HRUMC XXIII Hudson River Undergraduate Mathematics Conference Saint Michael’s College Colchester Vermont 2016-Apr-02 Schedule overview 8:30–9:50am Registration Dion Center 10:00–10:55am Parallel Sessions One St Edmunds and Jeanmarie Halls 11:05–12:15pm Welcome, Invited Address McCarthy Arts Center 12:25–1:30pm Lunch Dion Center 1:40–2:55pm Parallel Sessions Two St Edmunds and Jeanmarie Halls 3:00–3:25pm Coee and refereshments Dion Center 3:30–4:25pm Parallel Sessions Three St Edmunds and Jeanmarie Halls Wifi Use the network “SMC-Guest” with the password “PurpleKnights”. Abridged program A brief version of this program, not including the abstracts, is available at http://joshua. smcvt.edu/hrumc/hrumc2016short.pdf. If you need help Medical, fire, or police In an emergency call 911. For non-emergency security or safety issues, call campus security at 802-654-2911, or simply dial 2911 from any campus phone. There is a campus phone in each classroom. Classroom equipment To help with computers or other equipment there will be student assistants, wearing special tee-shirts, in or near each classroom. You can also call (802) 654-2959, or dial 2959 from the campus phone in each classroom. Rest rooms There are rest rooms on each floor of the academic buildings JEM and STE. There is a unisex rest room on the second floor of JEM, next to room 277. Welcome everyone! Welcome to the twenty third annual Hudson River Undergraduate Mathematics Conference (HRUMC). Whether you are a student at your first conference or an experienced speaker, we hope that you will find today beneficial, rewarding, and inspiring, and that you will make new friends. Our aim is to build an atmosphere that includes the message, “We are glad that you are joining the mathematics community!” This conference features fifteen minute talks by students and faculty, and a longer plenary address. Each year, we invite students and faculty from universities and colleges in New York and New England to send abstracts for the short talks. These describe research projects, independent study projects, or any other independent work by students and faculty. If you are a first time attendee then start by studying the short talks schedule to find some that grab your interest. Each of these is marked as Level 1 or Level 2: the Level 1 talks are accessible to everyone while Level 2 talks are aimed at faculty and advanced students. Note that each session has a Chair, who keeps all presentations strictly to the schedule. This means that you can easily move from room to room to see talks — you know that each talk that you attend will end on time, and each next one will start when it says it will. If you are a first time presenter then we especially say, “Welcome!” Giving a presentation can be daunting, but is also energizing. The session Chair will be able to help with any questions that you have, including any technology questions. The first HRUMC was held at Siena College in 1994, and now it is an annual tradition. For information about previous meetings, pictures from this year’s conference, as well as information about next year’s conference you can check out the web site: http://www.skidmore.edu/hrumc. Presenter or attendee, we hope that you enjoy the HRUMC and that you will you will learn a great deal. And, if you can, we hope to see you again, sharing your work, at next year’s conference, hosted by Westfield State University, on Saturday April 8th, 2017. This conference would not be possible without the generous financial support provided by the Oce of the Vice President for Academic Aairs at Saint Michael’s College, and by the Depart- ments of Mathematics and Computer Science. Support also comes from the NASA-VT Space Grant Consortium, and from the Pi Mu Epsilon national mathematics honor society. We also thank all of the student and faculty volunteers who contributed their time, talents, energy, and enthusiasm. HRUMC Committee Site Arrangements Lauren Childs, Williams College Jim Hefferon Paul Friedman, Union College Lloyd Simons Mohammad Javaheri, Siena College Jesse Johnson, Westfield State University Emelie Kenney, Siena College Joe Kirtland, Marist College Allison Pacelli, Williams College Alejandro Sarria, Williams College David Vella, Skidmore College Edward Welsh, Westfield State University William Zwicker, Union College Institutional Greeting and Invited Address Welcoming Remarks: Dr. Karen Talentino, VPAA, Saint Michael’s College Introduction of the Speaker: Celsey Lumbra, SMC ’16 Keynote address: The P vs. NP Problem Scott Aaronson MIT, University of Texas at Austin Abstract I’ll discuss the status of the famous P ?= NP problem in 2016, oering a personal perspective on what it’s about, why it’s important, why many experts conjecture that P != NP is both true and provable, why proving P != NP is so hard, the landscape of related problems, and crucially, what progress has been made in the last half-century. I’ll say something about diagonalization and circuit lower bounds; the relativization, algebrization, and natural proofs barriers; and the recent works of Ryan Williams and Ketan Mulmuley, which (in dierent ways) hint at a duality between impossibility proofs and algorithms. Biography Professor Aaronson holds a Ph.D. in Computer Science from University of California, Berkeley. He is currently Associate Professor of Electrical Engineering and Computer Science at the Massachussets Institute of Technology. Starting in July he will be the David J. Bruton Jr. Centennial Professor of Computer Science at the University of Texas at Austin. He has an international reputation as an expert in the Theory of Computation and Complexity Theory. Scott studies the fundamental limits on what can be eciently computed in the physical world. This en- tails studying quantum computing, the most powerful model of computation that we have. His work has included limitations of quantum algorithms in the black-box model; the learnability of quantum states; quantum proofs and advice; the power of postselected quantum computing and quantum computing with closed timelike curves; and linear-optical quantum computing. PARALLEL SESSIONS ONE Abstract Algebra I JEM 380 Chair: Blair Madore 10:00-10:15 The Amazing Connections Between Groups and Symmetric Subgroups (Level 2) Henry Young (Wheaton College) The history of group theory started with the development of per- mutation groups. However, our mathematical predecessors did not know they were dealing with such an essential algebraic structure. Arthur Caley proved that all groups are isomorphic to a sub- group of a permutation group. Thus, permutations groups contain the structure of all of the fa- mous groups that we know and love. By looking at these isomorphisms and their applications to the theory of groups, we may as well rethink what it is meant to be a group. 10:20-10:35 How Commutative are Dihedral Groups? (Level 1) Amanda Peterson (SUNY Potsdam) We will show the probability of any two randomly selected elements of a dihedral group com- muting through the use of Clifton, Guichard, and Keef’s work. We will explain how many and which elements in a dihedral group commute and use that information to compute the probability of any two elements commuting in a product of dihedral groups. 10:40-10:55 Subnormal Subgroups of M-Groups (Level 2) John McHugh (University of Vermont) A wealth of structural information about a finite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters — those induced from a linear char- acter of some subgroup — since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the class of M-groups, those groups all of whose irreducible characters are monomial. A classical the- orem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solv- able group can be embedded as a subgroup of an M-group. After discussing results related to M- groups, we will construct explicit families of solvable groups that cannot be embedded as subnor- mal subgroups of any M-group. Analysis JEM 389 Chair: Andrew McIntyre 10:00-10:15 The Equality in Young’s Inequality (Level 1) James Vees (Hamilton College) Young’s Inequality is an important result in functional analysis that relates the areas enclosed by a function and its inverse. This presentation will oer two dierent calculus-based proofs of the conditions that imply equality in Young’s Inequality. The talk will also give an example of an integral that becomes easy to compute if we use this result. 10:20-10:35 Limiting Distributions for Topological Markov Chains with Holes (Level 2) Mark F. Demers (Fairfield University), Chris Ianzano (Stony Brook University), Philip Mayer (Fairfield University), Peter Morfe (Cooper Union), Elizabeth C. Yoo (Columbia University) Open dynamical systems are models of physical systems in which mass or energy is allowed to escape from the system. Central questions involve the existence of conditional equilibria (measures that are invariant under the dynamics conditioned on non-escape) which can be realized as limiting distributions under the dynamics of the open system. We study this problem in the context of topological Markov chains, a class of symbolic dynamical systems with a wide variety of appli- cations. Under a combinatorial condition on the Markov chain, we study transfer operators as- sociated with positive recurrent potentials and prove the existence of a spectral gap on a natural function space. This implies the existence (and uniqueness in a certain class) of limiting distribu- tions which represent conditional equilibria for the open system. We also prove a relation between the escape rate from the system and the entropy on the survivor set (the set of points that never enters the hole).