewhere between the regularity of crystals and the disorder of random structures. In parallel with this scientific breakthrough, mathematicians have developed tools for understanding aperiodic order, such as Fibonacci substitutions and Penrose tilings. We will survey these mathematical models of quasicrystals, relying on linear algebra and graph theory. Eigenvalues play a central role, giving insight into how these exotic materials could behave. These problems can be subtle and surprising, opening opportunities for a wide range of mathematical contributions. We will describe our collaborative approach, which integrates numerical computation as a key tool in mathematical discovery, providing a bridge between pure and applied mathematics.
The field of Number theory is very diverse, as it has been studied for thousands of years. One of its newer subfields is known as Iwasawa/Tate Theory, first established in the 1950s by Kenkichi Iwasawa. This field was developed thanks to the discovery of p-adic numbers as well as Class Field Theory in the earlier parts of the century. In this talk we will give a brief introduction to Iwasawa Theory, and its relationship to traditional Number Theory. As well, we will discuss the history of the field and how it has changed since its inception.
The Riemann zeta function has enchanted mathematicians for over 150 years, and is the focus of one of the Clay Millenium Prize problems. In this talk, we will discuss some surprising geometry that arises from certain values of the zeta function, the genesis of which can be understood from solving some closely related linear ODEs. Emphasis will be placed on concrete computations that can be understood by students who have completed Calc II and ODEs.
Off-line states are periods during which the internal dynamics of the brain are relatively independent of external stimuli. The oscillatory dynamics that occur during these states are thought to be critical for learning and memory and are often disrupted by disease. An understanding of these oscillatory dynamics offers the possibility of both enhancing the cognitive capacities of healthy individuals and providing pharmacological and stimulation interventions for disease. In the first half of my talk, I will present a mechanistic model of alpha (8-13 Hz) oscillations during general anesthesia. In the induction of general anesthesia, behaviorally defined loss of consciousness coincides with anteriorization, the spatial shift of alpha power from posterior to anterior regions. We show that anteriorization can be explained by the differential effect of anesthetic drugs on thalamic nuclei with disparate spatial projections. In particular, we show that anesthetic drugs can disrupt the alpha activity generated at depolarized membrane potentials in posteriorly projecting thalamic nuclei while engaging a new, hyperpolarized alpha in frontally projecting thalamic nuclei. In the second half of my talk, I will present work examining oscillations during REM sleep. REM sleep, the period of sleep during which vivid dreams occur, is important for the processing of emotional memories. REM sleep is important, for example, in reducing the emotional charge of fear memories. Rhythmic interactions, especially in the theta band (4-8 Hz) between the medial prefrontal cortex (mPFC) and limbic structures, are known to play a role in reducing emotional charge, but the processing that occurs is largely unknown at the mechanistic and circuit levels. Using mathematical models, we show that theta inputs, but not other frequency inputs, from the mPFC are effective in producing synaptic changes that ultimately suppress the activity of fear expression cells in the amygdala associated with a given memory. We show how aberrant dynamics in this circuit may lead to the symptoms of Post-Traumatic Stress Disorder (PTSD). Our work also suggests potential neuromodulatory therapies for ameliorating PTSD symptoms.
Abstract TBD
This will be a two-part informational session on Math Department Scholarships and Undergraduate Research in Mathematics. To begin, Professor Childs will give a short overview of the Math Department Scholarship process including how to become eligible and important deadlines. This will be followed by Professor Palsson discussing opportunities for undergraduate research. This will include information about opportunities at VT (both during the semester and outside the semester) as well as ways to be involved in undergraduate research through summer programs such as REUs (Research Experiences for Undergraduates) run at a variety of schools across the country. There will be ample opportunity to ask questions.
From Cryptography to Quantum computing the cloud in essence seeks to solve an incredibly complex problem: how do we use the fewest resources for the most benefit? This goal requires subject matter expertise in several practical and theoretical domains one of the largest being mathematics. Coding and Software development utilizes skills in almost every branch of mathematics from linear algebra to functional analysis which makes a strong foundation in math crucial for proper understanding of the cloud. Math is also used in a variety of physical applications. How much energy are we using for a data center? How much water? How can we strive to be earth’s best employer? In this presentation I will guide you through some of the key examples of clouds use of mathematics and hopefully inspire you to ask questions and learn and be curious.
Mathematics Education Research is a systematic study of the teaching and learning of mathematics. In this talk, I will share methods and results from the NSF funded "Proofs Project," which investigates the persistent challenges that introductory proofs students experience as they transition from calculation-based mathematics to proof-based mathematics. Proof by mathematical induction is one such challenge. Prior research indicates a variety of related factors that contribute to students' difficulties with this proof technique and suggests a promising instructional approach, called quasi-induction, to support more effective learning. However, students still experience difficulty in bridging the gap between quasi-induction and formal induction. To better understand this cognitive gap, The Proofs Project collected video data of research-based instruction on proof by mathematical induction in two Virginia Tech Math 3034 classes. I'll share our methods for qualitatively analyzing this data, findings from the study, and implications for instruction. For more information on The Proofs Project, visit our website: https://math.vt.edu/proofs-project.html.
In calculus we learn about trig functions, integrals and many of the objects that harmonic analysis relies on. In this talk, I will connect to your calculus knowledge to give an idea of what harmonic analysis is about. Further, I will show some successes of the theory and give a glimpse of major open problems, such as the restriction conjecture and the Kakeya conjecture, that harmonic analysts are trying to solve.
What do you mean there is more math to learn??? This session will feature a panel of Math Department faculty and graduate students walking through some of the most commonly asked questions involving graduate school. Some topics include: How and when to apply, courses and activities and experiences to prepare, and letters of recommendation. Graduate school is a big next step in continuing one's math career dependent on many factors, including personal interests, career / academic goals, etc., so please bring any questions you have regarding the process and experience!
Image reconstruction, such as problems from medical imaging, and so-called inverse problems in general are often ill-posed. The main problem is that the solution does not depend continuously on the data, which means that the solution is extremely sensitive to small changes in the data. These problems generally involve reconstructing/computing a (very) large number of parameters, such as absorption of x-rays in tissue, combining a physical model with measurements to estimate the unknown parameters in the model. As measurements always involve noise, the ill-posedness creates serious problems for accurate reconstruction. Moreover, in many cases, we do not have enough data to uniquely determine the parameters. I will explain some of the problems, use mathematical tools like the singular value decomposition (SVD) to demonstrate and analyze these problems, and discuss some solutions and interesting research directions.
Many highly significant decisions—including in college admissions, hiring, insurance, criminal justice and more—are made with the assistance of algorithms and mathematical models. There are, of course, many technical questions about how to design such models. But the use of such models also raises ethical questions concerning fairness, transparency, and accountability. I'll talk about some of these ethical issues, using a variety of recent case studies.
We summarize some recent reduced order model (ROM) developments for the quasi-geostrophic equations (QGE) (also known as the barotropic vorticity equations). The QGE are a simplified model for geophysical flows in which rotation plays a central role, such as wind-driven ocean circulation in mid-latitude ocean basins. Since the QGE represent a practical compromise between efficient numerical simulations of ocean flows and accurate representations of large scale ocean dynamics, these equations have often been used in the testing of new numerical methods for ocean flows. ROMs have also been tested on the QGE for various settings in order to understand their potential in efficient numerical simulations of ocean flows. We survey the ROMs developed for the QGE in order to understand their potential in efficient numerical simulations of more complex ocean flows: We explain how classical numerical methods for the QGE are used to generate the ROM basis functions, we outline the main steps in the construction of projection-based ROMs (with a particular focus on the under-resolved regime, when the closure problem needs to be addressed), we illustrate the ROMs in the numerical simulation of the QGE for various settings, and we present several potential future research avenues in the ROM exploration of the QGE and more complex models of geophysical flows.
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Come listen to some of our members give talks on topics they are interested in or have worked on in the past! At this meeting, we will have Yash Agarwal giving a talk "Charting Hidden Dimensions", Levi Walker talking on "H_2 Optimal Model Reduction on GPUs", Aryan Palit talking on "Collaborative Filtering in Dating Apps", and Richard Morgan talking on "Uncovering non-linear gene interactions using Graph Embeddings". If you want to give a talk in the future, reach out to one of our officers! We plan on having another one of these meetings this semester.
The discoveries made by LIGO in gravitational wave astrophysics have been hailed as a triumph of experimental methods in physics – and they are. However, this paper will analyze the fact that – as the LIGO researchers know well – it would not have been possible to derive secure inferences from those experiments without significant formal advances (in numerical relativity, effective one-body systems and dynamics of two-body systems, linearization of the field equations, and the like). These advances, by researchers including Pretorius and Choquet-Brouhat, made it possible to move from the pioneering methods used to analyze the 1980s discoveries by Hulse and Taylor, to the even broader and more flexible LIGO framework. The paper will demonstrate that inferences about black holes and other binary systems in the LIGO project encode a complex set of choices involving higher-order formal relationships, which in turn reflect decisions about what is worth pursuing in experiment and theory-building. The answer to the question of whether LIGO is a triumph of experiment or theory is “Both – and neither: one truly novel feature of LIGO is the way they work with the relationship between theory, experiment, simulation, and testing”.
Mathematical models have long been useful tools for examining drivers of infectious disease emergence, spread, and control. In this talk, I will introduce population and infectious disease models and discuss my recent work in using models to study mosquito-borne diseases. Mosquito-borne diseases that have been historically endemic to areas with tropical climates have been spreading in temperate regions of the world with greater frequency in recent years. Numerous factors contribute to this spread, including urbanization; increases in global travel; and changes in temperature, precipitation, and humidity patterns leading to anomalies from historical averages. Mathematical, statistical, and computational modeling can help us understand how these different influences impact transmission and spread of pathogens, and models are useful for projecting how potential future changes in these factors could affect pathogen dynamics. Although models have been employed for years to study disease dynamics, diseases emerging in new regions present particular challenges. I will focus on here on recent modeling work for better understanding the emergence of dengue fever in Central Argentina. Dengue, caused by a virus transmitted by Aedes aegypti mosquitoes, first emerged in temperate Argentinian cities in 2009, and multiple outbreaks of increasing incidence have occurred since. With particular focus on the role of meteorological influences on dengue emergence, I present mathematical models designed to study seasonal Ae. aegypti and dengue dynamics in temperate Argentinian cities. I will show how different seasonal patterns influence the risk of outbreaks and how projected increases in average temperatures may influence future transmission risk. I will also discuss the implications of our work for dengue and mosquito mitigation strategies, and address some of the issues and areas for improvement in modeling emerging pathogens transmitted by mosquitoes.
Since the dawn of time, mathematicians have wondered about triangles. Only in the 19th and 20th centuries did they realize the set of all triangles itself forms a space! Master this space, and you understand triangles. We will explore a similar space of elliptic curves, a.k.a. parallelograms central in number theory. Time permitting, we will see other examples of moduli spaces and a first glimpse of the stack structure inherent to all these spaces. We look forward to seeing you there!
Interested in Data Science? Join us for an introduction to the new M.S. in Data Science offered by the Academy of Data science. During this presentation, we will highlight what makes this degree unique — including a curriculum that bridges the gap between theoretical knowledge and practical application, a wide range of electives, and instruction from industry experts with decades of experience who designed and now teach in the program.
We present a method of visualizing holomorphic functions and illustrate some basic notions from calculus within this visualization framework from tangent lines to Newton's Method.
TBD