# Mathematics Department

## Colloquium

#### Upcoming Seminar

March 13, 4:10 PM in Physics 123
Map making and vice versa
Zack Treisman, Department of Mathematics, Western State Colorado University

This talk is probably not about cartography.

Coordinate systems are descriptions of geometric objects, and there are ambiguities that can appear in these descriptions. I'll discuss one surprising example coming from a situation that we all think we understand, and how this example can help us understand some interesting interactions between algebraic geometry and string theory via motivic integration.

Most Thursday afternoons during the academic year, the Reed College Department of Mathematics hosts a math talk. The talks are directed to our mathematics majors but are usually accessible on a variety of levels. Refreshments are served before the talks. For more information, please email davidp at reed.edu.

### 2013-14 Schedule

#### Fall

4:10-5:00 in Physics 123 (unless marked otherwise). Directions to Reed.

Sept 5 Meeting with majors. No talk this week. Student PresentationsMaddie Brandt, Joseph Rennie, Qiaoyu Yang, and Kuai YuMaddie Brandt:  Packings of Four Equal Circles on Flat Tori Joseph Rennie:  Algorithms for Determining Zero Set Topologies Qiaoyu Yang and Kuai Yu:  Parking functions and tree inversions Maximally Intransitive DiceJoe Buhler, CCR/Reed CollegeIt's been known for a long time that there are three dice $$A$$, $$B$$, $$C$$ such that $$A > B > C > A$$ where "$$A> B$$" means that $$A$$ beats $$B$$ in the sense that the probability that a roll of $$A$$ is higher than a roll of $$B$$ is strictly bigger than 1/2. In fact there are dice such that $$A>B>C>A$$ and yet \(A^{[2]}

#### Spring

4:10-5:00 in Physics 123 (unless marked otherwise). Directions to Reed.

Jan 30 Asymptotics of Cyclic CodesMichael E O'Sullivan, Department of Mathematics & Statistics, San Diego State UniversityCodes for correcting errors that occur in transmission of data are found in numerous electronic devices. I will describe cyclic codes, which are a broad family of codes that are widely used and have interesting algebraic properties. The focus of the talk will be asymptotics: does there exist a "good" sequence of cyclic codes, one in which the length of the codes goes to infinity while the dimension and minimum distance don't approach zero. No talk this week. Nonnegative polynomials and sums of squaresGreg Smith, Department of Mathematics and Statistics, Queens UniversityA polynomial with real coefficients is nonnegative if it takes on only nonnegative values. For example, any sum of squares is obviously nonnegative. For a homogenous polynomial, Hilbert famously characterized when the converse statement holds, i.e. when every nonnegative homogeneous polynomial is a sum of squares. After reviewing the history of this problem, we will examine this converse in a new setting. In particular, we will provide new examples in which every nonnegative homogeneous polynomial is a sum of squares. This talk is based on joint work with Grigoriy Blekherman and Mauricio Velasco. Please note change in date.Please note change in location.Derandomization and Polynomial Identity TestingMatthew Anderson, University of CambridgeLocation: Library 204Randomness is a powerful computational resource that enables public-key cryptography, a cornerstone of the modern Internet, and powers extremely efficient data structures, like hash tables. In practice, randomness is algorithmically useful: There are many instances of problems that have simple efficient randomized algorithms but yet have no known efficient deterministic algorithm. Is randomness essential to efficient computation? In my talk I will survey several problems with natural and efficient randomized algorithms: Sorting, Primality Testing, and Polynomial Identity Testing. The first two problems also have efficient deterministic algorithms. However, an efficient deterministic algorithm for deciding whether a given polynomial identity holds has remained elusive. Indeed, derandomizing the general case of identity testing would imply circuit lower bounds, i.e., imply there is an explicit problem that is impossible for small Boolean circuits to compute, answering a central question in computational complexity theory that has been open for nearly half a century. Towards answering this question I will discuss some of my work on efficient deterministic identity tests for constant-read multilinear formulas, i.e., arithmetic formulas where each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Stronger isn't always betterAdam Groce, University of Maryland, College ParkToday, governments, universities, hospitals, and companies all maintain huge databases of private information. That information, if it could be analyzed, holds the key to a remarkable array of potential discoveries in medicine, social science, and other areas. However, the private nature of the data limits much of this analysis. The field of private data analysis seeks to provide tools for analyzing such data without violating privacy. However, before such tools can be created, it is necessary to formally define the desired notion of "privacy." I will survey existing definitions and then introduce a new definition, coupled-worlds privacy, proposed as part of my PhD work. The goal of coupled-worlds privacy is to formally model adversarial uncertainty in an effort to increase the accuracy of private data analysis. I will compare this definition with prior efforts toward the same end and discuss future directions for this definition and for private data analysis in general. The Little Remainder Theorem that CouldAnna Johnston, RaytheonOne theorem has the power to break cryptography, share secrets, correct errors, compute multiplicative inverses in finite fields, and enable countless other information theoretical and mathematical applications. This powerful little theorem is the da yen, commonly called the Chinese remainder theorem. The aim of this talk is to describe one of the least known of these applications: derivation of truncated Taylor series. This derivation gives an alternate perspective on Taylor series as an interpolated polynomial. On the way to this derivation the theory and history of the da yen will be described, including the origins of the names 'da yen' and Chinese remainder theorem. Map making and vice versaZack Treisman, Department of Mathematics, Western State Colorado UniversityThis talk is probably not about cartography. Coordinate systems are descriptions of geometric objects, and there are ambiguities that can appear in these descriptions. I'll discuss one surprising example coming from a situation that we all think we understand, and how this example can help us understand some interesting interactions between algebraic geometry and string theory via motivic integration. Spring break David Romano, Center for Interdisciplinary Brain Sciences Research, Stanford Luis David Garcia Melody Chan, Department of Mathematics, Harvard University