The Mathematics and Statistics department had another great year. Professor Susan Loepp was awarded the 2012 Haimo Award for Distinguished College or University Teaching by the Mathematical Association of America, a national teaching prize. Professor Steven Miller was awarded tenure and promoted to Associate Professor. Other accomplishments of our faculty are listed below. Our student team placed in the top ten and received an Honorable Mention out of 460 teams from 572 participating colleges and universities in the December 2011 William Lowell Putnam Mathematical Competition. This year we had 67 students sign up as new mathematics majors, our largest number so far. We also admitted 33 students to our SMALL summer research program, our largest number yet. This is the last year of Professor Silva as chair and the start of Professor Johnson’s tenure as chair of the department.
We hired a new statistician, Assistant Professor Wendy Wang, who comes to us from Pennsylvania State University. We also appointed Mark Mixer and Matt Gardner Spencer as new Visiting Assistant Professors of Mathematics. Four members of our department were on leave this past year. Ollie Beaver spent the fall at Williams. Edward Burger spent the year at Baylor University. Bernhard Klingenberg spent the fall at the University of Salzburg in Austria and the spring at the Institute of Statistics at the TU Graz in Austria. Susan Loepp spent the year at Williams. Steven Miller spent the year at Smith and Mount Holyoke. Ollie Beaver returned to teaching this spring. We look forward to the return of Ed Burger, Bernhard Klingenberg, Susan Loepp and Steven Miller in fall 2012. Next year Professors Colin Adams, Thomas Garrity and Mihai Stoiciu will be on leave for the year, and Professor Ed Burger will be on leave in the spring.
We are very proud of the accomplishments of our majors. The Rosenburg Prize for outstanding senior was awarded to Liyang Zhang ’12. Erik Levinsohn ’12 and Niralee Shah ‘12 received the Goldberg Prize for the best colloquium; this year the department awarded honorable mention in the Goldberg prize to Patrick Aquino ’12, Carolyn Geller ’12, David Gold ’12, Stephanie Jensen ’12, Andrew Nguyen ’12, Sidney Luc Robinson ’12, Tarjinder Singh ’12, and Matthew Staiger ‘12. Patrick Aquino ‘12 was awarded the Morgan Prize for teaching, and Gregory White ‘12 was awarded the Morgan Prize in applied mathematics. Hannah Hausman ’12 received the Robert M. Kozelka Award for outstanding students of statistics. The Witte Problem Solving Prize went to the members of this year’s Putnam Team: Carlos Dominguez ’13, Jared Hallett ’14 and Liyang Zhang ’12. Jared Hallett ’14 first, and Craig Corsi ’14 and Yang Lu ’14 second, were awarded the Benedict Prize for outstanding sophomore. Connor Stern ‘12 was awarded the Wyskiel Prize for a student who chooses a career in teaching. Finally, Liyang Zhang ’12 was awarded the colloquium attendance prize for seniors. Tara Deonauth ‘13 and Christina Knapp ’13 were awarded the colloquium attendance prize for non-seniors.
The members of our student advisory board, SMASAB (Students of Mathematics and Statistics Advisory Board), were Alexander Greaves-Tunnell ’13, Jared Hallett ’14, Hannah Hausman ’12, Christina Knapp ’13, Stephanie Jensen ’12, and Chansoo Lee ’12. They provided sage advice including help in our hiring process, in addition to organizing the department’s ice cream socials. This year one of our majors, Liyang Zhang ’12 was awarded Honorable Mention in the National Science Foundation graduate fellowships in mathematics.
In summer 2011, Colin Adams attended and spoke at the meeting on “Knots: Form and Function” at the Centro De Research di Giorgio in Pisa, Italy. Over the academic year, 2011-12, he gave a variety of talks. In particular, Tom Garrity and Colin Adams performed and recorded the Derivative vs. Integral Debate at Williams Family Days, with President Falk as moderator. The DVD is now on sale through the Mathematical Association of America. During spring break, Adams and Tom Garrity traveled to Wisconsin to give their humorous math debates at a variety of schools.
Adams published four papers, three with students from the SMALL program. He also worked with Tom Crawford ’12 who wrote a thesis on finding knots with disjoint totally knotted Seifert surfaces. He continued to serve as a co-principal investigator on a grant that supports undergraduate math conferences around the country. In the spring he taught knot theory to a record 45 students. In summer 2012, he will work with six students and a Korean postdoc on original research as part of the SMALL program.
In summer 2011, Professor Ollie Beaver taught in and coordinated the mathematics component of Williams Summer Science Program. During the academic year, Beaver continued her involvement in the Quantitative Studies program at Williams. She was again chair of the Winter Study Committee. In January Beaver attended the Joint Mathematics Meetings in Boston and was an invited panelist on the National Science Foundation Panel for Graduate Fellowships in Mathematical Sciences.
Professor Elizabeth Beazley has now completed her second year at Williams, and she has continued to find the Williams math department an exciting place to grow her research program. Following a workshop on “Algebraic Combinatorixx” at the Banff International Research Station in Alberta, Canada, she has initiated several research collaborations in the areas of quantum and equivariant Schubert calculus, including one which grew out of a Class of 60s visit to Williamstown in March 2012. She had one paper appear this year in the January edition of the Journal of Algebra titled “Affine Deligne-Lusztig varieties associated to additive affine Weyl group elements.” In addition, her paper “Maximal Newton polygons via the quantum Bruhat graph” was accepted to the Formal Power Series and Algebraic Combinatorics conference to be held in Nagoya, Japan in July 2012, resulting in a conference proceedings publication in Discrete Mathematics and Theoretical Computer Science. She was also especially pleased to take eight Williams undergraduates to two research conferences in September 2012 at Brown University and Smith College.
Professor Edward Burger spent the academic year 2011–2012 at Baylor University as Vice Provost for Strategic Educational Initiatives and Visiting Professor of Mathematics. He also served as an educational consultant for both The University of Texas at Austin and Winston-Salem State University. In the summer of 2011, Burger was a faculty member of Williams’ Summer Humanities and Social Sciences Program.
Burger’s appearance on the Today Show series “The Science of the Winter Olympics” (co-sponsored by the NSF), earned him a 2011 Telly Award (and the entire series won an Emmy Award). Also in 2011 he received a Distinguished Achievement Award from the Association of Educational Publishers, recognizing Fuse—his mathematics video textbooks for mobile devices, published by Houghton Mifflin Harcourt. In 2012 Burger was featured in three episodes of the NBC-TV/NSF series “The Science of NHL Hockey” (Vectors, Statistics & Averages, and Kinematics) shown on the Today Show as well as throughout the NHL 2012 season.
In 2012 Burger published A Generalization of a Theorem of Lekkerkerker to Ostrowski’s Decomposition of Natural Numbers with undergraduate co-authors David C. Clyde, Cory H. Colbert, Gea Hyun Shin ’11, and Zhaoning Wang ’11 in Acta Arithmetica 153 (2012) pp 217–249. With Michael Starbird, he completed a book on teaching, learning, and creativity entitled, The 5 Elements of Effective Thinking, to be published by Princeton University Press. Burger and Starbird also completed work on the fourth edition of their textbook, The Heart of Mathematics: An invitation to effective thinking. Also in 2012, he starred in Math on the Spot, the first math video app developed exclusively for the Apple iPhone, published by Houghton Mifflin Harcourt.
Burger delivered over 40 addresses in the past year including five keynote addresses at MAA and NCTM Sectional Meetings, one of the keynote addresses at the MAA 2011 MathFest in Lexington, KY, the closing keynote address at the 2012 NCTM Annual Conference in Philadelphia, PA, the 2011 Robert L. Moore Lecture at The University of Texas in Austin, and the “Charge” address at the Phi Beta Kappa induction ceremony at Baylor University. He also addressed all first-year cadets at West Point Academy at the opening of the 2011–2012 academic year, and spoke at the Benjamin Franklin Society in Vero Beach, FL.
Professor Satyan Devadoss had another year filled with wonder. His research is in the areas of discrete topology and geometry, on which he gave several invited talks from coast-to-coast, along with giving a two-day short-course on this subject at the national meeting for mathematicians in Boston. Several of his papers appeared in print this year, including one on the shape of associativity (Canadian Notes), one on the structure of trees (European Conference on Computational Geometry), and one on the nature of infinity (Esopus Magazine).
On the Williams front, Professor Devadoss gave several talks, to alums, to students, and to general audiences. With students, he supervised a SMALL research group on Geometric Origami in summer 2011, taking them to Toronto for a computational geometry conference. Devadoss also advised Brian Li’s senior thesis on Polygonal Linkages, taught his first tutorial on Phylogenetics, and offered a winter study course on Visualization (of the Williams curriculum). With faculty, he served as an official mentor, along with participating in his first Oakley center seminar (on Charles Taylor’s A Secular Age, the hardest reading imaginable). He’s looking forward to the next year.
Professor Richard De Veaux continued his work in data mining and gave a variety of talks, invited talks, keynote addresses and workshops on teaching and data mining throughout the United States and Europe. He advised Ville Satopӓӓ on his thesis. He was elected to the Board of Directors of the American Statistical Association in May 2012 for a three year term.
Professor Thomas Garrity has continued his research in number theory. His article, “Using Mathematical Maturity to Shape Our Teaching, Our Careers and Our Departments” appeared in the Notices of the American Mathematical Society. His DVD debate “Derivative vs. Integral: The Final Smackdown”, with Colin Adams and moderated by Adam Falk, was released through the Mathematical Association of America. In July 2011 he spoke to students at the Hampshire College Summer Studies in Mathematics. In September, he gave a colloquium at Bennington College. In March 2012, with Colin Adams, he gave various versions of their debates at the University of Wisconsin at Madison, at Ripon College and at the Madison Area Technical College. He gave a colloquium and a faculty seminar at TCU in April 2012. In May 2012, Colin Adams and he gave a debate at Framingham State University. In the summer of 2011 he led a group of seven students (Krishna Dasaratha from Harvard. Laure Flapan from Yale, Chansoo Lee ‘12 from Williams, Cornelia Mihaila from Wellesley, Nicholas Neumann-Chun ‘13 from Williams, Sarah Peluse from Lake Forest and the University of Chicago and Matt Stoffregen from the University of Pittsburgh) in SMALL. A number of papers will eventually result from this work. His book Algebraic Geometry: A Problem Solving Approach, with co-authors Richard Belshoff, Lynette Boos, Ryan Brown, Jim Drouihlet, Carl Lienert, David Murphy, Junalyn Navarra-Madsen, Pedro Poitevin, Shawn Robinson, Brian A. Snyder and Caryn Werner has been accepted for publication by the American Mathematical Society. This book is both innovative in its presentation of algebraic geometry and in how it was written (explaining in part the large number of co-authors). He learned quite a bit from his two thesis students, Stephanie Jenson ‘12 and Noah Goldberg ‘12. He has joined the steering committee for the Park City Mathematics Institute. Finally, he continues being the co-director of Williams’ Project for Effective Teaching (Project PET).
Professor Stewart Johnson continues his research in dynamical systems, modeling, and optimal control with a focus on systems that exhibit continuous and discrete behavior. He is currently developing computational methods for optimal control problems.
Professor Johnson remains active in the college-wide Quantitative Studies program which provides early identification and intervention for students with quantitative challenges.
Professor Bernhard Klingenberg spent the fall as a Visiting Professor at the University of Salzburg, Austria and the Spring as a Visiting Professor at the Institute of Statistics at the TU Graz, Austria. Over the year, he was invited to give short courses on Applied Statistics, Categorical Data Analysis, and Mathematical Statistics and he organized a special course on Ordinal Categorical Data. He presented his newest methodological results in the analysis of correlated binary data at the Austrian Statistics Days 2011. In addition, he offered consulting services to medical faculty at the Medical University of Graz, Austria. His paper on simultaneous inference for proportions appeared in the journal Computational Statistics & Data Analysis.
Professor Susan Loepp was a recipient of the 2012 Mathematical Association of America’s Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics. In January, she accepted the award and gave a talk at the National Mathematics meetings in Boston. In addition, in January, 2012, Loepp began a 5-year term as an associate editor for the Mathematical Monthly.
In summer, 2011, Loepp served as the SMALL director and advised the SMALL 2011 Commutative Algebra group. The group members, Ji Won Ahn ‘12, Elizabeth Ferme, Feiqi Jiang, and Gian Thi Huong Tran proved several original results in commutative algebra. They have written a manuscript based on their results and have submitted the paper to a refereed research journal in mathematics. Ji Won and Feiqi presented their results at the undergraduate research poster session at the National Mathematics meetings, and won an “Outstanding Presentation” award by placing in the top 15% of the poster presentations. In the spring of 2012, the paper written by the 2009 SMALL Commutative Algebra group, Nick Arnosti ‘11, Rachel Karpman, Caitlin Leverson, and Jake Levinson ‘11, appeared in the Journal of Commutative Algebra.
Last summer Professor Steven Miller supervised 9 summer REU students in the SMALL program. This led to 3 accepted papers, 3 submitted papers, 4 works in progress, and 25 talks by his students at various conferences, including an honorable mention at the Young Mathematicians Conference at Ohio State (his former thesis student, Jake Levinson ’11, received first place there for his thesis work).
Miller was on sabbatical last year, and is glad to be back. He continued his research in number theory, random matrix theory and probability, had 8 papers appear and several more accepted, and gave 19 talks. He also designed some new classes at Smith and Mount Holyoke to bring back to Williams; one of these, Advanced Applied Linear Algebra, will be taught in the fall.
Miller continues to be active in educational outreach activities. His math riddles page, http://mathriddles.williams.edu, is one of the top hits when googling ‘math riddles’, and is used by teachers in classes from K-12 all over the world. He gave a course on cryptography and Benford’s law to junior high and high school teachers in the Teachers As Scholars program, and gave talks at junior high and high schools.
Professor Frank Morgan has a new blog at the Huffington Post. He is continuing his study of minimal surfaces and densities with a number of collaborators and his undergraduate research Geometry Group. A joint paper with eight students on “Optimal Pentagonal Tiling” appeared in Notices of the American Mathematical Society, May 2012.
Professor Allison Pacelli was on maternity leave during the fall semester, and was thrilled to welcome her new son Andy into the world last July. Pacelli returned to campus this spring, teaching a senior seminar in Algebraic Number Theory and a tutorial in Galois Theory.
Professor Cesar Silva completed his third year as chair of the department. He continued his research in ergodic theory. He supervised the thesis of Praphruetpong (Ben) Athiwaratkun ‘12, and the research in the spring of Christina Knapp ’13 concerning a survey paper on the uncountability of the unit interval. Silva taught Calculus II in the fall, and Ergodic Theory in the spring, where he used his book on the subject. In December he participated in the MathBlast Williams workshop for 10th graders from Mount Greylock Regional High School and BART where he presented “Fractals and Natural Shapes.” He published a paper in Colloquium Mathematicum based on research with his students and had another paper submitted for publication. He was invited to present a lecture at a conference in memory of Jal Choksi at McGill University in June.
In the summer he supervised a research group in ergodic theory whose members were Ke Cai, Jared Hallett ’14, Lucas Manuelli and Sun Wei.
Professor Mihai Stoiciu taught “Groups and Characters” and “Differential Geometry” during the Fall Semester and two sections of “Applied Real Analysis” during the Spring Semester of the Academic Year 2011-2012. Also, Stoiciu supervised Gregory White ‘12, who wrote an undergraduate thesis titled “Stochastic Calculus and Applications to Mathematical Finance” and Liyang Zhang ‘12 whose thesis was titled “Spectral Theory for Matrix Orthogonal Polynomials on the Unit Circle.”
During the year, Stoiciu continued his research on spectral properties of random and deterministic operators, working with collaborators from UK and US. His paper “Some Spectral Applications of McMullen’s Hausdorff Dimension Algorithm,” written with collaborators from Durham University, UK, was accepted for publication in the journal “Conformal Geometry and Dynamics.” He was invited to present his research at the AMS Special Session on “Spectral Theory” in Tampa, FL and was an invited participant in the “Arizona School of Analysis and Mathematical Physics” at the University of Arizona in Tucson, AZ.
In October 2011 Stoiciu gave the plenary address at Mid-Hudson Mathematics Conference for Undergraduates at Bard College, Annandale-on-Hudson, NY. At Williams College, Stoiciu gave two talks at the Osher Lifelong Learning Institute in the “Frontiers of Science” Program, a talk for high school students at the Williams MathBlast 2011, and a faculty seminar on his recent research in spectral theory.
Mathematics Colloquia
Kobi Abayomi, Georgia Institute of Technology
“Statistics for Re-identification in Networked Data Models”
Colin Adams, Williams College
“The Geometric Degree of Knots”
“Triple Crossing Number of Knots”
Ivana Alexandrova, University of Albany
“Aharonov-Bohm Effect in Resonances of Magnetic Schrodinger Operators with Potentials with Supports at Large Separation”
Olga R. Beaver, Williams College
“Gleason’s Theorem Revisited”
Elizabeth Beazley, Williams College
“An Introduction to Quantum Schubert Calculus”
Ivor Cribben, Columbia University
“Dynamic Connectivity Regression: Determining State-Related Changes in Brain Connectivity”
Satyan Devadoss, Williams College
“Phylogenetic Trees and Origami Foldings”
“Packings, Partitions, and the Water Cube”
“Collapsing Spaces of Trees”
Thomas Garrity, Williams College
“On the Topological Hypothesis for Phase Transition in Statistical Mechanics”
Gyo Taek Jin, Korean Advanced Institute of Science and Technology
“Quadrisecant Approximation of Knots”
Stewart Johnson, Williams College
“The Return of the Banana Path”
Christine Kohnen, Duke University
“Disclosure Limitation: The Statistical Game of Protecting Waldo”
Susan Loepp, Williams College
“Completions of Local Rings”
“Completions and Complete Intersections”
Bethany McLean ‘92, famously exposed Enron, now at Vanity Fair and TV personality
“Why It’s Smart to be a Math Major Even if You’re Not Good at Math”
Christopher McMahan, University of South Carolina
“Topics in Heterogeneous Group Testing”
Steven Miller, Williams College
“Biases: From Benford’s Law to Additive Number Theory Via the IRS and Physics”
Luke Miratrix, University of California, Berkeley
“Adjusting Treatment Effect Estimates by Post-Stratification in Randomized Experiments”
Frank Morgan, Williams College
“Tilings, Densities, and Asia”
“Existence of Isoperimetric Regions with Density”
“Different Definitions of the Area Enclosed by a Closed Curve in Rn”
“Stokes Theorem”
“Soap Bubbles and Mathematics”
Allison Pacelli, Williams College
“Class Numbers & Class Field Towers”
Jennifer Quinn ‘85, University of Washington, Tacoma
“Digraphs and Determinants”
Stephen Sawin, Fairfield University
“South-Pointing Chariot: An Invitation to Geometry”
Jeffrey Schenker, Michigan State University
“The Problem of Quantum Diffusion”
Cesar Silva, Williams College
“Mixing Notions in Ergodic Theory”
“Sub-Sigma-Algebras of Standard Spaces in Ergodic Theory”
Aaron Smith, Stanford University
“Markov Chain Monte Carlo and (nearly) Exact Tests”
Rebecca Steorts, University of Florida
“Bayes and Empirical Bayes Benchmarking for Small Area Estimation”
Mihai Stoiciu, Williams College
“Spectral Applications of McMullen’s Hausdorff Dimension Algorithm”
Kaisa Taipale, St. Olaf College
“A Conjectural Quantum Puzzle Rule“
Qing Wang, Pennsylvania State University
“Topics in U-Statistics and Risk Estimation”
Mathematics Student Colloquia by 2012 Graduates
Ji Won Ahn ‘12 “Mathematics of Doodling”
Patrick Aquino ‘12 “The Jordan Curve Theorem: A Simple Proof”
Praphruetpong Athiwaratkun ‘12 “Local Solvability: The Lewy Example”
Eileen Becker ’12 “The abc Conjecture and Its Interesting Consequences”
Victoria Borish ’12 “Knotty Problems in Quantum Theory”
Luke Breckenridge ’12 “Thinking Outside the Box”
Hayley Brooks ’12 “A Drunk and His Dog: Cointegration and Developing Successful Investment Strategies“
Jack Chen ’12 “Bold Play: Go Big or Go Home”
Felipe Colina ’12 “Re-focusing with Fourier”
Thomas Crawford ’12 “Tropical Geometry”
Austin Davis ’12 “Little Boat, Big Trouble: The Jealous Husbands Problem”
Laura Dos Reis ’12 “Modeling the Spread of HIV in a Population with an Imperfect Vaccine”
Walter Filkins ‘12 “Encoding and Decoding Fractals”
Westcott Gail ‘12 “Newton’s Method and Beyond: From Convergence to Super Hayley”
Carolyn Geller ’12 “The Mathematics of Choosing a Spouse”
David Gold ’12 “Spacetime and Complex Geometry”
Noah Goldberg ’12 “A Priests’ Puzzle (and its Mathematical Solution)“
Benjamin Halbower ’12 “Chaos: Examining the Lorentz Water Wheel”
Hannah Hausman ’12 “The Kakeya Needle Problem”
Stephanie Jensen ’12 “Hyperbolic Geometry and Gieseking’s Manifold”
Samuel Jonynas ’12 “Sylow’s First Theorem for Finite Groups”
Aayush Khadka ’12 “Benford’s Law and Its Applications”
Murat Kologlu ‘12 “Functions that Blow Up: Several Variable Complex Analysis and Harthogs’ Theorem”
Josephat Koima ’12 “Resultants of Polynomials”
Andrew Kung ’12 “Are Your Friends Making You an Alcoholic?”
Pawel Langer ’12 “15-Puzzle Impossible?!”
Chansoo Lee ’12 “Finding Roots of Polynomials”
Erik Levinsohn ‘12 “Dude, Where’s My Convex Hull?”
Brian Li ‘12 “Carpenter’s Rules and Convexifying Polygons with Visibilities”
Timothy Lorenzen ‘12 “Do Traveling Salesmen Like Minesweeper?”
Shuai Ma ’12 “Irrationality of Pi”
Michael Mara ‘12 “True Lovers are Intrinsically Linked, But What About Graphs?”
Andrew Nguyen ’12 “Lost in Manhattan: The Problem of Self-Avoiding Walks”
Quoc Anh Nguyen ’12 “A Beautiful Mind Meets The Beautiful Game: A Game Theoretic Approach to Soccer”
Theo Patsalos-Fox ’12 “Sidestepping Difficult Differential Equations with Picard’s Theorem”
Katherine Rieger ’12 “Hip Hip Beret!”
Sidney Luc Robinson ’12 “Applications of Generating Functions”
David Samuelson ‘12 “The Burnside Counting Theorem”
Matthew Schuck ‘12 “Fractals and Hausdorff Dimension”
Niralee Shah ’12 “Hearing the Shape of a Drum”
Kevin Shallcross ’12 “Group Theoretical Approaches to Understanding the Rubik’s Cube”
Shara Singh ’12 “Mathematics of Sudoku Puzzles”
Tarjinder Singh ’12 “A Discussion of Markov Chains and Hidden Markov Models With Application to Gene Prediction and Motif Discovery”
Matthew Staiger ’12 “Sticking it to Negami’s Upper Bound on Stick Number for Knots”
Connor Stern ‘12 “Buffon’s Needle Problem”
Katherine Stevenson ‘12 “Shape Ninja: The Art of Slicing Polyhedra”
Daniel Tessier ’12 “Using Straight Lines to Divide the Projective Plane”
Philip Vu ’11 “The In-Between Worlds: Understanding Non-integer Dimensions with the Hausdorff Measure”
Norman Walczak ’12 “Hold the Applause: Pulse-Coupled Biological Oscillators”
Diqian Wang ’12 “Where Do You Draw the Line?”
Stephen Weiss ‘12 “The Projective Plane as Unifier for Conics”
William Weiss ‘12 “The End of Tarantino: How to Win a Mexican Standoff”
Gregory White ’12 “Regularity of Area-Minimizing Hypersurfaces”
Matthew Wyatt ’12 “Hausdorff’s Moment Problem”
Liyang Zhang ’12 “Grassmannian Variety”
Off-Campus Colloquia
Colin Adams
“Superinvariants and Indicatrices of Knots”
Centro De Research di Giorgio, Pisa, Italy
“The Great Calculus Debate”
University of Wisconsin
Ripon College
“The Great Pi/e Debate”
Madison area Technical College, Madison, WI
Framingham State University
“Blown Away: What Knot to Do When Sailing”
James Madison University
Kennesaw State University
“Making Math Fun”
Kennesaw State University
“Real Estate in Hyperbolic Space: Investment Opportunities for the Passive-Aggressive Investor”
Park City Mathematics Institute, Park City, UT
“Hyperbolic Knots”
Joint Mathematics Meetings, Boston, MA
“Mathematically Bent Theater”
Joint Mathematics Meetings, Boston, MA
Elizabeth Beazley
“Affine Flag Varieties in Positive Characteristic”
Brown University
“A Graph Encoding an Order on the Affine Symmetric Group”
Dartmouth College
“Quantum Cohomology and the Poset of Newton Polygons”
Universite du Quebec a Montreal
“Infinite Shuffling Between Matrix Varieties and Newton Polygons”
Haverford College
Satyan Devadoss
“Colloquium”
Vassar College
“Geometry Seminar”
Georgia Institute of Technology
“Phylogenetic Networks”
Mathematical Biosciences Institute
“Plenary Speaker”
Discrete Math Day Conference
Renaissance Weekend
“Veritas Forum”
Dartmouth College
Georgia Institute of Technology
University of Arizona
“Discrete and Computational Geometry”
MAA Two-Day Short-Course
Richard De Veaux
“The Seven Deadly Sins of Data Mining”
Cardinal Health Research Symposium, Chicago, IL
“JMP Explorers’ Series: Exploring Interactive and Visual Data Mining?”
Detroit, MI
Dallas, TX
Chicago, IL
New York, NY
Atlanta, GA
Los Angeles, CA
“Presentational Skills Workshop”
“Is the t-test Really Dead?”
Joint Statistical Meetings, Miami, FL
“Data Visualization for Large Data Sets
Workshop, Proctor and Gamble, Cincinnati, OH
“Data Mining: Fool’s Gold or the Mother Lode?”
Law Review Symposium, Thomas Cooley Law School, Lansing, MI
Math Association of America, Distinguished Lecture Series, Washington, DC
University of Kansas, Lawrence, KS
Pond Lecture, Chicago, IL
Pond Lecture, Milwaukee, WI
“Data Mining for Quality”
Fall Technical Conference, American Society for Quality, Kansas City, MO
“Robust Boosting”
Decision Sciences Institute, Boston, MA
“The Challenges of Big Data”
NSF Workshop, Bangalore, India
“Data Visualization for the Business Statistics Course”
Decision Sciences Institute, Columbia, SC
“Statistics and Data Mining for Computer Scientists, Engineers, Blacksmiths and Lawyers Workshop”
ITI, Cavtat, Croatia
Thomas Garrity
“On Writing Numbers”
Bennington College
Texas Christian University
“Derivative vs. Integral: The Final Smackdown”
with Colin Adams, University of Wisconsin, Madison
Ripon College
“On the Arrogance of Mathematicians”
Hampshire College
“The Great Pi/E Debate”
with Colin Adams, Madison Area Technical College
Framingham State University
“On a Thermodynamic Classification of Real Numbers”
Texas Christian University
Susan Loepp
“Where Algebra and Analysis Meet: What is the Distance Between Two Polynomials?”
Fairfield University
“The Key to Sending Secret Messages”
“Protecting your Personal Information: An Introduction to Encryption”
Frank S. Brenneman Lecture Series, Tabor College
“Teaching, Mentoring, and Advising Undergraduate Research: Lessons Learned on the Streets”
Haimo Award Talk, Joint Mathematics Meetings, Boston, MA
Steven Miller
“Cookie Monster meets the Fibonacci Numbers. Mmmmmm – Theorems!”
CANT 2011
Amherst College
University of Connecticut
Colby College
“Theory and Applications of Benford’s Law, or: Why the IRS Should Care About Number Theory!”
Hampshire College
“Finite Conductor Models for Zeros Near the Central Point of Elliptic Curve L-Functions”
University of Maine
Brown University
“Pythagoras at the Bat”
University of Massachusetts, Amherst
Fitchburg State University
Boston College
“Benford’s Law, Values of L-Functions and the 3x + 1 Problem, or: Why the IRS Should Care About Number Theory“
University of Massachusetts, Amherst
“Benford’s Law, or Why the IRS Cares About Number Theory”
Mount Holyoke College
“Eigenvalue Statistics for Toeplitz and Circulant Ensembles, Analysis and Probahility Seminar
University of Connecticut
“Distribution of Summands in Generalized Zeckendorf Decompositions”
George Washington University
“To Infinity and Beyond: Gaps Between Summands in Zeckendorf Decompositions”
CANT 2012
“Moment Formulas for Ensembles of Classical Compact Groups”
Joint Mathematics Meetings, Boston, MA
“Models for Engaging Undergraduate Students in Research”
with Dave Damiano, Dean Evasius, Joe Gallian, Ivelisse Rubio, Jake Levinson ’11, and Gina-Marie Pomann
Joint Mathematics Meetings, Boston, MA
“Distribution of Missing Sums in Sumsets”
CANT 2012
Frank Morgan
“SurpriZing Science: AmaZing Bubbles”
With the SMALL Geometry Group, Spring Street, Williamstown, MA
“Soap Bubbles and Mathematics”
St. Michael’s College
Pine Cobble High School, Williamstown, MA
MathCounts, Hartford, CT
CUNY, Staten Island
Tufts University
MIT ’87 Reunion Dinner
Edinburgh, ICMS Isoperimetric Problems, at Dynamic Earth for 60 high school students
“Existence of Isoperimetric Regions in Rn With Density”
Joint Mathematics Meetings, Boston, MA
“Densities from Geometry to the Poincaré Conjecture”
Washington State University
MAA Meeting, University of Portland
“Optimal Pentagonal Tilings”
MAA Meeting, University of Portland
“Optimal Tilings”
Undergraduate Prize Lecture, Northwestern University
Bridgewater State University
“The Isoperimetric Problem With Density”
Lehigh University
Allison Pacelli
“Algebraic Number Theory: An “Ideal” Subject”
Wellesley College
Summer Program for Women in Mathematics, George Washington University
“Algebraic Number Theory & Fermat’s Last Theorem”
Siena College
“Class Number Indivisibility”
Brown University
“Indivisibility of Class Numbers in Global Function Fields”
Canadian Number Theory Conference
“Math and Politics in the Undergraduate Colloquium”
Arcadia University
Cesar Silva
“Panel Presentation”
The Hotchkiss School, Lakeville, CT
“Métricas Compatibles y Sensitividad Medible”
IMCA, Lima, Peru
“Cantor’s Set Throughout Real Analysis”
Joint Mathematics Meetings, Boston MA
“On Mu-Compatible Metrics and Measurable Sensitivity”
Joint Mathematics Meetings, Boston, MA
“On Measurable Sensitivity for Nonsingular and Measure-Preserving Maps”
McGill University
Mihai Stoiciu
“Mid-Hudson Mathematics Conference, Plenary Address”
Bard College
“Spectral Theory”
University of South Florida
Postgraduate Plans of Mathematics Majors
| Ji Won Ahn | Attending medical school at Washington University School of Medicine in St. Louis |
| Praphruetpong Athiwaratkun | Working with Exeter Group in Boston |
| Victoria Borish | Researching at the Institute for Quantum Optics and Quantum Communication in Vienna, Austria. |
| Hayley Brooks | Pursuing a business with another Williams senior that facilitates fundraising in elementary schools via an online platform. |
| Thomas Crawford | Attending math graduate school at Boston University |
| Austin Davis | Teaching English at Phillips Andover as a Teaching Fellow and pursuing a Master’s Degree in English over the summers through the Bread Loaf School of English. |
| Westcott Gail | Working as an Investment Banking Analyst at Deutsche Bank in New York |
| Carolyn Geller | Working as an Analyst in the Health and Benefits Consulting Department at Mercer in New York |
| David Gold | Working at Mighty Food Farm in Pownal, VT. |
| Hannah Hausman | Doing a two-year fellowship at the Carnegie Foundation for the Advancement in |
| Chansoo Lee | Pursuing a Ph.D. in Computer Science at the University of Michigan |
| Erik Levinsohn | Teaching high school math at the BART Charter School in Adams, MA |
| Brian Li | Research at the National Institute of Health while applying to medical school |
| Andrew Liu | Investment Fellowship with T. Rowe Price in Baltimore, Maryland |
| Andrew Lorenzen | Working as a Software Engineer at Google in Mountain View |
| Andrew Nguyen | Working at an economic consulting firm in New York |
| Theo Patsalos-Fox | Working at Boston Consulting Group |
| Katherine Rieger | Working at Vanguard Group participating in the Vanguard Accelerated Development Program in Philadelphia, PA. |
| Niralee Shah | Mathematics Teaching Fellow at King’s Academy in Madaba, Jordan |
| Shara Singh | Working at the Williams College Investment Office |
| Tarjinder Singh | Pursuing a Master’s Degree (MPhil. in Biological Sciences) at the University of Cambridge |
| Connor Stern | Teaching secondary math for Teach for America in Camden, NJ |
| Katherine Stevenson | Working for ITT Exelis, a government contractor, at the Goddard Space Flight Center in Greenbelt, MD, on NASA’s Space Communications Network Services doing satellite communications work. |
| Stephen Weiss | Working at Exeter Group in Boston |
| Gregory White | Working as an Associate Consultant at Bain & Company in New York City |
| Matthew Wyatt | Working in the Model Review Group at JPMorgan in New York City, a subset of the Quantitative Research Division. |
| Liyang Zhang | Pursuing a Ph.D. in Mathematics at Yale University |
Mathematics and Statistics
Dynamics, Information, and Energy of Morris-Lecar Neurons
Ji Won Ahn
We studied the Morris-Lecar model, which is a mathematical model of a motor neuron. In particular, we studied the mutual information, metabolic energy cost, and energy efficiency of unidirectionally connected Morris-Lecar neurons, and compared our result to the work of Moujahid et al., who studied the mutual information, metabolic energy cost, and energy efficiency of unidirectionally coupled Hodgkin-Huxley neurons.
We found that unidirectionally coupled Morris-Lecar models behave differently from unidirectionally coupled Hodgkin-Huxley neurons in both information transfer and energy efficiency. Unlike Moujahid et al., we found that among groups of one, five, ten, and twenty postsynaptic neurons, the single Morris-Lecar neuron synchronizes with the presynaptic neuron the best and is the most energy efficient.
On Multiply Recurrent and Manifold Mixing Properties on Infinite Measure Preserving Transformations
Praphruetpong Athiwaratkun
We show an example of an infinite measure preserving transformation such that it is not 2-recurrent and not power weakly mixing. This example demonstrates the striking difference between measure preserving transformations in a finite and sigma-finite measure spaces.
Totally Knotted and Semi-Free Seifert Surfaces
Thomas N. Crawford
In 2005 Osamu Kakimizu determined the Kakimizu Complex, a simplicial complex whose vertices correspond to isotopy classes of Seifert surfaces of a given knot, for all knots with 10 crossings or fewer. We investigate a few properties the surfaces themselves. Specifically we show various combinations of semi-free and totally knotted surfaces, can be embedded in the same knot complement. We restrict ourselves to hyperbolic knots allowing us to also look at the maximal cusp diagrams of the resultant manifold.
Monkemeyer Map Analogues to Stern’s Diatomic Sequence
Noah N. Goldberg
Stern’s Diatomic Sequence is a well-studied sequence of integers which stems from continued fractions. The Monkemeyer Map is a type of multidimensional continued fraction. We will examine an analogue of Stern’s Diatomic Sequence for the Monkemeyer Map.
Ergodic Properties of TRIP Maps: A Family of Multidimensional Continued Fractions
Stephanie Jensen
We study the ergodic properties of several of the most relevant TRIP maps, a family of multidimensional continued fractions that encompasses many well-known algorithms. As a first step, we show these maps converge almost everywhere. From there, we are able to prove ergodicity.
Spaces of Planar Polygons
Brian Li
We introduce the space of convex planar polygons with different side lengths. We then consider the side lengths that produce valid linkages as well as the relation of this space to the associahedra and M_0,n.
Choose to Play: A New Take on the Spatial Prisoner’s Dilemma
Connor McKean Stern
The Prisoner’s dilemma is one of the most important models we have to study the evolution of cooperation in a world of self-interested individuals. Defecting is the only evolutionarily stable strategy, but from previous studies we know that in repeated games and in games with spatial effects cooperating becomes not only possible, but also preferable under certain conditions. In this paper we explore a new repeated model of the spatial prisoner’s dilemma game where a player can select which opponents to continue interacting with. By giving players this option we are rejecting the key condition of the repeated game that players cannot avoid interaction, yet we find that this new model shares the same underlying structure of the traditional spatial prisoner’s dilemma.
Stochastic Calculus and Applications to Mathematical Finance
Gregory White
In this paper, we review fundamental probability theory, the theory of stochastic processes, and Ito calculus. We also study an application of Ito calculus in mathematical finance: the Black-Scholes option pricing model for the European call option. We study the development of the model and the assumptions necessary to arrive at the Black-Scholes no arbitrage rational price for a European call option.
We supplement the simple Black-Scholes model by relaxing the assumption that trading can be performed continuously in time, and studying the deviation the Black-Scholes replicating portfolio exhibits from the self-financing characteristic of the continuous-time portfolio. We term this deviation the cumulative correction of the portfolio and explain in detail its construction. We study the cumulative correction of Black-Scholes portfolios by performing a numerical analysis of the cumulative correction for outcomes of the stock price stochastic process. While finding a closed form probability distribution representing the cumulative correction proves difficult and we do not pursue that route in this paper, the numerical analysis indicates that the second central moment of the distribution of cumulative corrections decreases as the number of discrete time steps at which the portfolio is rebalanced increases. Additionally, we analyze the cumulative correction required to replicate the European call option for the historical stock price data series of certain actual stocks, finding examples of a stock that would have required a positive cumulative correction and a stock that would have required a negative cumulative correction.
Spectral Theory for Matrix Orthogonal Polynomials on the Unit Circle
Liyang Zhang
In this thesis, we first introduce the classical theory of orthogonal polynomials on the unit circle and its corresponding matrix representations – the GGT representation and the CMV representation. We briefly discuss the Sturm oscillation theory for the CMV representation. Motivated by Schulz-Baldes’ development of Sturm oscillation theory for matrix orthogonal polynomials on the real line, we study matrix orthogonal polynomials on the unit circle. We prove a connection between spectral properties of GGT representation with matrix entries, CMV representation with matrix entries with intersection of Lagrangian planes. We use this connection and Bott’s theory on intersection of Lagrangian planes to develop a Sturm oscillation theory for GGT representation with matrix entries and CMV representation with matrix entries.
Mathematics and Statistics
The Complimentary Regions of Knot and Link Projections
Colin Adams, R. Shinjo and K. Tanaka
Annals of Combinatorics, Vol. 15, No. 4, 549-563 (2011)
An increasing sequence of integers is said to be universal for knots and links if every knot and link has a reduced projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are each universal for knots and links:
(3,5,7, …), (2,n, n+1,n+2,…) for each n ≥ 3, (3,n,n+1,n+2, …) for each n ≥ 4. Moreover, the finite sequences (2,4,5) and (3,4,n) for each n ≥ 5 are universal for all knots and links.
It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n+1 odd-sided faces if n is odd.
Planar and Spherical Stick Indices of Knots
Colin Adams, D. Collins, K. Hawkins ‘10, C. Sia, R. Silversmith ’11, B. Tshishiku
Journal of Knot Theory and Its Ramifications, Vol. 20, No. 5, 721-739 (2011)
The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants, we show that the spherical stick index distinguishes between the granny and square knots, and that composing a nontrivial knot with a second nontrivial knot need not increase its spherical stick index.
Stick Index of Knots and Links in the Cubic Lattice
Colin Adams, M. Chu, T. Crawford ‘12, S. Jensen ’12, K. Siegel, L. Zhang ‘12
Journal of knot Theory and Its Ramifications, Vol. 21, No. 5 (2012)
The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p+1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.
Duality Properties of Indicatrices of Knots
Colin Adams, D. Collins, K. Hawkins ‘10, C. Sia, R. Silversmith ’11, B. Tshishiku
Geometriae Dedicata (on-line publication), (September 1, 2011)
The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.
CSI: MSRI
Colin Adams
Mathematical Intelligencer, Vol. 33, No. 2, 18-21 (2011)
What happens when there is a crime committed at the Mathematical Sciences Research Institute?
The Book
Colin Adams
Mathematical Intelligencer, Vol. 33, No. 3, 107-109 (2011)
Paul Erdos hypothesized a book in God’s possession that contained all of the beautiful proofs ever discovered. What happens if you have access to that book.
Leonhard Euler and Seven Bridges of Konigsberg
Colin Adams
Mathematical Intelligencer, Vol. 33, No. 4, 18-20 (2011)
Many people attribute the birth of topology to Euler’s solution of the Konigsberg Bridge Problem. But what is the true story of what really happened?
The Dog Who Knew Calculus
Colin Adams
Mathematical Intelligencer, Vol. 34, No. 1, 16-17 (2012)
In a 2003 article, the author explained how his dog Elvis seemed to understand calculus, as he was so good at minimizing the time it took to get a ball thrown in the water. So let’s give him a job teaching.
Derivative vs. Integral: The Final Smackdown
Colin Adams, Thomas Garrity and Adam Falk
Mathematical Association of America (January 2012)
Which is better, the derivative or the integral? Recorded at Williams Family Days, Fall 2011.
A Generalization of a Theorem of Lekkerkerker to Ostrowski’s Decomposition of Natural Numbers
Edward B. Burger, David C. Clyde, Cory H. Colbert, Gea Hyun Shin ’11, and Zhaoning Wang ‘11
Acta Arithmetica, 153, 217-249 (2012)
Let a be a fixed, irrational real number and pk/qk its associated kth convergent. In 1921, Ostrowski proved that each natural number n can be expressed uniquely as a linear combination of the continuants of a, namely the qk’s, in which the integer coefficients satisfy certain natural diophantine conditions. Here we analyze the asymptotic behavior of the average number of summands required in such decompositions relative to the size of the corresponding natural numbers in the case for which a is a quadratic irrational. Our results generalize the work of Lekkerkerker, who in 1951 explicitly computed this asymptotic ratio for the particular case a = (1+√5)/2 and found it to equal (5–√5)/10 = 0.2763… .
The Shape of Associativity
Satyan Devadoss
Canadian Mathematical Society Notes, 44, 12-14 (2012)
Associativity is ubiquitous in mathematics. Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance. But over the past few decades, this concept has blossomed and matured. We show how to visualize the concept of associativity.
What Makes a Tree a Straight Skeleton?
Satyan Devadoss
Proceedings of the European Conference on Computational Geometry (2012)
Given any polygon, one can construct a geometric tree associated to it called its straight skeleton. This appears in the construction of roof and origami folding designs. We ask the inverse question: For what tree does there exist polygons with the tree as its skeleton?
Triple Infinity
Satyan Devadoss, Associate Professor of Mathematics
Esopus Magazine (2011)
A conversation between a mathematician, a cosmologist, and an artist about the meaning and nature of infinity in these three fields.
A Robust Boosting Algorithm for Chemical Modeling
Richard DeVeaux and Ville Satopӓӓ ‘11
Current Analytical Chemistry, Vol. 8, No. 2, 254-265 (2012)
Baggins and boosting have become increasingly important ensemble methods for combining models in the data mining and machine learning literature. We review the basic ideas of these methods, propose a new robust boosting algorithm based on a non-convex loss function and compare the performance of these methods to both simulated and real data sets both with and without contamination.
Using Mathematical Maturity to Shape our Teaching, our Careers and our Departments
Thomas Garrity
Notices of the American Mathematical Society, 1592 – 1593 (2011)
Derivative vs. Integral: The Final Smackdown
Thomas Garrity, Colin C. Adams and Adam Falk
Mathematical Association of America (January 2012)
Which is better, the derivative or the integral? Recorded at Williams Family Days, Fall 2011.
Semi-Local Formal Fibers of Minimal Prime Ideals of Excellent Reduced Local Rings
Susan Loepp, Nicholas Arnosti ’11, Rachel Karpman, Caitlin Leverson, and Jake Levinson ’11
Journal of Commutative Algebra, No. 1, 29-56 (2012)
Given a complete local ring T containing the rationals, and a positive integer m, the authors find necessary and sufficient conditions for there to exist an excellent reduced local ring A, whose completion is T, such that A has exactly m minimal prime ideals. In addition, the authors show that the formal fibers over the minimal prime ideals can be controlled.
Distribution of Eigenvalues for Highly Palindromic Real Symmetric Toeplitz Matrices
Steven J. Miller, Steven Jackson ’10 and Thuy Pham ‘11
Journal of Theoretical Probability, 25, 464-495 (2012)
Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d random variables chosen from a fixed probability distribution p of mean 0, variance 1 and finite higher moments. Previous work showed that the limiting spectral measures (the density of normalized eigenvalues) converge in probability and almost surely to a universal distribution almost that of the Gaussian, independent of p. The deficit from the Gaussian distribution is due to obstructions to solutions of Diophantine equations and can be removed by making the first row palindromic. In this paper, we study the case where there is more than one palindrome in the first row of a real symmetric Toeplitz matrix. Using the method of moments and an analysis of the resulting Diophantine equations, we show that the moments of this ensemble converge to a universal distribution with a fatter tail than any previously seen limiting spectral measure.
Rational Irrationality Proofs
Steven J. Miller and David Montague
Mathematics Magazine, 85, No. 2, 110-114 (2012)
Proving the irrationality of the square-root of 2 is a rite of passage for mathematicians. The purpose of this note is to spread the word of a remarkable geometric proof, and to generalize it. The proof was discovered by Stanley Tennenbaum in the 1950′s, and first appeared in print in John H. Conway’s article in Power. In the interest of space, we often leave out the algebra justifications for the lengths of the sides in our figures. The reader is encouraged to prove these expressions for themselves, or see the arxiv post for complete details.
Moments of the Rank of Elliptic Curves
Steven J. Miller and Siman Wong
Canadian Journal of Mathematics, 64, No. 1, 151-182 (2012)
Fix an elliptic curve E/Q, and assume the Riemann Hypothesis for the L-function L(E_D, s) for every quadratic twist E_D of E by D in Z. We combine Weil’s explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of E_D. We derive from this an upper bound for the density of low-lying zeros of L(E_D, s) which is compatible with the random matrix models of Katz and Sarnak. We also show that for any unbounded increasing function f on R, the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer conjecture) the number of integral points of E_D are less than f(D) for almost all D.
Generalized More Sums Than Differences Sets
Steven J. Miller, Geoffrey Iyer, Oleg Lazarev, Liyang Zhang ‘12
Journal of Number Theory, 132, No. 5, 1054-1073 (27 pp) (2012)
A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set A of Z such that |A+A|<|A-A|. Though it was believed that the percentage of subsets of {0,…,n} that are sum-dominant tends to zero, in 2006 Martin and O’Bryant proved that a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We prove that |ε1 A + … + εk A|>|δ1 A + … + δk A| a positive percent of the time for all nontrivial choices of εj,δj\in {-1,1}. Previous approaches proved the existence of infinitely many such sets given the existence of one; however, no method existed to construct such a set. We develop a new, explicit construction for one such set, and then extend to a positive percentage of sets.
We extend these results further, finding sets that exhibit different behavior as more sums/differences are taken. For example, we prove that for any m, |ε1 A + … + εk A| – |δ1 A + … + δk A| = m a positive percentage of the time. We find the limiting behavior of kA = A+ … +A for an arbitrary set A as k goes to infinity and an upper bound of k for such behavior to settle down. Finally, we say A is k-generational sum-dominant if A, A+A, …, kA are all sum-dominant. Numerical searches were unable to find even a 2-generational set (heuristics indicate that the probability is at most 10{-9}, and quite likely significantly less). We prove that for any k a positive percentage of sets are k-generational, and no set can be k-generational for all k.
Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
Steven J. Miller, Sidney Luc Robinson ’12 and Sean Pegado ‘11
Integers, 12, No. A30 (2012)
A More Sums Than Differences (MSTD) set is a set of integers A of {0, …, n-1} whose sumset A+A is larger than its difference set A-A. While it is known that as n tends to infinity a positive percentage of subsets of {0, …,n-1} are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman gave explicit constructions of a large family of MSTD sets; though their density is less than a positive percentage, their family’s density among subsets of {0, …,n-1} is at least C/n4 for some C>0, significantly larger than the previous constructions, which were on the order of 1/2{n/2}. We generalize their method and explicitly construct a large family of sets A with |A+A+A+A| > |(A+A)-(A+A)|. The additional sums and differences allow us greater freedom than in MOS, and we find that for any ε>0 the density of such sets is at least C/nε. In the course of constructing such sets we find that for any integer k there is an A such that |A+A+A+A| – |A+A-A-A| = k, and show that the minimum span of such a set is 30.
Models for Zeros at the Central Point in Families of Elliptic Curves
Steven J. Miller, Eduardo Duenez, Duc Khiem Huynh, Jon Keating and Nina Snaith
J. Phys. A: Math. Theor., 45, 115207 (2012)
We propose a random matrix model for families of elliptic curve L-functions of finite conductor. A repulsion of the critical zeros of these L-functions away from the center of the critical strip was observed numerically by S. J. Miller in 2006; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the one-level density of eigenvalues of orthogonal matrices after appropriate rescaling). Our purpose here is to provide a random matrix model for Miller’s surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this sub-ensemble of SO(2N) the excised orthogonal ensemble. The sieving-off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of L-functions implied by the formula of Waldspurger and Kohnen-Zagier. The cut-off scale appropriate to modeling elliptic curve L-functions is exponentially small relative to the matrix size on the order of N. The one-level density of the excised ensemble can be expressed in terms of that of the well-known Jacobi ensemble, enabling the former to be explicitly calculated. It exhibits an exponentially small (on the scale of the mean spacing) hard gap determined by the cut-off value, followed by soft repulsion on a much larger scale. Neither of these features is present in the one-level density of SO(2N). When N goes to infinity we recover the limiting orthogonal behaviour. Our results agree qualitatively with Miller’s discrepancy. Choosing the cut-off appropriately gives a model in good quantitative agreement with the number-theoretical data.
On the Number of Summands in Zeckendorf Decompositions
Steven J. Miller, Murat Kologlu ’12, Gene S. Kopp, and Yinghui Wang
Fibonacci Quarterly, 49, No. 2, 116-130 (2011)
Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it’s natural to ask how many summands are needed. Using a continued fraction approach, Lekkerkerker proved that the average number of such summands needed for integers in [Fn, F{n+1}) is n / (j2 + 1) + O(1), where j = (1+sqrt(5))/2 is the golden mean. Surprisingly, no one appears to have investigated the distribution of the number of summands; our main result is that this converges to a Gaussian as n tends to infinity. Moreover, such a result holds not just for the Fibonacci numbers but many other problems, such as linear recurrence relation with non-negative integer coefficients (which is a generalization of base B expansions of numbers) and far-difference representations.
In general the proofs involve adopting a combinatorial viewpoint and analyzing the resulting generating functions through partial fraction expansions and differentiating identities. The resulting arguments become quite technical; the purpose of this paper is to concentrate on the special and most interesting case of the Fibonacci numbers, where the obstructions vanish and the proofs follow from some combinatorics and Stirling’s formula.
From Fibonacci Numbers to Central Limit Type Theorems
Steven J. Miller and Yinghui Wang
Journal of Combinatorial Theory, Series A, 119, No. 7, 1398-1413 (2012)
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers {Fn}_{n=1}∞. Lekkerkerker \cite{Lek} proved the average number of summands for integers in [Fn, F{n+1}) is n/(j2 + 1), with phi the golden mean. This has been generalized: given nonnegative integers c1,c2,…,cL with c1,cL>0 and recursive sequence {Hn}_{n=1}∞ with H1=1, H{n+1} =c1Hn + c2H{n-1} + … +cnH1+1 (1 ≤ ∞ n < L) and H{n+1}=c1Hn+c2H{n-1}+ … +c_LH_{n+1-L} (n ≥ L), every positive integer can be written uniquely as a sum of aiHi under natural constraints on the a_i‘s, the mean and variance of the numbers of summands for integers in [H{n}, H{n+1}) are of size n, and as n tends to infinity the distribution of the number of summands converges to a Gaussian. Previous approaches used number theory or ergodic theory. We convert the problem to a combinatorial one. In addition to re-deriving these results, our method generalizes to other problems (in the sequel paper we show how this perspective allows us to determine the distribution of gaps between summands). For example, it is known that every integer can be written uniquely as a sum of the ± Fn‘s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The presence of negative summands introduces complications and features not seen in previous problems. We prove that the distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely -(21-2j)/(29+2j), which is approximately -0.551058.
Steiner and Schwarz Symmetrization in Warped Products and Fiber Bundles With Density
Frank Morgan, Sean Howe and Nate Harman
Revista Mat. Iberoamericana, 27, 909-918 (2011)
We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.
Isoperimetric Pentagonal Tilings
Frank Morgan, Ping Ngai Chung, Miguel Fernandez, Yifei Li, Michael Mara ’12, Isamar Rosa Plata, Niralee Shah ’12, Luis Sordo Vieira, and Elena Wikner ‘11
Notices Amer. Math. Soc., 59, 632-640 (2012)
We generalize the isoperimetric problem from geometry to numbers.
Alan Alda’s Flame Challenge and Kids’ Five Most Popular Science Questions
Frank Morgan
Huffington Post Blog, (March 16, 2012)
Can Math Survive Without the Bees?
Frank Morgan
Huffington Post Blog, (March 6, 2012)
Recent and new results on perimeter-minimizing tilings.
Soap Bubbles in Scotland
Frank Morgan
Huffington Post Blog, (March 23, 2012)
The latest progress on the century-old search for the least-perimeter way to partition space into unit volumes.
Math Finds the Best Doughnut
Frank Morgan
Huffington Post Blog, (April 2, 2012)
A report on the proof of the Willmore Conjecture.
Geometry Festival
Frank Morgan
Huffington Post Blog, (April 30, 2012)
A mathematics progress report from this annual meeting of geometers.
Function Fields With Class Number Indivisible by A Prime ℓ
Allison Pacelli, Michael Daub ’08, J. Lang, M. Merling and Natee Pitiwan ‘09
Acta. Arith., 150, 339-359 (2011)
In this paper, we prove that there are infinitely many function fields of any degree over the rational function field with class number indivisible by an arbitrary prime number.
On Mu-Compatible Metrics and Measurable Sensitivity
Cesar E. Silva, Ilya Grigoriev, Nate Ince, Marius Catalin ’09, and Amos Lubin
Colloquium Math. 126, 53-72 (2012)
We introduce the notion of W-measurable sensitivity, which extends and strictly implies canonical measurable sensitivity, a measure-theoretic version of sensitive dependence on initial conditions. This notion also implies pairwise sensitivity with respect to a large class of metrics. We show that nonsingular ergodic and conservative dynamical systems on standard spaces must be either W-measurably sensitive, or isomorphic mod 0 to a minimal uniformly rigid isometry. In the finite measure-preserving case they are W-measurably sensitive or measurably isomorphic to an ergodic isometry on a compact metric space.
