December 3, 2016

The William Lowell Putnam Mathematical Competition is an annual contest for college students established in 1938. Substantial cash prizes are awarded to the top five teams and to the top five individuals. In 1997, the Elizabeth Lowell Putnam Prize was established to honor the top woman competitor.

The 2016 Putnam exam will take place on Saturday, December 3, 2016. As usual, there will be two three-hour sessions starting at 10:00 a.m. and 3:00 p.m., held in the Math Dept Conference Room, 16 Colchester Ave. While the competition for the prizes is very strong, all students should enjoy and benefit from the experience of taking part in the most prestigious mathematical contest in the country.

Registration forms listing participants are due by October 7 (so we have to mail ours by October 4). Here is some additional information about the competition:

- The Department provides lunch for all participants on the day of the competition at a nearby restaurant.
- Sample problems from recent competitions are available from members of the committee.
- Only “regularly enrolled undergraduate students” may participate (students have a lifetime maximum of 4 times). Registering at the outset guarantees that we will have a slot for the student; but he/she may decide, even at the last minute, not to come (it’s no problem!). Some students stay only for the morning session and lunch (which is OK too). Unregistered “walk-ins” may also participate (unregistered students may take the place of registered students who don’t show up too – so there are usually ample “walk-in” slots).
- Students must arrive by 9:50 a.m. to participate in the Competition. A student may not take the afternoon portion without first having attended the morning session.
- Each session consists of six (challenging) problems, most of which can be answered with calculus and a little linear algebra (and a healthy dose of ingenuity).
*Advanced course work is not required.* - The committee will hold one or two
*optional*training sessions to review some potentially useful problem-solving skills. Solutions to past exams will be available for interested students. These sessions are low-key, fun and informal, where students take the lead in discussing practice problems (that we will circulate beforehand). - The official Putnam website is: http://math.scu.edu/putnam/index.html. Another very useful web site for problems from some previous years (reworded slightly) and their solutions is: http://web.archive.org/web/20080124125203/www.kalva.demon.co.uk/putnam.html

For more information or to register to participate, contact Prof. Richard Foote (richard.foote@uvm.edu)

]]>The first meeting will be

**Curvature: The Fundamentals**

At the backbone of differential geometry sits the notion of curvature, a quantity measuring how much a curve or surface "curves" at a given point. In this talk, we will introduce the basics of planar differential geometry, including parameterizations of curves, various notions of the curvature vector, and the topology of curves. This talk aims to build a working understanding of curvature with the goal of later applying that knowledge to understanding the mean curvature flow, or curve-shortening flow, of planar curves.

**Origami Knots in Graphs**

Inspired by a potential topological obstruction to the origami method of DNA nanostructure self-assembly design, we define and study origami knots in Eulerian graphs embedded on orientable surfaces in 3-space. We present a complete characterization of origami knots in naturally-defined infinite families of triangular and rectangular toroidal grids, and discuss origami knots in composites of both types of grids embedded on higher-genus surfaces.

]]>Mondays at 6:45 PM

Perkins 101

The Math Club will hold weekly events on Monday evenings at 6:45 PM in Perkins 101 this year. Our **Unofficial Kickoff** will be **Monday, September 5**. We will have opportunities to meet each other and to practice for our next event, the **Faculty vs. Undergrads Math Bee**.

The **Faculty vs. Students Math Bee** will take place on **Monday September 12**. There will be pizza at this event.

Everyone is welcome at all our events, so bring a friend!

We welcome all feedback so if there is anything you want to see become a part of Math Club or you have any questions, email us at mathclub@uvm.edu

]]>Sean Cleary

The City College of New york and the CUNY Graduate Center

Thursday, September 22, 2016

4:30 - 5:30 pm

Kalkin 002

**Abstract:**

Rooted binary trees are an important structure underlying many computational structures, such as binary search trees and more complex data storage methods. Having trees which are relatively balanced is important for having efficient searches. Modifying unbalanced trees to make them more balanced can be done in many ways, including a simple "rotation" operation which is a local move. Rotation distance between trees measures the minimum required number of rotations to transform one tree to another. There are no known efficient algorithms for calculating rotation distance exactly, but there are approximation algorithms. Rotation distance between a pair of trees is equivalent to an edge-flip distance between triangulations of regular polygons. Here, I discuss a number of properties of rotation distances, some of which are easier to recognize from the tree perspective and some from the polygonal perspective. This is based on joint work with Andrew Rechnitzer, Thomas Wong, and Katherine St. John, as well as with undergraduate student researchers at the City College of New York.

ADA: Individuals requiring accommodations please contact Doreen Taylor at 802-656-3166

]]>Samuel V. Scrapino, Ph.D.

Complex Systems

University of Vermont

Thursday, May 5, 2016

4:00 - 5:00 pm

Perkins 101

**Abstract:**

The primary tool for modern population genetic inference is coalescent theory, which provides a retrospective, mathematical framework for relating genetic variation to historical evolutionary processes. Because many pathogens mutate so rapidly, their evolutionary and population-level processes are inextricably linked. Therefore, studying epidemics requires models able to connect evolution to ecology. The emerging field of phylodynamics seeks to leverage the genetic variation of pathogens to investigate their complex, epidemiological dynamics through the use of mathematical transmission models. Linking these models with the genetic sequence data–now routinely collected during disease outbreaks–provides an unprecedented opportunity to advance our scientific understanding of epidemics and pathogen establishment. In this talk, I will present new results on the expected rate of coalescence for diseases spreading through social networks and demonstrate that an unbiased estimate of the coalescent rate can be obtained when only a subset of cases are reported. With these results, we will explore the utility of coalescent models during the 2014-15 ebola outbreak in West Africa and the ongoing whooping cough outbreak in the USA.

ADA: Individuals requiring accommodations please contact Doreen Taylor at (802) 656-3166.

]]>

Monday, May 2, 2016

6:30 PM

Perkins 101

Thanks for a great semester of Math Club! We hope to see at our last meeting of the semester. There will be FREE PIZZA, origami and games.

If you know you will be coming, please contact the Math Club President at rbayersd@uvm.edu so we can get an idea of how many pizzas to order. But if you decide to come at the last minute, that's okay too.

We hope to see you there!

Nick Allgaier, PhD

Postdoctoral Associate in Psychiatry, University of Vermont

Friday, April 29, 2016

4:00 - 5:00 PM

Kalkin 003

**Abstract:**

The neurological mechanisms underlying addiction in humans are still not well understood. Though the reward-based reinforcement learning circuitry of the limbic system has been implicated, it is not clear why many people who partake in recreational drug and alcohol use avoid getting hooked, while others succumb to addiction. In this talk we describe a neurodiagnostic methodology comprised of nonlinear functional mapping (NFM), a procedure developed at UVM for applying an evolutionary algorithm to functional magnetic resonance imaging (fMRI) data, and subsequent classification by support vector machine (SVM). NFM is a symbolic regression algorithm that searches for models relating the activity in different regions of interest (ROI) in the brain, as represented by fMRI signal, without assuming linearity. Summary statistics of the models inferred by NFM indicate levels of pairwise coordination among ROI, and diagnosis of addiction is accomplished by SVM based on these coordination levels.

We apply this methodology to resting-state fMRI time series from a cohort of 25 addicted cigarette smokers, and 30 control subjects. The resulting cross-validated classifier correctly diagnoses all 25 smokers while only miss-diagnosing 5 control subjects. Further, many of the top-ranking SVM features represent coordination among ROI in the prefrontal cortex, as well as coordination between these ROI and both cortical and subcortical ROI involved in the limbic system. The importance of coordination among these particular ROI in addiction diagnosis hints at a mechanism of prefrontal executive control over the limbic system, whose efficacy may be a key determining factor in a subject's risk of addiction.

ADA: Individuals requiring accommodations please contact Doreen Taylor at (802) 656-3166.

]]>**Abstract:** From the right perspective, everything is mathematical – even juggling. In this talk, Greg Warrington from the University of Vermont will describe how juggling relates to traditional mathematical fields such as graph theory and look at what happens when a person starts juggling randomly. In doing so, he’ll illustrate how these mathematical underpinnings can be useful even for accomplished jugglers. There will be numerous live demonstrations with the juggling of 0 to 7 objects.

We hope to see you there!

]]>Mary Beth Ruskai, Emeritus Prof. of Mathematics, University of Massachusetts Lowell

Research Prof., Tufts University

Associate member, Institute for Quantum Computing, Waterlook, Canada

Friday, April 15, 2016, 4:00 PM

Kalkin 003

**Abstract:**

Entanglement is both one of the most puzzling aspects of quantum theory and a key component of powerful new methods of computation and communication. In contrast to classical systems, the conditional information of a quantum system can be negative. It is now known that this can be interpreted in terms of quantum correlations that can be used to transmit information by such mechanisms as quantum teleportation.

In quantum information theory, quantities like mutual and conditional information are defined using the von Neuman entropy. Key properties of the von Neumann entropy are closely associated with operator and trace inequalities, which can be proved using extremely elementary methods.

This talk will give an overview of, and introduction to, the concepts mentioned above. No prior knowledge of quantum theory is required and the mathematics is accessible to anyone with a good knowledge of linear algebra.

]]>

Generalizations and Applications

Sagun Chanillo, Rutgers University

Friday, April 8, 2016, 4:00 PM

Kalkin 004

**Abstract:**

The Fundamental theorem of Calculus can be generalized in two ways in higher dimensions, via the Gagliardo-Nirenberg inequality with its close connections to the isoperimetric inequality and the Moser-Trudinger inequality which has powerful ramifications in Conformal Geometry. Recently a third inequality was discovered by Bourgain-Brezis which is very close to the fundamental theorem of Calculus in one dimension. We discuss generalizations of this inequality to Nilpotent Lie Groups, Riemannian Symmetric spaces and also a new proof of the original inequality of Bourgain-Brezis that allows us to extend the scope of the Bourgain-Brezis inequality. Lastly we provide appliactions of the Bourgain-Brezis inequality to the two dimensional Navier-Stokes equation of Fluid Mechanics and the Maxwell equations of Electromagnetism. This is joint work with Jean Van Schaftingen and Po-lam Yung.

]]>

A panel Discussion with UVM Alumni and Community Professionals

Jeffrey Young,

Brian Orleans,

John Stanton-Geddes,

Polly Ramsey,

Leah Shulman,

Wednesday, April 6, 2016, 7:00 - 8:15 pm

Waterman Memorial Lounge

The UVM Statistics program will host a panel presentation of careers in Statistics and Data Science on Wednesday, April 6 from 7:00 - 8:15 pm in Waterman Memorial Lounge. Five speakers will talk about career opportunities at Dealer.com, ICF International, Cigna Insurance, and the Vermont Health Department. A question and answer session will follow. Refreshments will be served.

Save the date for the last few meetings of the semester:

April 2 - Hudson River Undergraduate Math Conference at Saint Michael's College

April 4 - Jim Bozeman's talk on gerrymandering

April 18 - Greg Warrington's talk

May 2 - End of year celebration

]]>We will be building a modular origami creation as well as playing games such as Set.

Feel free to bring your favorite board game to play. We will provide some as well.

All are welcome. Bring a friend!

]]>**When:** Thursday, February 4, 2016, 12:00 - 1:00 PM

**Place:** Math Conference Room

Bring your lunch.

**Speaker:** Ian M. Wanless, Monash University

**Title:** *Embedding small partial Latin squares in Cayley tables*

**Abstract:**

Sudoku puzzles are examples of a combinatorial object called a partial latin square (PLS). In 1974 Keedwell asked for each n what is the smallest PLS that cannot embed into any group table of order n. In 2012 Hirsch and Jackson asked what is the smallest PLS that can be embedded in an infinite group but not into any finite group. We answer both these questions and a couple of related ones.

Other events planned for the semester include guest talks about actuarial math careers, cryptography, and juggling, as well as a trip to Saint Michael's College for the Hudson River Undergraduate Math Conference.

]]>

Nathan Dowlin

Princeton University

Thursday, December 10, 2015

4:00 – 5:00 PM

Kalkin 003

Abstract:

The last two decades have seen the construction of many knot invariants. The most powerful have been homology theories, of which there are two main types: those constructed using symplectic techniques, such as knot Floer homology, and those related to the representation theory of quantum groups, such as Khovanov homology and HOMFLY-PT homology. Despite the fundamental differences in these theories, they seem to be closely related. I will discuss both the known and conjectured relationships between them, with special attention to the conjectured spectral sequences from HOMFLY-PT and Khovanov homology to knot Floer homology.

ADA: Individuals requiring accommodations, please contact Doreen Taylor (802) 656-3166

]]>Talk: Fake Simple Random Walks.

Speaker: Ewa Infeld, Dartmouth

Abstract: If we let two tokens do simple random walks on a connected graph, they

will collide. Can we set them up so that they don't, but if you only see

either one of them it looks like it's doing a simple random walk? What

does that even mean? The answer is: sometimes. Let's see when.

**Robert D. Tortora, Ph.D.****Senior Fellow of Survey Methodology & Chief Methodologist****Survey Research, ICF International**

**Wednesday, September 23****12:00 PM****Perkins 101**

Abstract:

This talk will use the Total Survey Error (TSE) model to highlight a few of my experiences as a statistician working on surveys for the federal government and in the private sector. The TSE has four major components: sampling error, coverage error, nonresponse error and measurement error. After a brief overview of each component of error I will use the model to discuss my work experience conducting agriculture surveys for USDA, dissecting the biggest and most expensive statistical undertaking in the world, viz., the US Census of Population and Housing, serving as an expert witness in a case against the NCAA and starting and working on a worldwide survey conducted in over 125 countries each year with a special emphasis on sub Saharan Africa. A brief review of my current research on coverage error and the use of non-probability surveys will conclude the talk.

ADA: Individuals requiring accommodations, please contact the Office of Affirmative Action & Equal Opportunity at 656-3368

Title: Impact of Local Testing in a Targeted Therapy Setting with Companion Diagnostic Development

Abstract: In the pharmaceutical industry, one goal using pharmacogenomics information is to select patients who have certain characteristics that may respond to your treatment over the standard of care, or therapy of a competitor. Identification of patients is made through assays that are commonly DNA, RNA or protein based, may be multi-marker and/or may even be derived through a combination of technologies. To the health authority, assigning treatment to specific patients represents significant risk and requires the generation of a high quality companion diagnostic assay to ensure that a consistent patient population is robustly selected over time. This requires running a series of experiments to both verify and eventually validate that the assay performs according to strict specifications. In a clinical setting, there are several challenging factors that may negatively impact the quality of the diagnostic data one collects. In addition, the popularity of genomic screening presents new challenges that require open dialogue among academia, industry and regulators. These challenges, accompanied by actual examples, will be the focus of my initial presentation.

For the second part of my presentation, the focus will shift towards the role of the statistician in the pharmaceutical industry. There are many roles one may aspire to, including clinical trial statistician in early or late phase trials, clinical pharmacology, biomarker and diagnostics, and methods development. Although these roles are varied, there are commonalities in terms training, ability to communicate and collaborate, etc. that make one successful. A question and answer period will follow.

]]>Faculty are welcome, too! ]]>

Dr. Galia Dafni

Department of Mathematics

Concordia University

Montreal, Quebec

Thursday, April 30, 2015, 4:00 PM

Kalkin 003

**Abstract:**

The mathematical field of harmonic analysis, named for "harmonics" in music, is also known as Fourier analysis, after the Frenchman Joseph Fourier. It is concerned with decomposing signals into their frequency components and reconstructing the signal from this data, and has had numerous applications, from describing the motion of a vibrating string or heat propagation (as studied by Fourier) to modern-day techniques in image processing and medical imaging, such as wavelets and compressed sensing.

Current research in harmonic analysis is much broader and has deep connections to other fields, especially complex analysis, partial differential equations, mathematical physics and number theory.

We will focus on the theory of Hardy spaces and how it illustrates the ideas of decomposition and reconstruction.

ADA: Individuals requiring accommodations, please contact Doreen Taylor at (802) 656-3166

]]>This time, though, the mission was different. Cole and his fellow members of the Oncologic Drugs Advisory Committee would forge a new route through the realm of regulatory review of drugs in the United States. They would assist in the FDA’s first evaluation of a “biosimilar” product. Biosimilars are close copies of biological drugs, which are derived from living cells instead of the cocktail of chemicals that make up most medicines.

When chemically based drugs lose their patents, generic versions with identical components can easily reach the marketplace, typically lowering prices. Until now, the FDA had no mechanism to approve close copies of biologics, deemed too complex to ensure that similar but inexact alternatives were equally safe and reliable. The most popular biologics, used to treat cancer and autoimmune diseases, are also some of the most expensive drugs.

Under a mandate of the Affordable Care Act, with a goal to encourage competing products that could help lower costs, the FDA now has a way to evaluate biosimilars. The federal law permits approval of a product shown as “highly similar” to a specific drug on the market and with “no clinically meaningful differences” in safety or effectiveness as that existing drug.

In March -- relying on the recommendation of Cole and the advisory panel, which applied those federal guidelines for the first time -- the FDA approved Zarxio. The new drug mimics the well-established Neupogen to fight infections in cancer patients undergoing chemotheraphy and other treatment.

The FDA’s Center for Drug Evaluation and Research approves more than 100 new medications each year. Most never go through advisory committees.

Cole’s group only sees the tough cases, those that lack a “favorable risk profile” or raise alarm bells, making it unclear whether the potential benefits outweigh the risks, he says. “They only bring stuff to us if the question is difficult.”

Cole has participated in four reviews on the 13-member panel. His fellow members include oncologists and other medical professionals, a cancer patient or survivor and a consumer advocate. Cole is the only biostatistician.

“The level of the work isn’t that bad, but the level of responsibility is huge,” he says. “It would be terrible for a drug company to get a drug approved that doesn’t work out” or that causes harm, he says. Equally terrible is the prospect of denying, because of perceived risks, a drug that is actually safe and could help millions of sick people or maybe save lives.

“I think of it from the overarching public health perspective,” Cole says of his role. “Every time you make a drug available, you’re altering public policy.”

For 23 years, Cole has tied his biostatistics background to cancer. During a post-doctoral fellowship at the Dana-Farber Cancer Institute in Boston, he found he liked collaborating with oncologists and studying patient outcomes. It put him at the cross-section of numerical science and human medicine, where he could not only advance the analysis of data but also answer questions for cancer patients.

“I look for a well-designed study,” Cole says of his approach on the FDA panel. “The famous line is: ‘All studies have warts.’ None of them are perfect.”

Cole has his own system after reviewing the briefings provided by the FDA and the drug company. After he studies the information, he writes two statements -- one in support of approval, and one against it -- explaining his reasoning for each. The position that sounds most convincing tells him where to lean.

Then, Cole gets on a plane and heads to FDA headquarters in Silver Springs, Md., for the meeting. It’s more like a courtroom trial, and the committee is the jury.

Each side, the drug company and the FDA staff, presents its argument. Cole and the other committee members sit at a U-shaped table and can ask questions, and he revises his previous statements. The meeting, which usually lasts a day, also includes time for public comments.

“Oftentimes, they’re patients who come and tell us stories about what they’ve been through,” Cole says. “Hearing those statements really puts some perspective on what we’re doing.”

Unlike a courtroom jury, the panelists don’t come to a consensus on a verdict at the end of the day. They simply answer “yes” or “no,” to the FDA’s question: “Should we approve this drug?” They also can comment at that time, and Cole says he sometimes reads his written statement.

With Zarxio, he says, the answer was relatively clear. The vote in favor was unanimous.

The committee played a significant role in the biosimilars review process, ensuring its transparency to the public and bringing expertise to vet critical information, says Tim Irvin, a spokesman for the Center for Drug Evaluation and Research. For now, the committee will look at every upcoming application for a new biosimilar, he says.

“We appreciate getting their feedback,” Irvin says. “It gives everybody a chance to look at the data and make sure everything is as it should be.”

Cole doesn’t expect every biosimilar review to go as smoothly. A key difference is the narrower scope of the clinical trials to show that the characteristics of the new drug and the outcomes for patients are close to those of the existing drug. It’s foreseeable that the measurements might not line up, he says.

“If there’s one chink in the armor, it opens up a question,” he says. “And then you might need a big clinical trial to answer the question.”

The committee only makes recommendations, but the FDA usually follows them. Cole says he believes all the committee members feel the weight of their decision.

“You get a collection of people together, you have a collective wisdom,” he says. “So it’s not all on one person’s shoulders.”

]]>Professor Mike Wilson

Department of Mathematics & Statistics

University of Vermont

Thursday, April 9, 2015, 4:00 PM

Kalkin 004

**Abstract:** The astrolabe is a special purpose analog computer. One can think of it as a "circular slide rule" for astronomical calculations. In this talk I will say a little about its history, demonstrate its parts, and show how it can be used to: tell time, predict the length of the day or night, predict the positions of stars (including the sun) and find ascendants (very important in casting horoscopes—just ask Geoffrey Chaucer). I will show how to make one and describe the mathematics behind it.

ADA: Individuals requiring accommodations, please contact Doreen Taylor at (802) 656-3166

]]>Dr. John Voight

Department of Mathematics

Dartmouth College

Thursday, April 2, 2015, 4:00PM

Kalkin 004

**Abstract:**

There is a marvelous and deep connection between:

- surfaces obtained by gluing together copies of a triangle,

- triples of permutations whose product is the identity,

- finite index subgroups of triangle groups,

- bicolored graphs equipped with a cyclic orientation, and

- three-point branched covers of the complex projective line.

The consequences of these equivalences are myriad for geometry, arithmetic, combinatorics, and group theory.

In the first part of the talk, we introduce these connections via examples and pictures.

In the second part of the talk, we discuss their algorithmic aspects: specifically, we exhibit a numerical method that, given a permutation triple, computes equations for the associated three-point branched cover.

This is joint work with previous UVM graduate students Michael Klug, Michael Musty, and Sam Schiavone.

ADA: Individuals requiring accommodations, please contact Doreen Taylor at (802) 656-3166