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Counting points on algebraic curves over finite fields has numerous applications in communications and cryptology, and has led to some of the most beautiful results in 20th century arithmetic geometry. A natural generalization is to count the number of points over prime power rings, e.g., the integers modulo a prime power. However, the theory behind the latter kind of point […] |
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We review a beautiful 17th century result by the philosopher Rene Descartes: a univariate real polynomial with t monomial terms has no more than t-1 positive roots. We then see […] |
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TOPIC: The Mathematics of Information We are surrounded by information. Words in books, ones and zeros in computers, mathematical equations, and DNA sequences are all examples of information, but can we say something more about it? In this workshop, we will learn about the mathematics of information, see how it is related to concepts from […] |
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Markov chains are widely used throughout mathematics, statistics, and the sciences, often for modelling purposes or for generating random samples. In this talk I’ll discuss a different, more recent application of Markov chains, to developing distributed algorithms for programmable matter systems. Programmable matter is a material or substance that has the ability to change its […] |
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If $F$ is a finite field and $d$ is a positive integer relatively prime to $|F^\times|$, then the power map $x \mapsto x^d$ is a permutation of $F$, and so is called a power permutation of $F$. For any function $f: F \to F$, and $a, b \in F$, we define the differential multiplicity of $f$ with respect to […]
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I will discuss joint work with Jim Hoste, where we prove that a unique folded strip of paper can follow any polygonal knot with odd stick number. In the even […] |
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In 2007, Dr. Maria D'Orsogna learned of proposed oil activities in her home region of Abruzzo, Italy. Century-old wineries were to be uprooted to build clusters of oil wells, refineries […] |
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Clustering in image analysis is a central technique that allows to classify elements of an image. We describe a simple clustering technique that uses the method of similarity matrices, and an algorithm in which a collection of image elements is treated as a dynamical system. Efficient clustering in this framework is achieved if the dynamical system admits […] |
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Mathematicians like to count things. Often in very complicated and fancy ways. In this talk I will explain how we can use quantum Airy structures -- an abstract formalism recently proposed by Kontsevich and Soibelman, underlying the Eynard-Orantin topological recursion -- to count various interesting geometric structures. Quantum Airy structures can be seen as a […]
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This triple-header of topology talks will include three speakers: First, Hyeran Cho from The Ohio State University will speak about Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams. In this talk, we show that the dual (1, 1)-diagram of a (1, 1)-diagram (a.k.a. a two pointed genus one Heegaard diagram) D(a, 0, 1, […] |
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A fascinating fact on mathematics is that there are many interesting connections between seemingly different mathematical disciplines. In this talk, I will present a surprising formula counting integral points on […] |
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We review some recent developments in the study of biquandle brackets and other quantum enhancements.
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The bridge distance and the topological index are measures of the complexity of the bridge splitting of a knot. In 2016, Johnson and Moriah gave a formula for the bridge distance of the canonical bridge sphere of a knot in a highly twisted plat projection in terms of the height and the width of the […] |
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Silica-based glasses are increasingly becoming vital components in our current technology, from optical data transmission lines, to electronics, to optical lenses, to smartphone screens. These materials are inherently brittle and […] |
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Joan Ponce Purdue University Abstract: One of the main challenges of mathematical modeling is the balance between simplifying assumptions and incorporating sufficient complexity for the model to provide more accurate and reliable outcomes. For mathematical simplicity, many commonly used epidemiological models make restrictive modeling assumptions. Although models under such assumptions are capable of producing useful insights into […] |
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At the turn of the twentieth century, physicist Henri Bénard heated a shallow plate of fluid from below. For temperatures above a critical value, the fluid’s evenly heated state became unstable as thermal convection took hold; heated fluid rose in localized areas while cooler fluid fell nearby. The rising and falling fluid created hexagonal convection cells, […] |
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Domination in graphs has been an important and active topic in graph theory for over 40 years. It has immediate applications in visibility and controllability. In this talk we will […] |
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