# Mathematics Department

## Colloquium

#### Upcoming Seminar

February 8, 4:10 PM in Physics 123
Helping Software Exploit Hardware
Kelly Shaw, University of Richmond

Software developers frequently write code in a manner that abstracts away the details of the hardware on which the software will run.  While this software will execute on many computing platforms, it will not run as quickly or efficiently as it could if the software took advantage of hardware specific characteristics. Unfortunately, optimizing programs for specific hardware is a time intensive process, making it unlikely that applications will be optimized for many hardware configurations.

In this talk, I present two approaches we created to help developers write high performance software for graphics processors. Starchartenables developers to systematically and quickly understand how to tune important characteristics of their applications for different hardware.  The second approach, called MRPB, automatically prioritizes and reorders data accesses to increase the benefits obtained from data caching in graphics processors.  Both approaches ease the software developer's burden when writing and optimizing software for graphics processors.

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.

### 2015-16 Schedule

#### Fall

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

#### Spring

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

Jan 26 Please note change in date.Invariants of Singular SpacesJesse Gell-Redman, Johns Hopkins UniversityOne of the fundamental distinctions in mathematics is that between the local and global invariants of a space. A global invariant is a property of a shape that depends on how the space looks in its entirety, i.e. from far away. Examples include the number of sides a polygon has or the number of holes in a surface. (Does the surface look like a basketball, so zero holes, or like a doughnut, so one hole?) Local invariants come from the actual geometry itself, the main example being a numerical measure, called the Gauss curvature, of how much a space is curved at a given point. As I will explain, local and global invariants of spaces are related in a profound way.  Although this relationship has been known for a long time, it is not well-elaborated in the context of singular spaces, i.e. shapes like cones or horns which have a point or points that are not smooth. That would be fine, were it not the case that some of the most interesting geometries that arise in mathematics and physics are singular!  I will illustrate the latter statement with simple examples and explain how progress can be made understanding invariants of these spaces. An Inverse Problem in Spectral GeometryDavid Sher, University of MichiganConsider a physical object, such as a drumhead, that vibrates when struck. The motion of the drumhead may be understood in terms of its resonance frequencies and modes of vibration. In spectral geometry, we study the relationship between these resonance frequencies - mathematically, the spectrum of the Laplace operator - and the geometry of the drumhead. A key question is the *inverse spectral problem*: what can we deduce about the geometry from the spectrum? That is, if we know the resonance frequencies of a drumhead, what can we say about its shape? This question may be asked in a variety of different contexts. One such context involves the Dirichlet-to-Neumann operator, which appears in problems ranging from medical imaging to hydrodynamics. In this talk, I will first give an overview of the types of questions we ask in spectral geometry. I will then discuss several recent results on the spectral geometry of the Dirichlet-to-Neumann operator, with particular emphasis on the inverse spectral problem. Please note change in date.Fair Computation with Rational PartiesAdam Groce, Reed CollegeTraditional cryptography requires that regardless of what malicious parties might do, security still holds. Rational cryptography instead uses game theory to model behavior. It assumes that the parties in question have goals and will misbehave only if it furthers those goals. In this talk, I'll introduce rational cryptography and show how it can be used to accomplish the otherwise impossible. In particular, we’ll look at secure computation, which allows parties who each have one input to compute a function of those inputs without sharing them with each other. Depending on the function chosen, secure computation can allow electronic voting that keeps votes anonymous or allow companies to collaborate without sharing proprietary data. It is one of the most general and powerful primitives in all of cryptography. Unfortunately, traditional protocols are all "unfair," meaning that one party can get the answer but prevent other parties from seeing it. In contrast, I will show a fair rational-secure protocol, circumventing two separate impossibility results. Please note change in date.Please note change in location.Derandomization, Lower Bounds, and Polynomial Identity TestingMatt Anderson, Union CollegeLocation: Eliot 314Randomness 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 computational 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 (PIT).  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.  This is difficult question, in part, because providing such an efficient PIT algorithm would imply a strong circuit lower bound, that is, it would demonstrate a computational problem that cannot be solved efficiently by any deterministic algorithm.  The search for such lower bounds is a central, long-standing open question in Computational Complexity Theory.     One line of my research approaches answering these questions by examining them in more restricted models of computation.  I will discuss efficient deterministic identity tests and lower bounds I developed for several arithmetic models of computation.  I conclude by discussing possible extensions to these works and other research topics I am interested in, including a number of ideas for (summer) research projects and senior theses. Elliptic equations with vanishing coefficients and regularity of solutionsLyudmila Korobenko, McMaster UniversityAmong the central questions of Partial Differential Equations is existence and uniqueness of solutions and their regularity. It turns out that to define a solution to a second order partial differential equation it is not necessary to require a function to be twice differentiable, instead one can define the so called weak solutions''. The advantage is that it is relatively easy to show the existence and uniqueness of such solutions, however, the question arises whether they are differentiable or even continuous. We are interested in Elliptic equations that generalize well known Laplace's and Poisson's equations which arise in many areas of physics, such as electromagnetism, fluid dynamics, and mechanics. The study of solutions to elliptic equations goes back to 18th century and the properties of solutions are well known. If we consider equations with variable coefficients that can vanish at certain points the theory becomes much more complicated and thus much more interesting. I will talk about weak solutions to elliptic equations and a method to show their regularity called Moser iteration. I will then show how this method can be applied to study equations with vanishing coefficients, called degenerate elliptic equations. I will also talk about some new results in this area, including the results my collaborators and I have recently obtained for infinitely degenerate elliptic equations. Please note change in date.Helping Software Exploit HardwareKelly Shaw, University of RichmondSoftware developers frequently write code in a manner that abstracts away the details of the hardware on which the software will run.  While this software will execute on many computing platforms, it will not run as quickly or efficiently as it could if the software took advantage of hardware specific characteristics. Unfortunately, optimizing programs for specific hardware is a time intensive process, making it unlikely that applications will be optimized for many hardware configurations.   In this talk, I present two approaches we created to help developers write high performance software for graphics processors. Starchartenables developers to systematically and quickly understand how to tune important characteristics of their applications for different hardware.  The second approach, called MRPB, automatically prioritizes and reorders data accesses to increase the benefits obtained from data caching in graphics processors.  Both approaches ease the software developer's burden when writing and optimizing software for graphics processors. Please note change in date.Please note change in location.Improving Password Security and Usability through Data-Driven ApproachesBlase Ur, Carnegie Mellon UniversityLocation: Eliot 314Users often must make security and privacy decisions, yet are rarely equipped to do so. In my research, I aim to understand both technical systems and the humans who use them. Armed with this understanding, I design approaches and systems that help users protect their security and privacy. In this talk, I will describe how I applied this research approach to password security and usability. As understanding what makes a password good or bad is crucial to this process, I will first discuss my work on metrics for password strength. These metrics commonly involve modeling password cracking, and I found that the approaches often modeled by researchers vastly underestimate passwords' vulnerability to experts in password forensics. I instead propose combining a series of carefully configured approaches, which I found to conservatively model experts. I used these insights to design a Password Guessability Service, which I have opened to the community to enable more robust research. I will then highlight my work on another key step, investigating how humans create and perceive passwords. I will focus on the impact of password-strength meters, users' perceptions of password security, and my ongoing efforts to develop a better password-strength meter. By combining better metrics with an understanding of users, I am able to design approaches and systems that guide users towards better passwords. Please note change in location.Math curriculum discussion (for students only)Location: Psych 105 TBDAlbyn Jones, Reed College Homotopy, Cobordisms, and GeneraKyle Ormsby, Reed College TBD TBD Spring Break TBDJeremiah Heller, University of Illinois Urbana-Champaign TBDSteve Awodey, Carnegie Mellon TBD TBDVesna Stojanoska, University of Illinois Urbana-Champaign TBDHaynes Miller, Massachusetts Institute of Technology