Seminars archive

The excedance algebra, given by the noncommutative quotient Z<a,b>/(ba-a-b-ab), is motivated by a recurrence for a permutation statistic. We examine this algebra and a related matrix construction known as the excedance matrix. We then consider properties of the box polynomials, which arise from applying the finite difference operator to monomials and whose coefficients are the entries of the rightmost column of the excedance matrix. Evaluating these polynomials yields a variety of identities involving set partition enumeration. We extend these identities using restricted growth words and a new operator called the fast Fourier operator. This talk is based on joint work with Richard Ehrenborg and Dustin Hedmark.

An old open problem in Euclidean Ramsey theory asks if the points in $E^3$ are coloured in red and blue, does there exist either a red pair of points at unit distance or six collinear blue points separated by unit distances? After a short survey of related problems, I will give a positive answer to this problem and outline the proof. This talk is based on a joint work with Sergei Tsaturian.

mall, low-cost, low-power sensing devices are transforming science and society, making it possible to transform the deployment of new applications and allow the collection and analysis of data far beyond the scale of what was previously possible. A commercially available embedded platform, such as Arduino, helps developers to prototype embedded systems easier, faster, and in a fun way. One of Arduino's fun projects is autonomous car, where a racing car must be able to move without a remote control. The car should be able to move forward, avoid any obstacles, stop and reverse if necessary, without any user input apart from turning it on. In this talk, we are going to learn how to control the electronic speed control and the steering servo of the car using an Arduino.

Rachel Pries (4pm) \\ Title: Galois action on homology of Fermat curves \\ Abstract: We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren. \\ Gopal Prasad (5:15pm) \\ Title: Weakly commensurable Zariski-dense subgroups of semi-simple groups and isospectral locally symmetric spaces \\ Abstract. I will discuss the notion of weak commensurability of Zariski-dense subgroups of semi-simple groups. This notion was introduced in my joint work with Andrei Rapinchuk (Publ. Math. IHES 109(2009), 113-184), where we determined when two Zariski-dense S-arithmetic subgroups of absolutely almost simple algebraic groups over a field of characteristic zero can be weakly commensurable. These results enabled us to prove that in many situations isospectral locally symmetric spaces of simple real algebraic groups are necessarily commensurable. This settled the famous question "Can one hear the shape of a drum?" of Mark Kac for these spaces. The arguments use algebraic and transcendental number theory.

Modern search engines have made dramatic progress in answering many user questions, especially about facts, such as those that might be retrieved or directly inferred from a knowledge base. However, many other more complex factual, opinion or advice questions, are still largely beyond the competence of computer systems. For such information needs users still have to dig into the "10 blue links" of search results and extract relevant information. As conversational agents become more popular, question answering (QA) systems are increasingly expected to handle such complex questions and provide users with helpful and concise information. In my dissertation I develop new methods to improve the performance of question answering systems for a diverse set of user information needs using various types of user-generated content, such as text documents, community question answering archives, knowledge bases, direct human contributions, and explore the opportunities of conversational settings for information seeking scenarios.

Toric varieties are ubiquitous in algebraic geometry. They have a rich combinatorial structure, and give the simplest examples of log Calabi-Yau varieties. \\ We give a simple criterion for characterizing when a log Calabi-Yau pair is toric, which answers a case of a conjecture of Shokurov. This is joint work with James McKernan, Roberto Svaldi, and Runpu Zong.

Testing integers for primality and factoring large integers is an extremely important subject for our daily lives. Every time we use a credit card to make online purchases we are relying on the difficulty of factoring large integers for the security of our personal information. Similar encryption methods are used by governments around the world to protect their classied information, stressing the importance of the subject of primality testing and factoring algorithms to both personal and national security. Elementary number theory has been a key tool in the foundation of primality testing and factoring algorithms, specifically the work of Euler and Fermat, whose developments on modular arithmetic give us key tools that we still use today in the more complex primality tests and factoring methods. More recently people have used deeper ideas from geometry, namely elliptic curves, to develop faster tests and algorithms. In this thesis we continue this trend, and develop new primality tests that utilize previous theory of elliptic curves over nite elds. The primary point is that the points on these curves form a special group, which breaks down when working over Z/NZ, when N is not prime. Our theorems make use of the work of Kubert, Hasse, Mazur, and many more to yield a primality test that gives no false positives.