Website: hbsahana.com
Email: hassanba@lafayette.edu
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Group Actions on Hyperbolic Spaces. (UG,G,P)*
Group actions on hyperbolic spaces have proven to be a useful tool in geometric group theory. In particular, recent avenues of research have shown that many groups classically studied admit a large class of hyperbolic actions. I will describe the concept of groups, actions and hyperbolic spaces, and discuss interesting examples. I will conclude with a brief discussion of groups that are "inaccessible" via this strategy, thereby touching upon recent developments and directions for future research.
Non-recognizing Spaces for Stable Subgroups. (G,P)*
A popular avenue of research in geometric group theory is to produce a "recognising" hyperbolic space for stable subgroups. In this context, largest acylindrical actions have proved fruitful to study for certain well-known examples of acylindrically hyperbolic groups, including mapping class groups and right angled Artin groups. This prompts the question whether the largest acylindrical action (if it exists) always serves as a recognizing space. I will talk about recent work where we produce a counterexample to this claim. This is part of joint work with M.Chesser, A.Kerr, J.Mahangahas and M.Trin.
The Semi Simple Theory of Acylindricity in Higher Rank. (UG,P)*
In this talk, I will discuss forthcoming work with T.Fernos that explores the theory for countable groups acting (AU-) acylindrically on products of hyperbolic spaces. This theory draws inspiration from the theory of S-arithmetic lattices in linear algebraic groups as well as acylindrical actions on a single hyperbolic space (rank-1 case). We reprove and extend many results about acylindricity from rank-1 to higher rank, explore elementary subgroups in higher rank and prove results akin to the Tits alternative. We also give a partial answer to a recent conjecture of Sela concerning outer automorphism groups in this setting.
Website: http://depts.ycp.edu/~cbauer/
Email: cbauer@ycp.edu
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Cryptology on Campus During WW II.
Many colleges and universities were involved in offering cryptology classes in the 1940s. There was a tremendous amount of diversity in who delivered the classes. The professors were not all mathematicians, but rather came from departments that included Astronomy, Classics, English, Geology, Greek, Psychology, and Philosophy. Some of these classes were open to all students, while others were run with great secrecy and open only by invitation. These classes are detailed along with the backgrounds of the instructors and the role they played in the war.
The Cryptologic Contributions of Dr. Donald Menzel.
Dr. Donald H. Menzel is well known for his work in astronomy, but his cryptologic work (long secret) has attracted much less attention. This talk describes how his interest in this area was first sparked and provides the details of the secret class he led in "Naval Communications" (really cryptanalysis) at Radcliffe College during World War II. This class served as a prototype and was copied elsewhere. A sketch of some of the classified work he carried out during the war, and after, as a consultant, is also included, along with a brief over-view of Menzel’s personality and other interests.
Cracking Matrix Encryption Row by Row.
The Hill Cipher, also known as matrix encryption, uses matrices to encipher and decipher text. Various attacks, such as those found by Jack Levine, have been published for this system. This talk reviews a few previous results and presents a powerful new attack in which the rows of the matrix can be determined independent of one another, greatly reducing the amount of time needed for decipherment.
A New Connection Between the Triangles of Stirling and Pascal.
An enumeration problem involving matrices is seen to give rise to Pascal's triangle, Stirling's triangle, and infinitely many other new triangles. Some special properties of the better known triangles are related and analogous results for the new triangles are investigated. This is an open problem with much work still to be done and best of all, it is appropriate for undergraduate contemplation!
Website: https://sites.lafayette.edu/berkovee/
Email: berkovee@lafayette.edu
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Folding Mathematics into Origami. (HS,UG)*
Origami is the traditional Japanese art of paper folding. In the past 30 years investigations into folding properties have not only resulted in many stunning models, but also a surprising number of applications. In this talk we will provide an introduction to some of the interactions between mathematics of origami. We’ll describe what types of folds are mathematically possible, how "complicated" folding can be, and how mathematical ideas have helped further origami. We’ll also talk about a number of mathematical results inspired by origami.
Shining a Light on the Menger Sponge. (HS,UG)*
The Menger sponge is a 3-dimensional fractal that is formed from a unit cube by recursively removing collections of smaller and smaller subcubes. This talk’s title is literal: we will investigate what shadows are cast when you shine a light (at a point at infinity—it’s a bright light) on the Menger sponge from different angles. Along the way we’ll look at the Cantor set in a new light, and determine the structure of Menger sponge cross sections that are parallel to an outside face. These results were part of an REU project, and represent joint work with Max Auerbach, Adam Hodapp, Derek Smith, and Rebecca Whitman.
Website: https://sites.millersville.edu/rbuchanan/
Email: Robert.Buchanan@millersville.edu
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Turing Instability in Pioneer/Climax Species Interactions.
Aspects of Financial Mathematics: Options, Derivatives, Arbitrage, and the Black-Scholes Option Pricing Formula.
An Introduction to State Space Reconstruction: Applications and Mathematical Prerequisites.
Website: https://www.millersville.edu/math/faculty/retired-faculty/catepillan/catepillan.php
Email: ximena.catepillan@millersville.edu
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Ethnomathematics and Kinship Systems. (UG,G,P)*
In this talk, the following examples will illustrate how kin relationships and mathematics can be connected. The first two originate in the Warlpiri and the Aranda tribes. These tribes are the most traditional aboriginal groups in Northern Australia. The last example is from the tribes in the Malekula and Ambrym Islands of the republic of Vanuatu located in the southwestern Pacific Ocean.
Ethnomathematics: The Connection Between Culture, History, and Mathematics. (HS,UG,G,P)*
For over 16 years, I have traveled to remote places with Dr. Edwin Barnhart, founder and Director of the Maya Exploration Center and Ancient Explorations, and his team to do archaeological studies associated with ethnomathematics, the connection between culture, history, and mathematics. In this presentation, I will provide examples of ethnomathematics that I have taught in an undergraduate course, a graduate course, and a first-year seminar at Millersville University of Pennsylvania and at Universidad de Santiago de Chile USACH.
Maya Numbers and Computations. (UG,G,P)*
Mesoamerican calendars were many and complex. There have been a good number of studies done to decipher them. By the arrival of Hernan Cortes in 1519 in what is current day Mexico, there were 21 calendars in use while 4 of them were extinct. Using astronomical observations, the Maya developed an elaborate system of calendars, among them the Tzolkin Calendar, the Haab Calendar, the Round Calendar (a combination of the first two) and the Long Count. Which operations did the Maya use to perform their calendrical computations? While they used a vigesimal system to write the numbers, this system was never used in connection with days. No inscriptions use vigesimal numbers but rather quasi-vigesimal (chronological) numbers. In spoken numbers, a mix of decimal and vigesimal notation appears. Multiplication by 20 was the most common computation. They also needed to divide to do some of the calendar conversions. I’ll illustrate calendrical computation within and among calendars and conversion examples in which division is needed. This technique is quite simple using just a pencil and paper.
Website: https://www.albright.edu/faculty-detail/associate-professor-christopher-j-catone-ph-d-chair/
Email: catonec@gmail.com
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Bringing Calculus into Discrete Math via the Discrete Derivative - OR - Sequences Want To Be Differentiated Too!
Most mathematics majors take Calculus and Discrete Mathematics early in their college careers and often consider them disjoint. Although well known among mathematicians, the discrete derivative usually does not appear in either course. By making this exclusion, we miss an opportunity to emphasize the connections among mathematical disciplines. We present the discrete derivative as it could appear in a discrete math class and illustrate the parallels with the derivative studied in calculus. We give examples where this tool can be used to solve discrete problems.
The Calculus of Variations and the most important equation math students never learn.
The Calculus of Variations is the study of finding extremals of functionals. Usually these functionals take the form $J(y(x))=\int_{x_1}^{x_2} f(x,y,y’) dx$. This talk will discuss some examples which yield exactly this situation, some of which are the very problems that led to the development of our topic. We will derive the Euler-Lagrange equation and use it to solve these examples. This talk is aimed at students; only a basic knowledge of Calculus is necessary to delve into this subject.
An Introduction to Differential Forms.
What is a differential form and how can it be used in the study of geometric objects? In this talk we will define what differential forms are and explore their properties. We will see how they can be used to consolidate many theorems of Calculus in to a single concise theorem.
Website: marioncohen.net
Email: mathwoman12@gmail.com
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Coining a New Word: MathLIKE. (HS,UG,G,P)*
In this talk I’ll discuss “mathLIKE”, a word I’ve coined (which means, roughly and predictably, "LIKE math") and how it pertains to a course I’ve developed, Mathematics in Literature, as well as to people, both mathematicians and not, and to events and impressions from everyday life. Two of the more obvious mathLIKE things are dreams and fears, both of which the students and I have discussed extensively in that course. Maybe everything is mathLIKE. At any rate, for math-anxious people, mathLIKE things can be helpful in dealing with (and teaching) actual math.
Non-ordinary Associative Arithmetics. (HS,UG,G,P)*
A binary operation # on Z+ is said to be an associative arithmetic if both # and its iteration -- the binary operation * defined inductively by: x * 1 = x,
x * y = [x * (y-1)] # x -- are associative. E. Rosinger showed that, under reasonable conditions, an associative arithmetic must be ordinary addition. However, in the general case, there are associative arithmetics that are not ordinary addition. This paper gives examples of these, as well as results towards a structure theorem for associative arithmetics.
Website: https://www.fandm.edu/annalisa-crannell
Email: annalisa.crannell@fandm.edu
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Drawing Conclusions from Drawing a Square.
The Renaissance famously brought us amazingly realistic perspective art. Creating that art was the spark from which projective geometry caught fire and grew. This talk looks directly at projective geometry as a tool to illuminate the way we see the world around us, whether we look with our eyes, with our cameras, or with the computer (via our favorite animated movies). One of the surprising results of projective geometry is that it implies that every quadrangle (whether convex or not) is the perspective image of a square. We will describe implications of this result for computer vision, for photogrammetry, for applications of piece-wise planar cones, and of course for perspective art and projective geometry.
Double Take: Geometry, Perspective, and Optical Illusions.
Perspective geometry, so it's said, was the art that made paintings seem realistic and true to life. Yet the same artistic techniques---and by extension, the same geometry---can create images that astound and confound us. We provide a carnival of such examples, from special effects in the movies, to sculptures that seem to move as we move, to hidden objects, and to objects whose mirror images transform them into other beings.
Email: bdunham@brynmawr.edu
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The Infinitude of Primes: Euclid, Euler, Erdös.
The foundation of number theory lies among the primes. It thus seems fitting to examine three different proofs, from across history, of the infinitude of the prime numbers. We first look at Euclid’s argument from 300 BCE, which appears as Proposition 20 of Book IX of the Elements. Although "Euclid’s proof of the infinitude of primes" is a standard in every number theory textbook, some people might be surprised to see his argument in its original form. We next consider Euler’s analytic proof from 1737. Like so much of his work, this features a blizzard of formulas, manipulated with a maximum of agility and a minimum of rigor. But the outcome is spectacular. Finally, we examine Erdös’s combinatorial proof from the 20th century. This is an elementary argument, but it reminds us once again that "elementary" does not mean "trivial." Taken together, these proofs suggest that, to establish the infinitude of primes, it helps to have a two-syllable last name starting with "E." More to the point, they show mathematics as a subject whose creative variety knows no bounds.
An Eulerian Miracle.
This talk features some genuine Eulerian magic. In 1748, Leonhard Euler considered a modification of the harmonic series in which negative signs were attached to various terms by a rule that was far from self-evident. With his accustomed flair, he determined its sum, and the result was utterly improbable. There are a few occasions in mathematics when the term "breathtaking" is not too strong. This is one of them.
The Math Matriarchs of Bryn Mawr.
Over its first 50 years, Bryn Mawr College boasted three remarkable mathematicians who, one after the other, left deep footprints on the institution and on the U.S. mathematical community. They were Professors Charlotte Angas Scott (British), Anna Pell Wheeler (American), and Emmy Noether (German). In this lecture, we meet these women and flesh out their biographies with plenty of local color… not to mention a real-life assassin in a supporting role. From 1885 to 1935, they gave Bryn Mawr a record of women in mathematics unsurpassed by any college, anywhere.
Website: https://dr-michael-ecker.weebly.com/
Email: drmwecker@aol.com
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Mathemagical Black Holes.
Many of the best recreations have a common theme. Start with a number, perform some process over and over, and always arrive at one particular number - the mathemagical black hole.
Paradoxes.
How can you pass your body through a postcard? Can a surface really hold a finite volume of paint but not be painted? How does this last example relate to fractals, the von Koch snowflake, and the human intestines? What's the deal with time travel? How can removing balls from an urn lead to no balls or infinitely many?
Email: timothy.feeman@villanova.edu
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The ART of tomography.
We will look at how an algorithm from linear algebra, called Kaczmarz's Method, can be used to create a CAT scan image from X-ray data.
Website: https://www.dickinson.edu/info/20084/institutional_research/178/staff
Email: glassd@dickinson.edu
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Cryptography, Coding Theory, and Curve Interpolation.
The goal of cryptography is to use mathematics to communicate secrets so that only the people you want to hear the information do in fact hear it. The goal of coding theory is to use mathematics to build redundancy into messages so that they can withstand errors that may occur in the transmission. While much of the mathematics used in these areas is extremely deep, major advancements in both of these areas have come from the simple fact that two points determine a unique line and some generalizations of this fact. In this talk, I will describe these applications of curve interpolation, and discuss broader ideas that come up in the modern theory of communications.
Elliptic Curves and Poncelet's Porism.
Elliptic Curves have great interest to mathematicians for a variety of reasons, ranging from cryptography to the proof of Fermat's Last Theorem. In this talk, I will introduce the concept of elliptic curves, talk about some of their properties, and then show a modern proof of the classical result known as Poncelet's Porism, which uses Elliptic Curves in a surprising way.
Arithmetical Structures on Graphs.
In this talk, we will introduce the notion of an arithmetical structure on a finite connected graph. These structures were defined by Dino Lorenzini in order to answer some questions in algebraic geometry and have deep structure, but in this talk we will discuss how they can be described only in terms of elementary number theory and linear algebra, and how new results can be obtained with only a little bit of elementary school arithmetic and a lot of doodling. One goal is to count how many different structures one can place on a given graph as well as discuss finite abelian groups that are associated to arithmetical structures. We will fully answer this question for some families of graphs, and discuss why it is a hard but interesting question for other families.
Website: http://www.math.drexel.edu/~ahicks/
Email: ahicks@math.drexel.edu
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Controlling Ray Bundles with Reflectors.
How does one design a driver side mirror without a blind spot that does not distort the image? This is essentially the fundamental problem of optical design, which is to guide a given collection of light rays to some prescribed target points on a surface via a family of optical components, such as mirrors and lenses. We consider the problem of performing this task for a single 2-parameter ray bundle as typically is generated by a single source, show that a single reflector is not adequate to solve the problem, and give estimates on how bad the situation can. We will describe applications, including a driver-side mirror with no the blind spot problem and mirrors designed for panoramic imaging. Prototypes will be available for inspection.
Website: https://www1.villanova.edu/university/liberal-arts-sciences/programs/mathematics-statistics/faculty/biodetail.html?mail=vikram.kamat@villanova.edu&xsl=bio_long
Email: vikram.kamat@villanova.edu
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Erdos--Ko--Rado Theorems: New Proofs, Generalizations and Chvatal's Conjecture. (UG,G,P)*
The Erdos--Ko--Rado (EKR) theorem, a foundational result in extremal finite set theory, states that any family of k-subsets of a finite set containing at least 2k elements that is intersecting, i.e. the family has the property that no two subsets in it are disjoint, can be no bigger than the family of all k-subsets containing a fixed element x. The latter family is trivially intersecting and is called a star centered at x.
We discuss different proofs, new and old, of this classical result, uniqueness of the extremal star structures, and also recent work on a graph-theoretic generalization of the theorem that is partly inspired by a longstanding conjecture of Chvatal. We also consider a related problem of finding the optimal stars in this graph-theoretic setting, a problem that highlights some of the underlying challenges posed by Chvatal's conjecture.
Website: https://www.gettysburg.edu/academic-programs/mathematics/faculty/employee_detail.dot?empId=04007274120013369&pageTitle=Benjamin+Bartlett+Kennedy
Email: bkennedy@gettysburg.edu
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A short route to chaos: playing with irregular sequences at home. (UG)*
We look at a simple family of tent-like maps on the unit interval and give an explanation of where "chaotic" behavior comes from.
The Kakeya Needle Problem. (UG)*
An ink-covered needle of length 1 sits on a sheet of paper. We rotate and slide the needle on the paper until it sits where it did originally, but with its endpoints switched. What is the smallest area that the resulting ink blot can have? We discuss the (astonishing) answer, discovered by Besicovitch almost 100 years ago.
Email: Alevine@fandm.edu
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A Slice of Pi. (UG,G,P)*
During the past 4000 years or so, many mathematicians have derived a variety of methods for calculating Pi. We will look at several of these, including Archimedes, Euler, Gauss, Ramanujan, among others.
Calculus of Probabilities, by A. A. Markov. (UG,G,P)*
In 1900, Markov wrote a book entitled “Calculus of Probabilities”. In this talk, we’ll explore some of the main concepts in this book, which has been translated into English by the speaker. In many ways, the book resembles modern probability texts. In other ways, it is very different.
Website: https://sites.temple.edu/lpeilen/
Email: luke.peilen@temple.edu
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Local Laws and Fluctuations for Log and Riesz Gases. (P)*
We study the statistical mechanics of the log gas, an interacting particle system with applications to random matrix theory and statistical physics, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on a novel next order energy quantity that are valid down to microscopic length scales. Simultaneously, we exhibit a control on fluctuations of linear statistics that is also valid down to microscopic scales. Using these local laws, we exhibit for the first time a CLT at arbitrary mesoscopic length scales. The methods we use generalize well to the study of higher dimensional Riesz gases; we will discuss some generalizations of the above approach and results on local laws and fluctuations for the Riesz gas in higher dimensions. This is joint work in progress with Sylvia Serfaty.
Poisson behavior for Coulomb Gases. (P)*
We are interested in the behavior of the microscopic point process for Coulomb gases at intermediate temperature regimes β->0, βN -> ∞. The behavior at high temperatures βN ~1 is well understood (cf. Lambert '21) for general Riesz gases and at intermediate temperatures for Gaussian β-ensembles (Benaych-George, Péché '15), but the intermediate temperature behavior for Coulomb and Riesz gases is not well understood. In our work, we establish convergence of the microscopic point process for the Coulomb gas in d=2 to a Poisson point process for a strictly larger intermediate temperature regime than was previously known. Our approach relies on a precise asymptotic description of the correlation functions and overcrowding estimates due to Thoma '25. I will also discuss how many of these technical challenges can be circumvented in the case that the interaction kernel has an integrable Fourier transform; there, we can establish convergence to a Poisson point process more easily and in a wider temperature regime. This is joint work with David Padilla-Garza and Eric Thoma.
Website: https://webspace.ship.edu/msrenault/
Email: msrenault@ship.edu
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Infinity: It’s Really, Really Big.
In this talk we’ll consider some common notions of infinity, and see how mathematicians and philosophers have attempted to understand it. We’ll look at some of the ideas of Georg Cantor (1845-1918), the first mathematician to put the infinite on a solid foundation, and we’ll see that there are in fact different “levels” of infinity!
Website: https://divisbyzero.com/
Email: richesod@dickinson.edu
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Tales of Impossibility—On the Problems of Antiquity.
"Nothing is impossible!" It is comforting to believe this greeting card sentiment; it is the American dream. Yet there are impossible things, and it is possible to prove that they are so. In this talk, we will look at some of the most famous impossibility theorems—the so-called "problems of antiquity." The ancient Greek geometers and future generations of mathematicians tried and failed to square circles, trisect angles, double cubes, and construct regular polygons using only a compass and straightedge. It took two thousand years to prove conclusively that all four of these are mathematically impossible.
A Romance of Many (and Fractional) Dimensions—A History of Dimension.
Dimension seems like an intuitive idea. We are all familiar with zero-dimensional points, one-dimensional curves, two-dimensional surfaces, and three-dimensional solids. Yet dimension is a slippery idea that took mathematicians many years to understand. We will use pictures, animations, and analogies to describe our three dimensions and help us visualize a fourth dimension. We will discuss the history of dimension, the public's and artistic community’s infatuation with the fourth dimension, time as an extra dimension, the meaning of non-integer dimensions, and the unexpected properties of high-dimensional spaces.
Euler’s Beautiful Polyhedron Formula.
In 1751 Euler discovered that any polyhedron with V vertices, E edges, and F faces satisfies V-E+F=2, but the proof he gave was flawed. Many rigorous proofs followed, but it took 150 years for mathematicians to fully understand this simple formula. Today Euler's formula is held aloft as one of the most beautiful theorems in all of mathematics and the first great theorem of topology. In this talk, we will give the history of Euler's theorem and give a sampling of its many applications.
Website: http://www.unix.bucknell.edu/~ncr006/
Email: ncr006@bucknell.edu
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Elliptic Curves: New Solutions to Old Problems. (UG)*
Introducing the Pennsylvania Mathematics Alliance. (UG)*
Analysis of carceral algorithms: history, fairness, biases and outcomes. (UG)*
Email: savitzr@neumann.edu
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Super shoes: how super are they? (HS,UG,G,P)*
This talk examines the effect of carbon plated “super shoes” on the performance of elite male marathoners. Prototypes of these carbon plated shoes with extraordinary cushioning were used beginning in 2016, and became widely available in 2017. In order to quantify the effect of these shoes on elite athletes, we analyzed data on the number of athletes who ran a marathon in under 2:08:00 each year from 1985 through 2021. A multiple linear regression model was constructed that controlled for the non-“super shoe” related upward time trend in the number of sub-2:08 times, utilizing a Cochrane-Orcutt transformation to correct for autocorrelation. This model shows that this new shoe technology is responsible for an additional roughly 24 additional sub-2:08 times per year. Estimated from this, we find a shoe-related time reduction of 1 minute and 31 seconds, or a 1.174% decrease in time.
Covid-19 Vaccination and Decreased Death Rates. (HS,UG,G,P)*
Introduction: In this talk, we examine the relationship between vaccination against COVID-19 and both the death rate from COVID-19 and the rate of COVID-19 spread. Our goal is determine if vaccination is associated with reduced death and/or spread of disease at the local level. Methods: This analysis was conducted at the county level in the state of Pennsylvania, United States of America, with data that were collected during the first half of 2022 from the state of Pennsylvania’s Covid Dashboard (COVID-19 Data for Pennsylvania (pa.gov). Results: This study finds the vaccines to be highly effective in preventing death from Corona virus, even at a time when there was a mismatch between the vaccines and the prevalent variants. Specifically, a 1% increase in vaccination rate was found to correspond to a 0.751% decrease in death rate (95% confidence interval (0.236%, 1.266%)). Given that, during this time period, the vaccines being used were not geared specifically toward the common variants at that time, we found no statistically significant relationship between disease spread and vaccination rate at the county level. Conclusions: These results support previous findings from across the world that Covid vaccination is highly efficacious in preventing death from the disease. Even during a time when vaccine design was not optimally matched with the prevailing strains, vaccination was found to reduce death rate. Hence, improving global vaccine availability is vitally important, in order to achieve necessary outcomes.
Predicting the outcome of karate matches using statistics. (HS,UG,G,P)*
This research project introduces a model that addresses the way competitors in Shotokan karate martial arts competitions are evaluated. This new model is a combination of multiple variables that determine the winner of a sparring competition in Shotokan karate. We constructed the model using the forward LR variety of logistic regression in order to predict the winner of these matches with a near 100% success rate. Due to some issues of multicollinearity, this model was then refined using the author’s knowledge of karate. We also begin to investigate applications of artificial intelligence (AI) in predicting the outcome of Shotokan karate matches.
Website: https://www1.villanova.edu/university/liberal-arts-sciences/programs/mathematics-statistics/faculty.html
Email: robert.styer@villanova.edu
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Bouncing balls, Jenga blocks, and infinity.
If a ball bounces an infinite number of times, will it bounce an infinite amount of time? We use some elementary physics of a bouncing ball to lead to geometric series. Next, can you pile up Jenga blocks so the top one overhangs the bottom one? Some elementary physics leads to the harmonic series. Along the way we enjoy some paradoxes of calculating with infinite series.
Website: https://www1.villanova.edu/university/liberal-arts-sciences/programs/mathematics-statistics/faculty.html
Email: klaus.volpert@villanova.edu
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A Dynamic Kelly Criterion.
We start with a gambling question: Suppose you had $100 starting capital and had the opportunity to repeatedly play a game with favorable odds. Let's say you have a 60% of winning each time. How much of your capital (in percent) would you bet each time?
Paradoxically, if we want to maximize our expected winnings, the math tells us that we should bet everything each time. But surely, that can’t possibly be the best strategy. Going all in each
time would almost certainly lead to ruin in the long run. . .In 1956, J. Kelly proposed a solution to this paradox, using a logarithmic utility function, resulting in a fixed percentage to bet, now well-known as “Kelly’s Criterion”.
Suppose now that we had the opportunity to play the game with the same favorable odds, but that there was a maximum payout M imposed, and a finite number N of games to reach it. We will show that in this case, Kelly’s criterion becomes “dynamic”, in other words, it becomes a function of both M and N. Furthermore, we will explore the impact of loosening the logarithmic utility assumption on a crucial metric of success in finance, the risk-reward ratio or “Sharpe ratio”.
Math and Magic of Financial Derivatives. (This talk can be tailored to any level.)
Are Financial Derivatives the "Engine of the Economy," as declared by Alan Greenspan, or "Weapons of Mass Destruction," as Warren Buffett views them? Over the last 30 years, financial derivatives have overtaken stocks and bonds as the investment vehicle of choice for many large investors. Derivatives are often behind the spectacular profits of investment banks as well as the mind-boggling losses (e.g. at Citigroup) that we read about in the papers. While CEO’s and hedge fund managers profit handsomely when things are going well, the losses are mostly born by shareholders and small investors. Pension funds, even school districts and townships have suffered from disastrous deals in derivatives. It is therefore no exaggeration to say that taxpayers and investors can no longer afford to not understand derivatives. So what are derivatives? Simply put, they are contracts between two parties that stipulate some cash flow over a certain period of time. The size of that cash flow depends on what happens to some underlying asset, such as a stock prices, interest rates, currency exchange rates or commodity prices. The uncertainty in the development of the underlying creates the key difficulty, which is to properly evaluate the price and the risk inherent in a derivative. In this talk I will give an overview of the three main methods to price derivatives and discuss the advantages and short-comings of each method:
The analytic method by Black and Scholes, based on PDE’s
The discrete approach by Cox-Ross-Rubinstein, based on binomial trees.
Monte-Carlo Methods, which average information obtained from simulating a large number of random walks of the underlying.
The Mathematics of Income Inequality. (This talk will be accessible to all with some knowledge of calculus.)
The Gini-Index based on Lorenz Curves of income distributions has long been used to measure income inequality in societies. This single-valued index has the advantage of allowing comparisons among countries and within one country over time. However, being a summary measure, it does not distinguish between intersecting Lorenz curves, and may not detect certain sociological and economic trends over time. We will discuss a new two-parameter model for the Lorenz curves. We will present theoretical and empirical evidence for this model, discuss its mathematical properties, and its potential to discern hidden trends.
Website: https://sites.psu.edu/mry3/
Email: mry3@psu.edu
☑︎ Face-to-face presentation
☑︎ Online presentation
This talk explores references to mathematics within episodes of “The Simpsons” and “Futurama.” While most serve as quick jokes or as “sight gags,” we also discuss an example of a mathematical proof that was created (and published) specifically for an episode of Futurama. A version of this talk is available to view through Delaware County Community College's STEAM Speaker Series at https://www.youtube.com/watch?v=kojDFSpMlFo.