### 2016

**Ellen Gasparovic**, Union College,*Multi-Scale Modeling for Stratified Space Data*

**Abstract:**Geometric and topological methods in data analysis are capable of exposing essential shape information that may be hidden in the original data. Our goal is to introduce an algorithm for producing multi-scale models for data using an adaptive cover tree methodology, techniques from persistent homology, and multi-scale local principal component analysis. Our method takes in a point cloud sampled from a stratified space, and produces a model for the data that is fundamentally based on its local geometric properties and that captures how the different pieces of the dataset fit together. We will consider applications of our methods to musical audio data.

April 22, 2016.**Steve Miller**, Williams College,*Benford's Law: Why the IRS cares about the 3x+1 problem and the Riemann zeta function*

**Abstract:**TBA

Pi Mu Epsilon (PME) ceremony and colloquium, April 20, 2016.**Jeff Weeks**,*The Shape of Space*

**Abstract:**When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion. During the first half of the presentation, computer games will introduce the concept of a “multiconnected universe”. Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space. Finally, we'll see how satellite data provide tantalizing clues to the true shape of our universe. The only prerequisites for this talk are curiosity and imagination.

Leonard C. Sulski Memorial Lecture, 4:30 p.m., April 11, 2016.**Charles (Chip) E. Lawrence**, Brown University,*A Probabilistic Inference Algorithm, Repetitive RNA Sequences, and Lou Gehrig’s Disease*

**Abstract:**RNA editing by the adenosine deaminase acting on RNA (ADAR) enzymes has been associated with many human neurological diseases including: epilepsy; suicidal depression; autism; pediatric glioblastoma; and ALS (Lou Gehrig’s disease). RNA editing is ubiquitous in the animal kingdom. ADAR deaminates the RNA base adenosine (A) to inosine (I) in dsRNA molecules. Inosine is recognized by all cellular machineries as guanosine (G). ADAR specifically edits, recodes, a small number of adenosines in messenger RNA (mRNA) to such “Gs”. However, hyper editing acts more generally on perfect or nearly perfect double-stranded RNA (dsRNA). Within long dsRNA (>30bp), over 40% of adenosine residues are modified on both strands, generating numerous I-U mismatch pairs, and structurally destabilizing dsRNA. Dicer is an enzyme that cleaves near perfect long dsRNAs, and thus competes with ADAR. As a consequence, ADAR’s hyper editing has downstream consequences on Dicer products including gene expression silencing involving micro RNAs and endogenous small RNAs, endo-siRNAs. Increasingly evidence supports the notion that hyper editing may be rampant. However, because long dsRNA targets are very often largely repetitive transposable elements (TEs) and occurring primarily in neuronal nuclei, assessment of the scope of this process has been difficult. With our experimental collaborators we have been developing technologies to address these two challenges in studies using high throughput sequencing of Drosophila, fruit fly, RNA. Specifically, our collaborators have successful employed a technology, INTACT, that permits sampling RNAs directly from neuronal nuclei, and my group has developed a probabilistic algorithm for drawing Bayesian inferences on RNA sequences from repetitive genome loci. I this talk I’ll briefly summarize RNA editing, describe this probabilistic algorithm, and report results of its application including our team’s discovery of hyper editing of a TE in a gene, pur alpha, associated with ALS.

March 16, 2016.**Tingting Huan**,*Traveling wave solutions to reaction diffusion equations*

**Abstract:**Reaction diffusion equations are used to describe the spread of populations in space, modeling biochemical reactions and predicting the trend in the stock markets. The traveling wave solution is an important approach to the study of the reaction diffusion equations. In this talk, I will introduce the different types of diffusion process arising in different areas of research, especially the abnormal diffusion process - α−stable L´evy process. I will use population models as an example to illustrate the approaches of finding the traveling wave solutions to the reaction diffusion equations.

February 24, 2016.**Ryota Matsuura**, St. Olaf College,*What do 5 and 179424697 have in common?*

**Abstract:**Let p be a prime number with p > 2. When can we write p as a sum of two squares? (If p = 5, for example, we can write p = 12 + 22 .) In this talk, I will answer this question and describe a geometric (?!) proof by Minkowski that relies on such familiar shapes as parallelograms and circles. The talk should be accessible to mathematics students of all levels—no prior knowledge of number theory is necessary.

February 4, 2016.

### 2015

**Sean Smith**, Dartmouth College,*Security Circumvention: why do good people do bad things, and what can we do about it?*

**Abstract:**The field of computer security implicitly believes that it's possible to write down a correct policy of which subjects can do what to which objects, and that all good users will abide by this policy. However, in real-world enterprises, this is simply not the case: users systematically circumvent security controls (and other aspects of IT) in order to get their jobs done. This talk will survey such circumvention scenarios, explore their underlying structure, and consider some ways to improve the situation by reducing the gap between the de facto and the de jure.

December 2, 2015.**Mansoor Haider**, North Carolina State University,*Mathematical modeling of extracellular matrix regeneration in cartilage tissue engineering (An undergraduate Research Project*

**Abstract:**Articular cartilage physiology is regulated by specialized cells that are sparsely distributed within its extracellular matrix (ECM) and maintain a state of homeostasis in healthy tissue. ECM degeneration due to osteoarthritis can lead to complete degradation of cartilage surfaces, necessitating total joint replacement. Chondrocytes or stem cells can be used to regenerate cartilage via tissue engineering approaches in which these cells are seeded in biocompatible and degradable scaffolds. In such systems, biosynthetic activity of the cells results in regeneration and accumulation of ECM constituents along with degradation of the scaffold material. In this talk, mixture models are presented for interactions between biosynthesis of ECM constituents and ECM linking in cell-seeded scaffolds. Both ODE-based (temporal) models for evolution of average apparent densities and PDE-based (spatio-temporal) models will be presented for variables including unlinked ECM, linked ECM and scaffold. These models provide a quantitative framework for assessing and optimizing the design of engineered cell-scaffold systems and guiding strategies for articular cartilage tissue engineering.

November 18, 2015.**Jon Root, HC'10**,*TBD*

**Abstract:**We analyze the probability that a fixed $s$-sparse vector $x$ is recovered from $y=Ax$ via $\ell_1$-minimization using a random draw of an $m \times n$ Gaussian matrix $A$. In this setting the proof is relatively elegant and has the advantage of providing good constants in the estimate for the required number $m$ of measurements. Using Gordon's lemmas from the field of Gaussian processes as well as concentration of measure, we derive the sufficient condition $m > 2s \ln(N/s)$ which guarantees the fixed $s$ sparse vector $x$ is recovered from $y=Ax$ via $\ell_1$ minimization.

November 3, 2015.**Brianna Heggeseth**, Williams College,*The relationship between baseline factors and change over time: How easy it is to get it wrong!*

**Abstract:**In longitudinal studies, we have the opportunity to study the variability in both the starting levels as well as the change over time. Additionally, the relationship between the two could highlight important underlying mechanisms. For example, does having a high birthweight mean that you are destined to have a high growth rate throughout childhood? Or, does high in-utero chemical exposure result in a troubling development pattern? In this talk, I discuss why a simple and often used approach can result in incorrect conclusions and discuss how more complex approaches also get it wrong when trying to answer this question about baseline factors and change over time.

October 22, 2015.**Gideon Bahir-Maschler**, Clark University,*Distinguished metrics in conformal classes.*

Clavius Group Differential Geometry Seminar, July 16, 2015.**Lawrence Conlon**., Washington University,*An overview of the Handel-Miller theory of endperiodic maps.*

Clavius Group Topology and Foliations Seminar, July 16, 2015.**Patrick J. Ryan**, McMaster University,*Kimura’s Theorem: The classification of Hopf hypersurfaces with constant principal curvatures in complex projective space III.*

Clavius Group Differential Geometry Seminar, July 16, 2015.**Javier Leach, S.J.**, U. Complutense de Madrid, Spain,*Public and personal causation of believing processes.*

Clavius Topics Seminar, July 15, 2015.**Patrick J. Ryan**, McMaster University,*Kimura’s Theorem: The classification of Hopf hypersurfaces with constant principal curvatures in complex projective space II.*

Clavius Group Differential Geometry Seminar, July 15, 2015.**Martin Magid**, Wellesley College,*Minimal surfaces in the product of two dimensional real space forms endowed with a neutral metric.*

Clavius Group Differential Geometry Seminar, July 14, 2015.**Carlos E. Vasco**, Universidad Distrital de Bogota, Colombia,*Grundlagen der Pfeilkettenlehre. A model-theoretic approach to Chrono*topy beyond Cantorian points and sets towards arrows and chains.*

Clavius Group Topics Seminar, July 14, 2015.**Paul A. Schweitzer, S. J**., Pontificia Universidade Catolica, Rio de Janeiro,*Can exotic 4-manifolds be leaves of foliations?*

Clavius Group Topology and Foliations Seminar, July 14, 2015.**Patrick J. Ryan**, McMaster University,*Kimura’s Theorem: The classification of Hopf hypersurfaces with constant principal curvatures in complex projective space I*

Clavius Group Differential Geometry Seminar, July 13, 2015.**Carlos E. Vasco**, Universidad Distrital de Bogota, Colombia,*Was sind und sollen die Ur*Zahlen? A chrono*topic reframing of Arithmetic beyond Peano, Dedekind and Russell towards Pythagoras, Kant and Brouwer.*

Clavius Group Topics Seminar, July 13, 2015.**Ellen B. Ryan**, McMaster University,*Aging with Spirit Through Poetry*

Clavius Group Topics Seminar, July 13, 2015.**Nelson Velandia, S.J.**, Javeriana University,*Equations of motion of a spinning test particle in a Kerr spacetime - numerical solution*

Clavius Group Topics Seminar, July 9, 2015.**Jacob Binoy, S.J.**, Loyola Research Institute of Peace and International Relations, Kochi, India,*Randomness versus Fuzziness*

Clavius Group Topics Seminar, July 9, 2015.**Tommy Murphy**, California State University, Fullerton,*Complex Riemannian foliations of Hermitian symmetric spaces*

Clavius Group Differential Geometry Seminar, July 9, 2015.**Paul A. Schweitzer, S. J**., Pontificia Universidade Catolica, Rio de Janeiro,*Classical Invariants of Legendrian Knots on the 3-torus, II - Calculations using projections*

Clavius Group Topology and Foliations Seminar, July 8, 2015.**Andrew Hwang**,*Paper Surface Geometry*

Clavius Group Differential Geometry Seminar, July 8, 2015.**Palle Yourgrau**, Brandeis University,*On Time and Intuitionism: Some Questions for Brouwer*

Clavius Group Topics Seminar, July 7, 2015.**Carlos E. Vasco**, Universidad Distrital de Bogota, Colombia,*Chrono*topy before and after Arithmetic, Geometry and Analysis: Back to Pascal, Kant, and Brouwer?*

Clavius Group Topics Seminar, July 7, 2015.**Tommy Murphy**, California State University, Fullerton,*Curvature-adapted Foliations of Symmetric Spaces II*

Clavius Group Differential Geometry Seminar, July 7, 2015.**Tommy Murphy**, California State University, Fullerton,*Curvature-adapted Foliations of Symmetric Spaces I*

Clavius Group Differential Geometry Seminar, July 6, 2015.**Lawrence Conlon**,Washington University,*What is a lamination and why should I care?*

Clavius Group Topology and Foliations Seminar, July 6, 2015.**Tom Banchoff**, Brown University,*A Straight Shot at the Extrinsic Gauss-Bonnet Theorem*

Clavius Group Differential Geometry Seminar, July 2, 2015.**Tom Cecil**,*Clifford Algebras and Isoparametric Hypersurfaces in Spheres, II*

Clavius Group Differential Geometry Seminar, July 2, 2015.**Paul A. Schweitzer, S. J**., Pontificia Universidade Catolica, Rio de Janeiro,*Classical Invariants of Legendrian Knots on the 3-torus*

Clavius Group Topology and Foliations Seminar, July 1, 2015.**Tom Cecil**,*Clifford Algebras and Isoparametric Hypersurfaces in Spheres, I*

Clavius Group Differential Geometry Seminar, July 1, 2015.**Nicholas Horton**, Amherst College,*The Increasing Role of Data Science in the Mathematical Sciences*

**Abstract:**In a world of increasingly complex and sophisticated data, there is additional demand for graduates who are able to “think with data” and undertake computation in a nimble fashion, in order to extract actionable information. Gaining practice in utilizing all steps of the scientific method is vital for students in mathematics and statistics, in order to tackle real research questions. This talk will explore the statistical analysis process, which involves formulating good questions, considering whether available data are appropriate for addressing a problem, choosing from a set of different tools, undertaking the analyses in a reproducible manner, assessing the analytic methods, drawing appropriate conclusions, and communicating results. The talk will include examples and models from the speaker’s own research, including research with undergraduates.

Pi Mu Epsilon (PME) ceremony and colloquium, April 22, 2015.**Catherine Roberts**,*Environmental Mathematics*

**Abstract:**

Leonard C. Sulski Memorial Lecture, 4:30 p.m, April 16, 2015.**Cristina Ballantine**,*Partitions and divisors*

**Abstract:**The partition function counts the number of ways in which a positive integer can be written as a sum of positive integers without regard to order. There are interesting connections between the partition function and the number of divisors function. In particular, I will present several convolutions involving some variations of these two functions. I will also present some combinatorial interpretations of these results. (This is joint work with Mircea Merca.)

April 8, 2015.**Jeroen Sisling**, Dartmouth College,*Galois descent in geometry*

**Abstract:**Classical Galois theory tells us that if a complex number is invariant under all automorphisms of the complex field, then it in fact belongs to the rational field. We can also ask this question for geometric objects X over the complex numbers, as follows: If, upon transforming the defining equations of X by an automorphism of the complex numbers, one always ends up with an object that is isomorphic to the old X, can X then be defined by equations with rational coefficients? Surprisingly, the answer to this question is sometimes no. This talk will consider some different ways of looking at the problem in the case where X is a curve. The methods used include Galois cohomology and families of curves. Many concrete examples are given throughout.

March 26, 2015.**Robert Nazarian, HC'12**, PhD Candidate, Princeton University,*Around the World in 2880 Grid Cells: A Mathematical and Physical Approach to Studying and Modeling the Ocean, Atmosphere and Climate Systems*

**Abstract:**Over the past 50 years, elements of physics and mathematics (as well as computer science, geosciences, chemistry, biology, etc.) have allowed us to gain greater understanding of the ocean, atmosphere and climate systems. While in situ measurements, such as flight campaigns and oceanographic cruises, among others, are crucial to our understanding of these systems, the scientific community has been increasingly turning to complex numerical models to further our knowledge. These models, at their core, are comprised of equations representing the fluid dynamics (fluid evolution with time) and thermodynamics, with varying levels of complexity added to incorporate additional physical phenomena. We’ll first take a look at what fluid dynamics and thermodynamics go into building such a complex model (which include interesting problems in differential equations, numerical analysis and statistics), as well as the cutting-edge science that can be done with such intricate models. Such scientific topics may include paleoclimates, future climates, ocean currents and waves, weather patterns and extreme weather events, El Nino/La Nina and more based on time. This presentation will be a great opportunity to apply the physics and mathematics learned in the classroom to Earth-related phenomena we all regularly observe!

March 16, 2015.

**Dorothy Wallace**, Dartmouth College,*Building the in silico cancer tumor*

**Abstract:**Cancer researchers have observed that response of tumor cells to treatment varies depending on whether the cells are grown in monoculture as in vitro spheroids, or in vivo as mouse xenografts or human clinical trials. Solid tumors, whether in vitro or in vivo, are not an undifferentiated mass of cells. They include necrotic regions, regions of cells that are in a quiescent state (either slowly growing or not growing at all) and regions where cells proliferate rapidly. A range of observed qualitative properties of in vitro tumor spheroids allows mathematicians to test models for realistic qualitative behavior, and validates the role of TNF (tumor necrosis factor) in halting spheroid growth. I will discuss a model of in vitro spheroids based on data sets used to predict the effects of treatment on the tumor spheroid. I will also discuss a model of tumors grown in vivo as mouse xenografts. These have the additional feature of increased vascularization.

February 19, 2015.

### 2014

**Christine Sample**, Emmanuel College,*Population Structure and the Molecular Clock of Neutral Evolution*

**Abstract:**Evolution is driven by genetic mutations. While some mutations affect an organism's ability to survive and reproduce, most are neutral and have no effect. For almost 50 years, neutral mutations have been used as a “molecular clock” to estimate the timing of evolutionary events. We introduce a mathematical model to study how the rates of these molecular clocks are affected by the spatial arrangement of a population in its habitat. This graph-theoretic model can be applied to a variety of population structures. In one example, we investigate the accumulation of genetic mutations in the small intestine. In another application, we analyze Twitter networks to study the effect of network topology on the rate at which new ideas replace old ones.

October 23, 2014.**Andrew Niles**,*Moduli of elliptic curves via twisted stable maps*

**Abstract:**I will review some of the key ideas in the functorial algebraic geometry, focusing on moduli spaces of elliptic curves. I will then describe two ways of "compactifying" moduli spaces of elliptic curves and the relationship between the two.

October 1, 2014.**David Damiano**,*Handle Decompositions of Low Dimensional Manifolds*

**Abstract:**An n-dimensional manifold is a space that has the local structure of an n-dimensional Euclidean space. There are many different constructions and representations of manifolds, some of which depend on the dimension n. In this talk, I will give an intuitive introduction to handle decompositions. Although this is a technique that works in all dimensions, the focus will be on dimensions 2, 3, and 4. Based on Morse theory, this approach was first developed by Smale.

July 9, 2014.**Andrew Hwang**,*High-dimensional Euclidean Geometry*

**Abstract:**In Linear Algebra we study the vector space R^n of ordered n-tuples of real numbers. Familiar concepts of distance and angle make sense, but the high-dimensional analogs of balls and cubes have counterintuitive properties. We'll examine a selection of "weird" high-dimensional phenomena, including: (1) A plane does not separate R^4, just as a line does not separate R^3. (2) The diagonal of a high-dimensional cube is nearly orthogonal to the edges. (3) "Most" points of a cube lie far from the center. (Particularly, a ball inscribed in a cube occupies vanishingly small volume.) (4) An arbitrarily large 3-dimensional cube fits inside a 1-cm cube of sufficiently large dimension.

July 2, 2014.**John Little**,*Algebra by Another Name? Book II of Euclid's Elements*

**Abstract:**The Elements of Euclid (about 300 BCE) quickly established itself as the standard treatment of basic Greek mathematics and has been immensely influential ever since then. In this talk we will discuss some of the contents of Books I and II with specific attention to the question whether Book II contains a sort of "geometric algebra." We will see how different historians have proposed different answers to this question and follow a very heated and theatrical controversy that erupted in the late 1970's over different interpretations of the evidence. Drawing on these interpretations we will conclude with some general observations regarding the difference between mathematics itself and the history of mathematics.

June 25, 2014.**Cristina Ballantine**,*Polynomials over finite fields vs. polynomials over the reals (and the integers).*

June 18, 2014.**Susan Loepp**, Williams College,*Polynomials, Power Series, and Confessions of a Commutative Algebraist*

**Abstract:**Students discover the beauty and usefulness of polynomials early in their mathematical careers. Then, in Calculus, they are introduced to the intriguing idea of power series. In this talk, we will start by defining a new arithmetic for polynomials, followed by a discussion of several surprising relationships between polynomials and power series. Research results proved by undergraduates will be mentioned. The talk will be peppered with stories of the speaker’s own mathematical journey.

Pi Mu Epsilon (PME) ceremony and colloquium, April 30, 2014.**Julie Blackwood**, Williams College,*Rabies Persistence in Vampire Bats: Immunity Pathogenesis and Immigration*

**Abstract:**Vampire bats have a high incidence of carrying rabies virus. Each year, they cause thousands of livestock deaths and a few human fatalities. Dr. Blackwood and her colleagues developed four mathematical models of rabies transmission, each representing a different hypothesis for the biology of the rabies infection in a bat colony. The aim is to use their models ot develop more effective ways of controlling the spread of bat-borne pathogens among vampire bats in Peru.

Co-sponsored by Biology and Environmental Studies, March 25, 2014.**Joseph H. Silverman**, Brown University,*Dynamical Systems from a Number Theorist's Perspective*

**Abstract:**Discrete dynamics is the study of iteration. A primary objective of dynamics is the classification of points in a set S according to their orbits under repeated application of a self-map f:S→S. Classically, S is taken to be Rn or Cn, and real and complex dynamics are mature and thriving fields of study. But for a number theorist, it is natural to take S to be a set of arithmetic interest such as Z or Q. The past 25 years has seen the development of the field of Arithmetic Dynamics, in which one studies dynamical analogues of classical results in number theory and arithmetic geometry. Here are two illustrative problems: Let f(z)∈Q(z) be a rational functions. (1) There are always infinitely many complex numbers with finite forward orbit under iteration of f, but how many of these complex number can be rational numbers? (2) If we take a rational starting point α∈Q, when is it possible for infinitely many points in the orbit of α to be integers? In this talk I will discuss these and other problems in arithmetic dynamics. As is typical in number theory, there are many questions that are easy to state, but difficult to solve.

Leonard C. Sulski Memorial Lecture, 4:30 p.m., March 20, 2014.**Jen Paulhus**(HC'99), Grinnell College,*Decomposition of Jacobian Varieties*

**Abstract:**Every curve has an object connected to it called its Jacobian variety which is intimately connected to the points on the curve. The Jacobian has a natural group structure on it, analogous to the group structure of points on an elliptic curve. My thesis work involved developing a technique to decompose Jacobian varieties using the action of the automorphism group of the curve on the Jacobian, and my research program currently centers on questions about how Jacobian varieties factor. I'll assume some algebra but, depending on the audience, will not assume much algebraic geometry.

2:00 p.m., March 20, 2014.**Gina-Maria Pomann**, PhD candidate, North Carolina State University,*A Two Sample Test for Comparing Magnetic Resonance Images of Multiple Sclerosis Patients and Healthy Controls*

**Abstract:**A number of magnetic resonance (MR) imaging modalities can be used to measure the diffusion of water in the brain. An important question is which of these modalities are most useful for differentiating between MR images of patients with multiple sclerosis (MS) and those of healthy controls. We propose a hypothesis test that reduces the dimension of the testing problem in a way that enables the application of traditional nonparametric univariate tests. This results in a procedure that is computationally inexpensive. Simulation studies are presented to demonstrate the strength and validity of our approach. We also provide a comparison to a competing method. The proposed test is then illustrated by applying it to a state-of-the art diffusion tensor imaging (DTI) study where the objective is to compare white matter tract profiles in healthy individuals and multiple sclerosis (MS) patients.

February 10, 2014.**Matthew Gardner Spencer**,*The Nottingham Group*

**Abstract:**The Nottingham group is a well-studied finitely generated pro-p group. We will define the Nottingham group both as a group of power series and as a group of automorphisms, reviewing relevant concepts as needed. We'll then give some idea of why it is studied. Finally, we will discuss how I use the Nottingham group as a tool to help study and classify other sets of power series.

January 29, 2014.

### 2013

**Kathi Fisler**, WPI,*Could Software-Analysis Detect Usable Security Flaws?*

**Abstract:**Human-Computer Interaction studies how to create computing systems that people can use effectively and reliably. Usable security focuses on the usability of features that affect security or privacy. Usable security is hard to achieve and expensive to measure. Automated software-analysis tools have helped locate subtle design flaws in several domains. Could we create software-analysis tools that point designers towards potential usable-security flaws?

This talk gives an overview of software analysis (based on model checking), presents a definition of usable security, and explores how model-checking techniques might help designers evaluate usable security in sharing-oriented web applications. The talk assumes no prior background in either model checking or usable security, though experience using social networks or cloud-based data-sharing systems would be useful.

November 20, 2013.**David Dumas**, University of Illinois at Chicago,*Examples of Moduli Spaces.*

**Abstract:**Through a series of examples we will introduce the concept of a moduli space (also known as "shape space" or "configuration space") and show how these spaces provide powerful tools for studying families of related geometric objects. No background in this subject will be assumed, and we will start with simple and familiar objects like lines in the plane. We will see that even the simplest examples of moduli spaces exhibit rich and interesting geometry, and that a wide variety of mathematical techniques contribute to their study.

November 13, 2013.**Nancy R. Cook**(HC '76), Harvard Medical School,*Statistics in Medicine: Risk Prediction Models for Cardiovascular Disease in Women*

Leonard C. Sulski Memorial Lecture, April 11, 2013. PDF

### 2012

**Martin A. Nowak**, Harvard University,*The Evolution of Cooperation*

Pi Mu Epsilon Ceremony and Colloquium, May 2, 2012.**Victor Moll**, Tulane University,*The Evaluation of Integrals. There is More in it Than You Suspect.*

**Abstract:**The question of providing a closed-form for the evaluation of definite integrals is one of the standard topics in elementary Calculus courses. This talk will show relations of this elementary question to a variety of areas in Mathematics. There will be some Number Theory, some Combinatorics and a surprising connection to Dynamical Systems. Many possible projects for students will be described.

April 23, 2012.**Ed Soares***Statistical Evaluation of Pre-clinical Data: Power Analysis, Ratio Distributions, (h,phi) Divergence Measures, and ANOVA*

**Abstract:**In the first part of the talk, I will discuss some recent work on performing a power analysis using a one-way layout Multivariate Analysis of Variance (MANOVA). This is the multivariate equivalent of the two-sample t-test under the assumption of equal variances between the two populations under consideration. I will present the theoretical underpinnings as well as curves that relate sample size required to identify a statistically significant difference versus the effect size. The curves will be parametrized by the correlation between the features. In the second part, I wish to discuss some open problems that need to be addressed in the evaluation of pre-clinical data. These include: 1) Ratio distributions - useful when generating confidence intervals when longitudinal data have been normalized to the day 0 measurement 2) (h,phi) divergence measures - useful when determining if two histograms describe the same probability distribution. I will focus on the use of the Bhattacharyya distance with application to segmented MRI images, used to evaluate muscle growth. 3) If there is time, i will also discuss analysis of variance.

April 19, 2012.**Davide Cervone**, Union College,*The Hypercube and Hypersphere: Breaking Them Down and Building Them Up*

Leonard C. Sulski Memorial Lecture, March 2, 2012.**Tom Cecil**,*Dupin Hypersurfaces With Four Principal Curvatures*

**Abstract:**A hypersurface M embedded in the sphere S^{n}is proper Dupin if the number g of distinct principal curvatures is constant on M, and each principal curvature is constant along each leaf of its corresponding principal foliation. This property was shown to be invariant under the group of Lie sphere transformations by Pinkall in 1981. Thorbergsson proved in 1983 that for a compact, connected proper Dupin hypersurface M ⊂ S^{n}, g must be 1, 2, 3, 4 or 6, the same as Muenzner’s restriction for isoparametric hypersurfaces in S^{n}. In 1985, P.J. Ryan and the author conjectured that every compact, connected proper Dupin hypersurface M ⊂ S^{n}is equivalent to an isoparametric hypersurface by a Lie sphere transformation. The conjecture is true for g = 1, 2 and 3, but it was shown to be false in the case g = 4 in 1989 by in- dependent constructions due to Miyaoka-Ozawa and Pinkall-Thorbergsson. The construction of Miyaoka-Ozawa also provides counterexamples to the conjecture in the case g = 6. These counterexamples do not have constant Lie curvatures (invariants introduced by Miyaoka), which are the cross-ratios of the principal curvatures taken four at a time. A revised conjecture with the additional assumption of constant Lie curvatures is still open, and we will discuss recent progress on the revised conjecture in the case g = 4 by Q.-S. Chi, G. Jensen and the author.

February 20, 2012.**Gideon Maschler**, Clark University,*Kahler Metrics Conformal to Curvature-Distinguished Metrics*

**Abstract:**On certain manifolds admitting a complex structure there exist Riemannian metrics with nice geometric properties, and these are called Kahler metrics. We describe various attempts to link Kahler metrics to metrics with distinguished curvature characteristics. Some applications will also be given if time permits.

February 9, 2012.**Andrew Hwang***Constructing Kahler Metrics on the Total Space of a Line Bundle*

**Abstract:**Explicit examples of metrics having specified scalar curvature are generally difficult to construct. Starting nearly from scratch, we'll review the basics of metrics and curvature, holomorphic manifolds and line bundles, and an ansatz (due to Calabi) for Kahler metrics on the total space of a line bundle, in which the scalar curvature becomes an explicit linear second-order differential expression in a single unknown function of one real variable.

February 2012.

### 2011

**Matthew Demers**, (Ph.D. Candidate) Northwestern University,*Pattern Formation in Multicomponent Membranes*

**Abstract:**Multicomponent fluid membranes are found in many interesting physical systems. The presence of multiple components is an important feature, as it allows for the coupling of chemical and geometric properties, which can in turn lead to the formation of interesting shapes and patterns. I'll look at the example of three-component lipid vesicles and show some of the beautiful structures they can form.

November 30, 2011.**Giuliana Davidoff**, Mount Holyoke College,*Representations of Integers by Quadratic Forms and Generalizations*

**Abstract:**One of the questions that drove the development of modern number theory was that of representation of integers by quadratic forms. For example, Fermat found, among other results, that a prime could be represented as a sum of two squares if and only if it is of the form p = 4n + 1, for some integer n. In modern language we write this as follows: p=x^{2}+y^{2}if and only if p ≅ 1 (mod4). With such statements, Fermat brilliantly connected the representation of integers by certain quadratic equations to the existence of primes in particular arithmetic progressions arising naturally from these equations. Further work by Euler, Legendre and Lagrange was brought into its fullest form by Gauss, who studied the most general form of quadratic equations in two unknowns, written as f(x,y)=ax^{2}+bxy+cy^{2}. Gauss developed a rich theory of the representation of integers by such forms, which is still the most useful way of studying what eventually came to be called the arithmetic of quadratic extensions of the rational numbers. We will discuss this work and some recent extensions of it by Bhargava to number fields of higher degree.

November 21, 2011.**William B. Thompson**, University of Utah,*Visual Perception from a Computer Graphics and Virtual Environments Perspective*

**Abstract:**Computer graphics produces images intended to be seen by people, yet relatively few practitioners in the field know much about the specifics of human vision. The first part of this talk will argue the importance of understanding human visual perception for computer graphics, scientific and information visualization, virtual environments, and related areas. The second part of the talk will provide an overview of current work in my research group involving embodied perception in virtual environments and computer animation.

November 3, 2011.**Catherine Roberts**,*Improving our Children's Achievement in Mathematics, One Teacher at a Time*

**Abstract:**This talk will describe the Intel Math Program, a professional development course for K-8 school teachers reaching thousands of teachers across the US, and my personal experience as an instructor for the program here in Massachusetts.

October 20, 2011.**Ben Coleman**and Kevin Hartshorn, Moravian College,*Game, SET, Math*

**Abstract:**The elegance of the card game SET® is found in the connection between simple game play and deep mathematical ideas. A portion of this talk will discuss the game and a variety of fascinating mathematical properties of the deck and game play. We will also discuss our own research to determine the number of structurally distinct ways to take 1 ≤ n ≤ 81 cards from the 81 cards of the deck. We will describe not only the technique that ultimately produced the answer, but also tell the story of our successes and failures along the way. In doing so, we will show how topics in discrete mathematics, linear algebra, geometry, computational complexity, and group theory can be seen in this fascinating game.

September 29, 2011.**John Little**,*Toric Surface Codes -- Some New Observations*

**Abstract:**One of the central problems in coding theory is to find ways of constructing linear codes (that is, vector subspaces C of F^n -- F a finite field) that have good minimum distance properties (that is, such that any two distinct elements of C differ in as many coordinates as possible -- this is closely related to the error-correction capacity when the elements of C are used to encode information sent over a noisy channel). One construction that has received attention from a number of authors recently is the toric surface code construction. In concrete terms, this starts from a collection of integer lattice points (a,b) in the ordinary Euclidean plane. We construct elements of a code C over a finite field F by evaluating the corresponding monomials x^a y^b at all (x,y) in the (algebraic) torus (F^*)^2. In this talk, we will describe this construction in more detail, discuss some very good examples of codes that have been constructed this way, and summarize what is known about the minimum distances of these codes.

Then we will consider what happens when we fix the collection of lattice points (a,b), but change the field F (either by taking algebraic extensions, or by just going to fields of larger size but different characteristic). We will see that in some cases, the codes we get for all sufficiently large F have the same minimum distance as the codes from the set of all lattice points in the convex hull of the original set. On the other hand, there are cases where some interesting arithmetic properties of algebraic (specifically elliptic!) curves over finite finite fields come into play and give unexpectedly different results

September 8, 2011.**John Anderson**,*The Hull of Rudin's Klein Bottle*

**Abstract**: In 1981 Walter Rudin exhibited a totally real imbedding of the Klein bottle into C^{2}. I'll talk about the problem of finding holomorphic images of the unit disk in C with boundaries on Rudin's Klein bottle, and show that there are enough such disks so that their interiors fill an open subset of C^{2}. I'll also attempt to explain why this is interesting.

May 2, 2011.**Alisa DeStefano**,*Universal Observability of Dynamical Systems*

**Abstract**: A general question in control theory is whether the solution of a dynamical system is uniquely determined by a set of measurements of the system. If this is the case, the system is said to be observable. There are different ways to observe a dynamical system and some systems may be observable using only certain types of measurements. One might ask if there is a dynamical system that can be observed by any continuous function from the state space to the real numbers? If such a system exists, it is said to be universally observable. It might seem unlikely that such a system could exist. However, there are two known examples of such systems. We will discuss the history of the problem and then give the motivation and a brief outline of the construction of an elementary universally observable system on the two-dimensional torus. The ideas involved come from topological dynamics and small divisor problems. We also discuss an equivalence between universal observability and certain properties in the field of topological dynamics.

April 18, 2011.**Keith Ouellette**,*On the Fourier inversion theorem for PGL(2,Qp)*

**Abstract**: We will demonstrate our proof of the Fourier inversion and Plancherel formulae for spherical unramified principal series for the p-adic group PGL(2, Qp) where Qp is the quotient field of Zp, the p-adic integers. If time remains, we will apply the Fourier inversion theorem to transforms of some smooth compactly supported functions of PGL(2, Qp) for illustration.

March 21, 2011.**Carolyn Gordon**, Dartmouth College,*You Can't Hear the Shape of a Drum*

**Abstract:**In spectroscopy, one attempts to recover the shape or chemical composition of an object from the characteristic frequencies of sound or light emitted. Mark Kac's question "Can one hear the shape of a drum?" asks whether two membranes (drumheads) which vibrate at the same characteristic frequencies must have the same shape. We answer Kac's question negatively by constructing a pair of exotic shaped sound-alike drums. We also listen to a computer simulation, produced by Dennis DeTurck, of the sounds of these exotic drums.

Leonard C. Sulski Memorial Lecture, March 9, 2011.**Harold Connamacher**, Albion College,*The Satisfiability Threshold of Random Problems*

**Abstract**: A group of researchers attempting to classify the average difficulty of some notoriously hard problems (such as the NP complete problems) made an interesting discovery. For many problems, if you consider random instances with an increasing number of constraints, the probability that the instance has a solution undergoes a sudden transition at a specific constraint density. This threshold in the probability of having a solution attracted the attention of physicists and mathematicians because it seems similar to other thresholds in physical and mathematical structures such as thresholds encountered in random graphs and the transitions in an iron magnet as it cools. Despite over a decade of work on the subject, the major questions are still unresolved. This talk will survey the subject and describe some recent results.

February 10, 2011.**Kevin Walsh**, Cornell University,*Authorization in Nexus*

**Abstract**: We have little reason to trust computer systems. We do not know what software lurks on the other side of a network connection, or even what software runs on our own machines. We have few means to specify or reason about why we might trust a piece of software. And we do not have adequate authorization mechanisms in place to limit the damage that rogue software can inflict. In this talk I revisit the problem of authorization, and show how a combination of new secure hardware and a formal logic can result in a more trustworthy computer system.

February 9, 2011.**Audrey Lee-St.John**, Mount Holyoke College,*Rigidity Theory: from Foundations to Applications*

**Abstract**: I will present foundations for understanding rigidity properties of mechanical structures. Bar-and-joint frameworks, made up of rigid bars connected by universal joints, arise in important current applications; in engineering, they can model robots or buildings and, in biology, they can model proteins. Although such structures are the simplest type in a wide range of constraint systems, their mechanical properties (rigidity and flexibility) are still not fully understood.

For certain types of constraint systems, combinatorial results have led to an elegant and efficient algorithm called the pebble game. I will present this algorithm and several generalizations, using Java applets and physical models for demonstration. In addition, I will describe current work that addresses constraint systems arising from the widely-used CAD (Computer Aided Design) software SolidWorks as well as applications in computational biology.

February 7, 2011.**David Kauchak**, Pomona College,*Learning to Simplify Sentences Using Wikipedia*

**Abstract**: The complexity and readability of text can vary widely. Compare, for example, a research article and a popular media article on the same topic. The articles may discuss the same material, but their usefulness for the reader can vary widely depending on the reader's background; while the fundamental concepts of the research article may be accessible, the language and structure of the article may prohibit the lay-reader from understanding these concepts.

In this talk I will examine the problem of sentence simplification which aims to develop models to automatically reduce the reading complexity of a sentence. I will introduce a new data set that we have generated consisting of over a hundred thousand aligned Sentences from English Wikipedia and Simple English Wikipedia. Using this data set, we have developed a simplification system by extending work in statistical language translation. I will conclude with initial performance results.

January 31, 2011.

### 2010

**Megan M Kerr**, Wellesley College,*Symmetries in Geometry: Exploring (different) Constant Curvature Spaces*

November 30, 2010.**Sarah Wright**,*Graph C*-Algebras and Aperiodic Paths*

**Abstract**: Associating a visual object with a complicated mathematical structure is a popular and successful technique for teaching and discovering new ideas. We'll introduce the idea of a graph-algebra, where we associate a C*-algebra to some type of "graph". We'll discuss the "how's", "who's" and "why we cares" of this field of study. Once we have the basics, we can move on to the idea of aperiodicity. The condition "every cycle has an entry" first appeared in the literature in Kumjian, Pask, and Raeburn's paper on Cuntz-Krieger algebras of directed graphs, where it was called Condition (L). It provides a necessary condition for simplicity of the associated graph algebra. This condition has been generalized to aperiodicity conditions in the theory of topological graphs (Katsura), k-graphs (Kumjian, Pask), and the unifying theory of topological k-graphs (Yeend). We'll discuss the details of these generalizations as well as the theorems associated with them. We'll then introduce a Condition (F) on the finite paths of a topological k-graph that is equivalent to the corresponding aperiodicity condition. Hence we obtain a condition which is much easier to check than the aperiodicity of infinite paths.

November 15, 2010.**Cristina Ballantine**,*Powers of the Vandermonde determinant, Schur functions and the dimension game*

**Abstract**: Vandermonde determinants are ubiquitous in mathematics. Since every even power of the Vandermonde determinant is a symmetric polynomial, we would like to understand its decomposition in terms of the basis of Schur functions. Besides its obvious importance in mathematics, such a decomposition would shed light on the quantum Hall effect, in particular on the Slater decomposition of the Laughlin wave function. While we will not explore the problem from the point of view of physics, we will investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give an inductive approach of computing some of the coefficients by building them up from tetris type shapes.

November 8, 2010.**Keith Ouellette**,*On the Fourier Inversion Formula for the Full Modular Group*

**Abstract**: We offer a new proof of the Fourier inversion and Plancherel formulae for Maass-Eisenstein wave packets. The proof (presented in the case of the full modular group) uses truncation, basic analysis, and classical Fourier theory. Brief sketches of the proofs due to Langlands, Lapid, and Casselman are then presented for comparison.

October 25, 2010.**Annalisa Crannell**, Franklin and Marshall College,*Fibonacci Harps and a Shift of Finite Type*

October 19, 2010.**Dan Kennedy**, The Baylor School (Chattanooga, TN)

September 24, 2010.**John Little**,*An Euler-Maclaurin Formula for Polytopes and Hirzebruch-Riemann-Roch for Toric Varieties*

Abstract: In its basic form, the classical Euler-Maclaurin summation formula relates the integral of a smooth function over an interval to the sum of the values of the function at points in the interval, with an error term involving Bernoulli numbers and derivatives of the function at the endpoints. We will start out by looking at this elementary result and some of its applications. In the 1990's, two (transplanted) Russians, Khovanskii and Pukhlikov, published a generalization of Euler-Maclaurin where the interval above is replaced by a convex polytope in a higher dimensional Euclidean space. We will look at their generalization, again in essentially elementary terms. All of this is essentially generating function "magic." But the generating function of the Bernoulli numbers, f(x) = x/(1 - e^{-x}), also quickly leads to Todd classes and the Hirzebruch-Riemann-Roch theorem. Polytopes lead to projective toric varieties. Khovanskii-Pukhlikov gives an amazing proof of the Hirzebruch-Riemann-Roch theorem in this case. I hope to show you how it all works!

September 13, 20, 27, and October 4, 2010.**Barbara Reynolds, S.D.S.**, Stritch University,*Explorations in Plane Geometry with The Geometer's Sketchpad (GSP 5)*

Clavius Group Seminar, July 21, 2010.**Tom Banchoff**, Brown University,*Piecewise Circular Space Curves*

Clavius Group Seminar, July 20, 2010.**Javier Leach, S.J.**, Universidad Complutense de Madrid,*Mathematics and Reli*

*gion: Our Language of Sign and Symbol*

Clavius Group Seminar, July 20, 2010.**Paul A. Schweitzer, S. J**., Pontificia Universidade Catolica, Rio de Janeiro,*The Structure of Reeb Components: An Elementary Presentation with Pictures*

Clavius Group Seminar, July 20, 2010.**Pedro Suarez**, Barry University,

Clavius Group Seminar, July 19, 2010.>**Richard Freije**, Sun Life Company,*The Problem with the Internet*

**Abstract**: Some of the earliest legal cases involving the internet addressed liability of internet service providers. In particular, the cases asked if Compuserve, Prodigy, AOL and Netcom should be responsible for harassment, defamation or copyright infringement that occurred through use of theses services. After almost twenty years, and intervention by Congress, courts and commentators continue to argue the correct balance between freedom of speech and an open internet versus protection of intellectual property and the right to be free from cyberbullying or defamation.

*The Problem with the Internet* is that it remains a platform for bad behavior; the biggest players in the technology world including Google, YouTube, Facebook and even Apple are benefiting; and no one can seem to figure out a reasonable balance between competing interests. This talk will address the history and current state of this ongoing concern.

Clavius Group Seminar, July 15, 2010.**Marty Magid**, Wellesley College,*Timelike Minimal Surfaces via the Split-Complex Numbers*

Clavius Group Seminar, July 15, 2010.**Nelson Velandia, S.J.**, Universidad Nacional de Colombia,*Space, Time and Gravity*

Clavius Group Seminar, July 15, 2010.**Nelson Velandia, S.J.**, Universidad Nacional de Colombia,*Particles in Schwarzschild Coordinates and Eddington - Finkelstein Coordinates*

Clavius Group Seminar, July 13, 2010.**John Little**,*A Geometric View of Continued Fractions*

**Abstract**: "Ordinary" and "Hirzebruch-Jung" continued fraction expansions of rational numbers have very nice geometric interpretations involving sets of integer lattice points in polygons in the Euclidean plane. (This goes back ultimately to work of Felix Klein, but has been rediscovered independently several times.) We will develop this connection "from scratch" in the first hour, and then briefly indicate some applications in the theory of plane curve singularities and toric surfaces from algebraic geometry in the second.

Clavius Group Seminar, July 12 and 13, 2010.**Gareth Roberts,***Cyclic Central Configurations in the Four-Body Problem*

**Abstract:**We investigate the set of central configurations lying on a common circle in the Newtonian four-body problem. Such a configuration will be referred to as a cyclic central configuration (ccc). A central configuration is a special choice of positions where the gravitational force on each body is a scalar multiple of that body's position. Such a configuration leads to homothetic and homographic periodic solutions in the N-body problem. Using mutual distances as coordinates, we show that the set of ccc's with positive masses is a two-dimensional surface, a graph over two of the exterior side-lengths. Two symmetric families, the kite and isosceles trapezoid, are examined extensively. It is conjectured that if a ccc exists for a particular choice of masses, then it is unique. The techniques utilized in our work range from classical geometry (eg. the Cayley-Menger determinant and Ptolemy's Theorem) to modern computational algebra (eg. Grobner bases and Sturm's Theorem.)

Clavius Group Seminar, July 8, 2010.**Mike Ackerman**, Bellarmine University,*Results on Graphs of Semigroups*

Clavius Group Seminar, July 7, 2010.**Tom Banchoff**, Brown University,(part 1)*Self-Linking, Inflections, and Normal Euler Classes for Polygons and Polyhedra*

Clavius Group Seminar, July 6, 2010.,**Ockle Johnson**, Keene State College*Self-Linking, Inflections, and Normal Euler Classes for Polygons and Polyhedra*(part 2)

Clavius Group Seminar, July 6, 2010.**Pat Ryan**, McMaster University,*Isoparametric Hypersurfaces in Complex Space Forms*

Clavius Group Seminar, July 6, 8, 9, 12, 13 and 16, 2010.**Tom Banchoff**, Brown University,**Mike May, S.J.**, St. Louis University, and**Barbara Reynolds, S.D.S.**, Stritch University,*Pedagogy Meets Technology*

Clavius Group Seminar, July 2, 2010.,**Thomas Cecil***Isoparametric Hypersurfaces in Spheres*

Clavius Group Seminar, June 29, July 1 and 2, 2010.**Paul A. Schweitzer, S. J**., Pontificia Universidade Catolica, Rio de Janeiro,*Invariants of Legendrian Knots in T^3 and Similar 3-manifolds*

Clavius Group Seminar, July 1, 2010.**Solomon Friedberg**, Boston College,*Stitching Primes Together*

**Abstract:**Whole numbers that are two or more have factorizations: they are products of a finite number of primes. Can we find a way to do something interesting with*all*the prime numbers at once?

Pi Mu Epsilon Ceremony and Colloquium, April 14, 2010.**Rick Miranda**, Colorado State University,*Musical, Physical, and Mathematical Intervals - how fretting a guitar is more complicated (and more simple) than one might think*PDF

**Abstract:**The intervals between frequencies of musical notes on a scale have been studied for millennia, and the Pythagorean method is to relate these intervals to ratios of small integers (e.g., an octave has frequency exactly twice the base note, and the fifth (e.g. a G above a C) has frequency 3/2 times the base note). This leads to problems in transposing music, and the 'equal temperament' solution (from Renaissance times) finds a single ratio between half-notes (or adjacent frets on a guitar) that works fairly well. In 1743 Daniel Strahle (a craftsman with no apparent mathematical training) designed a simple geometric construction for placing the frets on a guitar that approximated the proper ratio. It was wrongly dismissed at the time as being inaccurate (by mathematicians no less!), but over 200 years later, in the second half of the 20th century, it was shown to be related to two simple and important approximation techniques (for functions and for irrational numbers), and to be extremely accurate.

In this lecture I'll review the basics from music theory, discuss a bit of history (Galileo is involved!) and then explain why Strahle's construction is so good.

Leonard C. Sulski Memorial Lecture, March 12, 2010.

### 2009

**Siman Wong**, UMASS-Amherst,*How many primes are there?*

: For many of us, Euclid's argument that there are infinitely many primes is probably one of the first proofs we encountered. In this talk we will discuss variations and refinements of this well-known proof, both as a starting point of further investigations and as an illustration of how mathematicians tackle research problems.**Abstract**

December 3, 2009.**Stephen Abbott**, Middlebury College,*A Brief History of Integration from Cauchy to Riemann to Lebesgue to...Riemann*

**Abstract:**In the first half of the 19th century there was significant ambiguity about the proper definition of the integral--was it an area or an anti-derivative? Riemann's familiar integral from 1850--the one we still teach in calculus--was actually a modification of a proposal by Cauchy intended to divorce the integral from the derivative, but it was not without shortcomings. In particular, the collection of functions that could be integrated was lacking (i) limits of some convergent sequences and, more surprisingly, (ii) an entire class of derivatives. In 1901, Henri Lebesgue introduced a new definition of the integral whose elegant resolution to problem (i) propelled it to become the undisputed industry standard. There is, however, a modern and much less well-known integral that is more powerful than Lebesgue’s, simpler to define, and solves problem (ii) by providing the world’s shortest proof of the Fundamental Theorem of Calculus.

Pi Mu Epsilon Ceremony and Colloquium, April 22, 2009.**Joe McKenna**, University of Connecticut,*Suspension Bridges Behaving Badly*

**Abstract:**The Tacoma Narrows Bridge disaster in 1940 is one of the most spectacular and well documented bridge collapses. Originally it was thought that the collapse was due to resonance, a phenomenon associated with linear differential equations. However, new insights from modern analysis reveal that the collapse is better explained by properties of solutions of nonlinear differential equations.

April 1, 2009.**Paul A. Schweitzer, S. J.**, Pontificia Universidade Catolica, Rio deJaneiro,*Asymptotic linking of volume-preserving R*^{k}actions on domains in R^{n}

**Abstract:**V. Arnol'd studied asymptotic linking of the flowlines of volume-preserving (i.e. divergence-free) smooth flows on a compact convex domain in R^{3}, related to the linking of closed curves in R^{3}and the helical twisting of the two flows. I shall describe Arnol'd's fascinating theory, with his two equivalent definitions of the asymptotic linking invariant: first, as a probabilistic integral involving Gauss' formula for linking of two disjoint simple closed curves in R^{3}, and second, in terms of the differential forms dual to the vector fields that generate the flows. Then I shall briefly describe a generalization discovered by my doctoral student Jose Luis Lizarbe Chira to asymptotic linking of volume-preserving actions of R^{k}and $R^\ell$ on a convex domain in R^{n}, where $k + \ell + 1 = n$.

March 26, 2009.**Paul A. Schweitzer, S. J.**, Pontificia Universidade Catolica, Rio deJaneiro,*Surfaces and 3-dimensional Manifolds: How Geometry Comes to the Aid of Topology*

**Abstract:**The talk will begin with a short outline of the classification of closed surfaces, their Euler characteristic and curvature of metrics, and how the metric can flow to a constant curvature metric. We next talk about 3-manifolds, their geometries and Thurston's geometrization conjecture. Finally, we consider the idea of Perelman's proof of the geometrization conjecture, with a flow making the geometry as homogeneous as possible, distributing the curvature equally. The mathematical details are lengthy and difficult, so we shall emphasize the intuitive ideas with pictures, rather than giving proofs.

Leonard C. Sulski Memorial Lecture, March 24, 2009.**John Little**,*The Spectral Sequence of a Filtration*

**Abstract:**Spectral sequences are an important and useful tool in modern algebraic topology and algebraic geometry. But, unfortunately, their algebraic "overhead" has given them something of a bad reputation. The goal of these talks is to partially de-mystify the idea by considering a rather concrete situation that leads to a commonly-encountered type of spectral sequence. Namely, we will consider computing the homology of a topological space X with a (finite) filtration by subspaces: X_0 \subset X_1 \subset \cdots \subset X_n = X. (CW complexes are especially nice examples.) We will construct spectral sequences from such a filtration which converge to the homology or cohomology of the whole space and consider some examples from the theory of toric varieties.

March 19, 2009.**Helen Wong**, Bowdoin College,*Quantum invariants for 3-dimensional manifolds*

**Abstract:**Growing out of the Jones polynomial for links, the quantum invariant for 3-dimensional manifolds are an exciting and relatively new development in topology. Its construction is based on deep theorems in classical topology, its inspiration is from quantum mechanics, and the theory has applications from physics to quantum computation to combinatorics. A brief overview of how it is defined and give some applications to topological questions will be given.

February 25, 2009.**Reinier Broker**, Worcester State College,*Constructing cryptographic elliptic curves*

**Abstract:**Elliptic curves play a key role in Wiles' proof of Fermat's Last Theorem. Besides being of key importance in abstract mathematics, they have become increasingly important for crytography. They are being used in fast algorithms to factor integers in prime numbers, and they form the basis of an entire industry of cryptographic protocols.

In this talk I will explain the basic ideas behind elliptic curve cryptography. As we will see, there are `good' crypto curves and `bad' crypto curves. I will present an efficient way to construct a `good' cryptographic curve. Many examples will be given.

February 19, 2009.**Gareth Roberts**,*On Central Configurations, II*, February 12, 2009.**Gareth Roberts**,*On Central Configurations, I*

**Abstract:**Central configurations (c.c.'s) play an important role in the Newtonian n-body problem. Physically speaking, a c.c. is a configuration of masses for which the gravitational force on each body points toward the center of mass (with an appropriate scaling). Mathematically, a c.c. is a solution to a complicated system of nonlinear algebraic equations. Three bodies of any mass placed at the vertices of an equilateral triangle is perhaps the most famous c.c. (Lagrange, 1772). C.C's play a vital role in spacecraft transport as ideal locations for observational spacecraft as well as helping determine low-energy trajectories to explore our solar system. In these talks, I will present an overview of the main results on central configurations, including the many different mathematical approaches to the subject (analytical, algebraic, topological, etc.) as well as the major open questions.

February 05, 2009.

### 2008

**Alex Popa**,*Applications of Modular Forms, II*,

December 02, 2008.**Gabe Weaver**,*Computing with Diagrammatic Content: Lessons Learned from Archimedes*

**Abstract:**The Episteme Project focuses on developing tools for increasing accessibility to and awareness of Ancient Scientific and Mathematical works. Unique to Episteme is its focus on representing and computing with diagrammatic information. Rather than treating diagrams as images, Episteme treats diagrams as a unique data type capable of representing a dynamic process in static form. Using this approach, both person and computer can interact with diagrams in terms of what they represent rather than how they are drawn. More specifically, Episteme explores the traditional operations of navigation, production, and logical assertion on diagrams through the lens of computation. In addition, non-traditional modes of interacting with diagrams such as querying become feasible.

Particular attention will be paid to the development of the Diagram Markup Language (DML) and the architecture of the DML compiler. The compiler is unique in that programs written in DML are not compiled into assembly code, but rather into SVG images, annotated with logical content encoding their meaning.

More information is available at http://www.episteme.ws/

November 19, 2008.**Alex Popa**, Applications of Modular Forms, I,

November 18, 2008.**Casey Douglas**, Rice University,*Perturbed, Genus-One Scherk Surfaces and Their Limits*

**Abstract:**The singly periodic, genus one helicoid is conjectured to be the limit of a one parameter family of doubly periodic minimal surfaces referred to as Perturbed Genus One Scherk Surfaces. Using both classical elliptic function theory and more recent handle-addition techniques for minimal surfaces, we prove the existence of these perturbed surfaces and verify their conjectured limit.

November 11, 2008.**Ed Soares**,*Evaluation of pharmaceutical treatments for cancer models, II*,

October 28, 2008.**Ed Soares**,*Evaluation of pharmaceutical treatments for cancer models, I*

**Abstract:**The talk describes joint work with Jack Hoppin and his new company inviCRO (along with Northeastern U, UNM, etc). The idea is to evaluate pharmaceutical treatments for cancer models. One acquires SPECT data from mice (control and treatment groups), derives activity and concentration measures of cancerous tumors, and evaluates these measures longitudally. A multivariate analysis of variance is used to test the null hypothesis that there is no difference between control and treatment groups. I am running this methodology on some real data to see if the statistical test will bear out what we see visually. Again, PCA is also used.

October 21, 2008.**John Anderson**,*Hardy spaces*

**Abstract:**I will give a brief introduction to the theory of classical Hardy spaces and their modern counterparts, with particular attention to the duality between the Hardy space H^1 and the space BMO of functions of bounded mean oscillation.

October 07, 2008.**Jack Hoppin**, Co-Founder InviCRO, LLC,*The Role of SPECT in Pre-Clinical Molecular Imaging*,

**Abstract:**The recent development of micro-imaging systems has resulted in new research opportunities for scientists and businesses. Using animal models of disease, coupled with increased image spatial resolution, these entities use small-animal imaging to better identify disease pathways and the effectiveness of drug treatments. By fusing structural x-ray computed tomographic (CT) images to emission images, such as single-photon emission computed tomography (SPECT), one can better identify regions of increased tracer uptake, which is common to cancerous tissue.

October 6, 2008.**Tom Cecil**,*Lie Sphere Geometry and Dupin Hypersurfaces, III*, September 30, 2008.**Tom Cecil**,*Lie Sphere Geometry and Dupin Hypersurfaces, II*, September 23, 2008**Tom Cecil**,*Lie Sphere Geometry and Dupin Hypersurfaces, I*, September 16, 2008.**Robert Benedetto**, Amherst College,*The abc Conjecture: An Introduction*, Department of Mathematics and Computer Science Pi Mu Epsilon Ceremony and Colloquium,

April 30, 2008.**Donal B. O'Shea**, Holyoke College,*The Shape We're In: The Poincare Conjecture*, Leonard C. Sulski Memorial Lecture,

April 22, 2008.**Gareth Roberts**,*The Planar, Circular, Restricted Four-Body Problem*

**Abstract:**An interesting case of the Newtonian four-body problem arises when one infinitesimal body interacts with three larger ones (primaries) fixed at the vertices of a rotating equilateral triangle. This problem, a planar, circular, restricted four-body problem (PCR4BP), can be characterized by a special potential function V. We investigate the number and location of the critical points of V (equilibria or "parking spaces") as a function of the masses of the three primaries. In particular, we prove that the number of critical points is finite for any choice of masses. A version of Saari's conjecture, that the only solutions to the n-body problem with a constant total size are rigid rotations, is also investigated for the PCR4BP. Techniques from computational algebraic geometry are employed such as Gröbner bases, resultants and BKK theory.

April 17, 2008.**Cristina Ballantine**,*Expander graphs - hard and easy constructions*

**Abstract:**I will present a construction of an infinite family of Ramanujan bigraphs (i.e., bipartite, biregular graphs which are asymptotically optimal expanders) based on representation theory. This provides the first known infinite family of Ramanujan bigraphs. I will also discuss the zig-zag product of graphs due to Reingold, Vadhan and Wigderson. This is a purely combinatorial construction that gives infinite families of constant degree expander graphs which are close to being Ramanujan. The first part of the talk is joint work with Dan Ciubotaru.

February 29, 2008.

### 2007

**Burt Tilley**, Olin College,*You Pick: Using Popular Science in a Mathematical Modeling Course*, Department of Mathematics and Computer Science Colloquium,

November 26, 2007.**Arvind Bhusnurmath**, University of Pennsylvania,*Flows and Cuts in Computer Vision*, Department of Mathematics and Computer Science Colloquium,

November 14, 2007.**Viraj Kumar**,*A sublinear space, polynomial time algorithm for directed s-t connectivity*

**Abstract:**For many algorithmic software verification techniques (model checking, conformance testing, etc.), the central problem reduces to exploring graphs efficiently. A software program can be viewed as a graph, where each vertex represents the state of the program (a tuple consisting of the program counter, the values of all variables, and the stack configuration) at a given time, and each edge from vertex u to vertex v represents the possibility of switching from state u to state v in one step. Since the number of states is exponential in the number of variables, the entire graph cannot be represented in memory and the challenge is to explore the graph using as little memory as possible.

Undirected graphs with n vertices can be explored very efficiently (in time polynomial in n, using O(log n) space) via random walks. Graphs that model software programs, however, are usually directed since computation steps are typically nonreversible. Random walks perform poorly on directed graphs with n vertices (they can require time exponential in n). This talk will present a deterministic sub-linear space, polynomial time algorithm by Barnes et. al. (1992) for the directed s-t connectivity problem, where one is given a graph and asked to find a (directed) path from s to t if one exists. Details of the paper are given below:

*A sublinear space, polynomial time algorithm for directed s-t connectivity*; G. Barnes, J. F. Buss, W. L. Ruzzo and B. Schieber; Structure in Complexity Theory (1992), pp. 27 -- 33.

November 01, 2007.**Steve Levandosky**,*Stability of Solitary Waves of a Fifth-Order KdV Equation II*,

October 26, 2007.**Steve Levandosky**,*Stability of Solitary Waves of a Fifth-Order KdV Equation*

**Abstract:**For a certain class of fifth-order KdV-like equations, one can use a constrained minimization problem to prove the existence of a 2-parameter family of solitary waves. Their stability is determined by the concavity of a function of one of the parameters (the wave speed c). Unfortunately, the explicit form of this function is known only in special cases. To remedy this situation, numerical calculations are required. I'll give an overview of the existence and stability theory, and then explain how an important scaling identity can be used to greatly reduce the scope of the numerical calculations. This scaling property also leads to some partial analytical results.

October 19, 2007.**Andy Hwang**,*Extremal Kähler metrics*

**Abstract:**This talk will introduce and briefly survey a differential-geometric variational problem whose solutions include the metrics of constant Gaussian curvature on a compact Riemann surface. In higher dimension, existence of solutions is not guaranteed, and leads to fascinating, subtle criteria. We'll see a geometric hint of what can go wrong in a minimizing sequence by looking at explicit complex surfaces.

September 28, 2007.**Richard P. Stanley**, Massachusetts Institute of Technology,*Plane Tilings*, Leonard C. Sulski Memorial Lecture,

April 24, 2007.**Paul Blanchard**, Boston University,*Newton's Method: Complex Numerics and Complex Dynamics*, Department of Mathematics and Computer Science Pi Mu Epsilon Ceremony and Colloquium,

April 19, 2007.**Tanya Leise**, Amherst College,*The $25,000,000,000 Eigenvector: The Linear Algebra Behind Google*, Department of Mathematics and Computer Science Colloquium,

March 28, 2007.**Noam D. Elkies**, Harvard University,*Canonical Forms: A Mathematician's View of Musical Canons*, Interdisciplinary Colloquium,

February 19, 2007.

### 2006

**Steven Sherwood**, Dept. of Geology and Geophysics, Yale University,*Global Climate Change: What We Know and Don't Know*, Interdisciplinary Colloquium, November 20, 2006.**Dave Damiano and Andrew Hwang**, HC,*The Poincare Conjecture*, Department of Mathematics and Computer Science Colloquium, October 20, 2006.**Michele Intermont**, Kalamazoo College,*On Calculus and Derivatives . . . the Goodwillie Kind*, Clavius Group Seminar, July 14, 2006.**Jennifer Beineke**, Western New England College,*Great Moments of the Riemann Zeta Function*, Department of Mathematics and Computer Science Pi Mu Epsilon Ceremony and Colloquium, April 26, 2006.**Andrew Swift**, Dept. of Mathematical Sciences, WPI,*An Introduction to Bayesian Statistics*, Department of Mathematics and Computer Science Colloquium, April 5, 2006.**Michele Intermont**, Kalamazoo College,*The Sound of Algebra*, Leonard C. Sulski Memorial Lecture, March 22, 2006.**David Vella**, Skidmore College,*The Higher Chain Rule and Composite Generating Functions*, Department of Mathematics and Computer Science Colloquium, February 15, 2006.

### 2005

**Robert L. Devaney**, Boston University,*The Fractal Geometry of the Mandelbrot Set*, Department of Mathematics and Computer Science Colloquium, December 1, 2005.**John Little**, HC,*Symmetry in Music*, Department of Mathematics and Computer Science Colloquium, November 16, 2005.**Catherine A. Roberts**, HC,*How does the solution grow?*, Department of Mathematics and Computer Science Colloquium, October 20, 2005.**Edward Burger**, Williams College,*Conjugate Coupling: The Romantic Adventures of the Quintessential Quadratic*, Department of Mathematics and Computer Science Pi Mu Epsilon Ceremony and Colloquium, April 13, 2005.**Frank Morgan**, Williams College,*Soap Bubble Geometry, 200 B.C. - 2005 A.D*, Leonard C. Sulski Memorial Lecture, April 7, 2005.**Neil Heffernan**, WPI,*Learning about Learning*, Department of Mathematics and Computer Science Colloquium, March 30, 2005.**Marian Batton**, GIS Senior Analyst, Johnson, Mirmiran and Thompson,*GIS: Solving Problems With a Geometric Network*, Department of Mathematics and Computer Science Colloquium, March 21, 2005.**William J. Martin**, WPI,*The Math Behind the Compact Disk*, Department of Mathematics and Computer Science Colloquium, January 26, 2005.

**2004**

**Allison Pacelli**, Williams College,*Polynomials, Primes, and Fermat's Last Theorem*, Department of Mathematics and Computer Science Colloquium, December 8, 2004.**Andrew Hwang**, HC,*Topology and Curvature of Polyhedra*, Department of Mathematics and Computer Science Colloquium, November 17, 2004.**John Geddes**, Olin College of Engineering,*Oscillations in Microvascular Bloodflow*, Department of Mathematics and Computer Science Colloquium, November 6, 2004.**Tom Hull**, Merrimack College,*The Secrets of Super-Complex Origami Design*, Department of Mathematics and Computer Science Colloquium, April 19, 2004.**Gil Pontius**, Clark University, International Development, Community and Environment (IDCE) Department & Graduate School of Geography, Environmental Studies, Educational Technology, and the Department of Mathematics & Computer Science Colloquium, March 25, 2004.**Frank Farris**, Santa Clara University,*The Edge of the Universe: Non-Euclidean Wallpaper*, Leonard C. Sulski Memorial Lecture, March 22, 2004.

**2003**

**Robert Devaney**(Holy Cross '69), Boston University,*The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence*, Department of Mathematics and Computer Science Pi Mu Epsilon Ceremony and Colloquium, April 28, 2003.**Kresimire Josic**, University of Houston,*Singularly Perturbed Quadratic Maps*, Department of Mathematics and Computer Science Colloquium, April 25, 2003.**Mark Lawrence**, HC,*Analytic Continuation in Several Complex Variables*, Department of Mathematics and Computer Science Colloquium, April 11, 2003.**Gregory Buck**, Saint Anselm College,*Physical Knot Theory*, Pi Mu Epsilon Ceremony and Colloquium, April 9, 2003.**Kathleen Shannon**, Salisbury University,*Pascal's Triangle, Cellular Automata and Serendipity: A Mathematical Tale*, Leonard C. Sulski Memorial Lecture, April 2, 2003.**Dave Damiano**, HC,*Knot Theory: Ribbon = Slice?*, Department of Mathematics and Computer Science Colloquium, March 19, 2003.**Diana M. Thomas**, Department of Mathematical Sciences, Montclair State University, A*Characterization for the Length of Cycles of the N Number Ducci Game*, Mathematics Colloquium, February 24, 2003.**Anne Schwartz**, Westfield State College,*Modular Functions and Almost Modular Functions and Why We Care*, Department of Mathematics and Computer Science Colloquium, February 21, 2003.**Andy Hwang**, HC,*An Introduction to Kaehler Geometry, and a Survey of Einstein and Extremal Kaehler**Metrics, III*, Department of Mathematics and Computer Science Colloquium, February 12, 2003.**Andy Hwang**, HC,*An Introduction to Kaehler Geometry, and a Survey of Einstein and Extremal Kaehler Metrics, II*, Department of Mathematics and Computer Science Colloquium, February 5, 2003.**Andy Hwang**, HC,*An Introduction to Kaehler Geometry, and a Survey of Einstein and Extremal Kaehler Metrics*, Department of Mathematics and Computer Science Colloquium, January 29, 2003.

**2002**

**Carlos Morales**, Worcester Polytechnic Institute,*A Rough Introduction to Roughness Estimation*, Mathematics Colloquium, December 5, 2002.**Stanzi Royden**, HC,*Seeing Through our Illusions: How the Brain Mechanisms Underlying Heading Perception Explain a Visual Illusion*, Department of Mathematics and Computer Science Colloquium, November 20, 2002.**Ann Trenk**, Wellesley College,*Professors who Snooze and those who Steal:An Introduction to Interval Graphs*, Mathematics Colloquium, November 14, 2002.**Gareth Roberts**, HC,*Central Configurations and Their Importance in the N-Body Problem, continued*, Department of Mathematics and Computer Science Colloquium, November 13, 2002.**Gareth Roberts**, HC,*Central Configurations and Their Importance in the N-Body Problem*, Department of Mathematics and Computer Science Colloquium, November 6, 2002.**John Little**, HC,*Fast Fourier Transforms and Polynomial Multiplication*, Department of Mathematics and Computer Science Colloquium, October 30, 2002.**Cristina Ballantine**, HC,*Rolle's Theorem for Local and Finite Fields*, Department of Mathematics and Computer Science Colloquium, October 9, 2002.**John Little**, HC,*Primes in P continued*, Department of Mathematics and Computer Science Colloquium, September 25, 2002.**John Little**, HC,*Primes in P*, Department of Mathematics and Computer Science Colloquium, September 18, 2002.**Dennis DeTurck**, University of Pennsylvania,*Coiling and Writhing in Geometry, Biology and Physics*, Leonard C. Sulski Memorial Lecture, April 18, 2002.**Jack Hoppin**'98, University of Arizona,*Comparing Estimation Tasks in Medical Imaging Without a Gold Standard*, Mathematics Colloquium, February 28, 2002.**Elizabeth Sweedyk Dimacs**, Rutgers University,*Approximating the Shortest Common Superstring*, Department of Mathematics and Computer Science Colloquium, February 26, 2002.

### 2001

**Laurie King**, HC,*Runtime Execution of Reconfigurable Hardware in a Java Environment*, Department of Mathematics and Computer Science Colloquium, November, 2001.**Rosalind Picard**, MIT Media Lab,*Toward Computers that Recognize and Respond to Human Emotion*, Department of Mathematics and Computer Science Colloquium, April 10, 2001.**Keith Miller**, University of Illinois-Springfield,*Minimum Testing Standards for Commercial Software: Technical and Ethical Issues*, Department of Mathematics and Computer Science Colloquium, April 5, 2001.**Jane Hawkins**, University of North Carolina,*Smoothing Out the Rough Edges of Fractals*, Leonard C. Sulski Memorial Lecture, March 29, 2001.**Donald Gotterbarn**, Professor,*Computer and Information Sciences, East Tennessee State University*, "Code Doesn't Count", Department of Mathematics and Computer Science Colloquium, March 15, 2001.**Cem Kaner**, J.D., Ph.D., Professor of Computer Science, Florida Institute of Technology,*Legislation and Embedded Software: Fundamental Ambiguities in Law and Computer Science*, Department of Mathematics and Computer Science Colloquium, March 12, 2001.

### 2000

**Richard Stallman**,*The Free Software Foundation, The Free Software Movement and GNU*, Department of Mathematics and Computer Science Colloquium, December 6, 2000.**Richard Freije**,*Corporate Attorney, Global Knowledge, The First Amendment, the Internet and the Regulation of Speech on Campus*, Department of Mathematics and Computer Science Colloquium, November 30, 2000.**Dan Kennedy**, The Baylor School,*Mathematical Paradoxes*, Leonard C. Sulski Memorial Lecture, March 23, 2000.**Fred Schneider**, Cornell University,*The Non-Technical Take on Computing System Trustworthiness*, Department of Mathematics and Computer Science Colloquium, March 3, 2000.

### 1994-1999

**Tracy Camp**, Colorado School of Mines,*WANs to Worms to the Web: A History of the Internet*, Department of Mathematics and Computer Science Colloquium, April 22, 1999.**Lawrence Conlon**, Washington University,*Sighting the Infinite Loch Ness Monster in the 3-sphere*, Leonard C. Sulski Memorial Lecture, April 15, 1999.

**Colin Adams**, Williams College,*Bus Tours of the Universe and Beyond*, Leonard C. Sulski Memorial Lecture, April 16, 1998.

**Peter Bushell**, University of Sussex,*Prime Numbers: The Search for a Formula Giving Every Prime Number. A Story with a Happy Ending?*, Leonard C. Sulski Memorial Lecture, April 9, 1997.

**Michael Rosen**, Brown University,*Historical Remarks on Fermat's Last Theorem*, Leonard C. Sulski Memorial Lecture, April 18, 1996.

**Thomas Banchoff**, Brown University,*Visualizing Across Dimensions: From Flatland to Interactive Hypergraphics*, Leonard C. Sulski Memorial Lecture, April 20, 1995.

**Robert Devaney**, Boston University,*The Fractal Geometry of the Mandelbrot Set*, Leonard C. Sulski Memorial Lecture, April 14, 1994.