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.