Welcome to 2019 Fall term Math Student Seminars

We will continue meeting **during common hour on Thursdays** in Bailey 207, with light lunch served at 12:30pm in Bailey 204, **unless otherwise noted**.

**Fall term 2019**

**Featuring: Student summer research**

**Dan Resnick** (class 2021)

**Sam Kemp** (class 2021)

**Friday, September 20th, 1:00pm, Bailey 207**

**Dan Resnick** (class 2021)

**Stellar Wind Collisions: Thin Shell Geometry**

I will be talking about the theory behind the stellar wind problem and the assumptions/setup to where we started working. I will discuss the methods we used and where we stopped on the problem, and where we plan to go forward with the project.

**Sam Kemp** (class 2021)**Simulation of Forces Between Inclusions on a Lipid Bilayer**

A program was written in C++ to simulate the forces between two inclusions on a lipid bilayer using the finite element method. This was first done using Dirichlet boundary conditions and then with mixed boundary conditions at the edge of the inclusions.

**Previous Talks**:

**Spring term 2019**

**The Historical Roots of Gödel’s Theorems**

**Professor Andrea Pedeferri - ****Philosophy Department @ Union College**

**Tuesday, May 28th, 1:00pm, Bailey 207**

At the beginning of “On formally undecidable propositions of Principia Mathematica and related systems” Gödel writes: “The development of mathematics toward greater precision has led [...] to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules.” In just few sentences Gödel summarizes a century of key developments in mathematics that made that period one of the most exciting and optimistic for the discipline. As Hilbert wrote in 1925 and in 1930 “we are all convinced that [...] in mathematics there is no *ignorabimus*”, “We must know. We will know”.

Then comes 1931. Gödel writes: “One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question [...]. It will be shown below that this is not the case [...]. The precise analysis of this curious situation leads to surprising results concerning consistency proofs for formal systems.” The impact of Gödel’s results was immense.

In my talk I will follow the main developments of mathematics highlighted by Gödel in order to show how they are crucial to understand the impact and the reach of Gödel’s theorems and to fully appreciate their dramatic but revolutionary nature.

**Rationalizable Voting Rules and the S-Correspondence**

**Professor Ashley Piggins****, **Department of Economics, National University of Ireland, Galway (joint work with Conal Duddy, Department of Economics, University College, Cork)

Tuesday, May 21st, 4:00pm, Bailey 207

Perhaps the most important feature in Kenneth Arrow’s famous “Impossibility Theorem” (which helped him earn a Nobel prize in Economics) is **IIA** – the Independence of Irrelevant Alternatives condition on voting rules, aka *social choice*. We introduce a related but new social choice property, the S-independence condition. We characterize the “S-correspondences” – those social choice rules that satisfy S-independence along with three more standard axioms: strong Pareto optimality, neutrality and anonymity. This class is closely related to the S-rules of Bossert and Suzumura (2008a, J. Econ. Theory 138, 311-320). S-independence can be justified by a new rationalizability argument that shows it is equivalent to assuming **IIA** in a somewhat different framework.

**The History of Fermat's Last Theorem**

**Professor Ravi Ramakrishna, Cornell University**

Thursday, May 16th, 1:00pm, Bailey 207

In 1994, Wiles, assisted by Taylor, finally settled the 340-year-old question of Fermat's Last Theorem. In this talk I will give some of the history of this problem, with particular focus on events of 1847 and the 1980-90s. I'll then talk a little about developments off the last 25 years in the subject.

This talk is NOT aimed at faculty in number theory. It is intended for undergrads that like math, but have not taken many upper level courses.

**Mathematical Modeling in Computational Neuroscience**

**Paulina Volosov, **Rensselaer Polytechnic Institute

Thursday, May 2nd, 1:00pm, Bailey 207, with lunch at 12:30pm in Bailey 204

Computational neuroscience is a field of active research for applied mathematicians. The brain has been actively studied since the second half of the 19th century, but there are still an endless number of unanswered questions about how and why the brain works the way it does.

This presentation will give a basic introduction to some of the questions asked in neuroscience and show how a mathematical model can be built and used to shed light on this complicated biological system. Namely, how can a mathematician attempt to learn about the structure of the brain and networks of neurons? What is it that a mathematical model tries to accomplish? And why is this useful?

**Who Discovered Integral Calculus?**

**Professor Emeritus Julius Barbanel, Union College**

Thursday, April 25th, 1:00pm, Bailey 207, with lunch at 12:30pm in Bailey 204

Isaac Newton and Gottfried Leibniz (both working in the late 17th and early 18th centuries CE) are generally considered to be the inventors of calculus. We will argue in this talk that part of the credit for this discovery should be given to the ancient Greek mathematician Archimedes (who worked in the 3rd century BCE). Archimedes pursued two lines of research involving areas and volumes. These approaches, known as the Method of Exhaustion and the Mechanical Method, can be viewed as early examples of ideas that we think of as being part of integral calculus. We will focus on Archimedes’ Method of Exhaustion.

**Presenting Mathematics - an open discussion**

**Professor Leila Khatami and Professor Jue Wang, Union College**

Thursday, April 18th, 1:00pm, Bailey 207, with lunch at 12:30pm in Bailey 204

As Steinmetz symposium approaches, many students are contemplating their presentations and are thinking about ways to make an engaging presentations to effectively share their work with fellow students, family members, and professors. Even those who are not presenting this year, will most probably present their work in the future. In this seminar, we will talk about some of the characteristics of a good mathematics presentation, as well as some of common missteps to avoid. As part of the discussion, we will also watch and analyze a short sample presentation.

**Math, ECBE, Physics and Astronomy joint seminar**

**Thursday**** April 11, 2019**, **during common hour in ****Karp 005**

**Towards Cyber-Physical Electrical Power Systems: where the laws of nature and the rules of algorithms collide!**

**Luigi Vanfretti****Electrical, Computer, and Systems Engineering, RPI**

Electrical power networks are undergoing unprecedented changes. On one hand, the adoption of distributed energy resources (DER) and renewable energy sources (RES), both of which have a large degree of variability in small time-scales, puts challenges to the traditional, historical-and-experience-based design and operation of electrical power networks. On the other hand, digitization and automation, opens opportunities for a more carbon neutral electrical energy system by helping to harmonize these new energy sources with the rest of the power grid, not without also bringing along the potential threats of the cyber world. This talk aims to give an overview of these challenges, and to present different research efforts conducted by the presenter to address how to transform today’s electrical grid into a cyber-physical power system. This includes the development of an experimental facility to conduct, real-time hardware-in-the-loop simulation experiments of power networks with “cyber” assets. This approach allows to characterize how the interaction of systems governed by the laws of nature will interact with engineered systems governed by rules of algorithms. Finally, with the rise of electrification in transport, and in particular aircraft, and the rise of more autonomous machines, the talk will also discuss the need for development of a new course on modeling and simulation for cyber-physical systems (CPS) and the teaching approach adopted which brings a “digital” toolbox and know-how to the next generation of electrical engineers that will have to increasingly deal with complex CPS.

**Winter term 2019**

**The Spectacular Spectral Theory**

**Ehssan Khanmohammadi, Union College**

Thursday, March 7th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Do you know the reason for the collapse of the Tacoma Narrows Bridge in 1940? Have you ever wondered about the mathematical idea behind Google’s page ranking and Netflix’s movie recommendation? Do you know why scientists believe that distant stars are largely composed of hydrogen?

This talk is an invitation to the spectacular spectral theory, which is the key to answering all of these questions.

**Using linear algebra to understand knots**

**Cynthia Curtis, Professor of Mathematics at The College of New Jersey – Union College Class of 1987**

Thursday, February 14th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Knots are prevalent in nature, and the study of knotting is important in diverse areas such as DNA, bonding of molecules, and statistical mechanics. Understanding knots has been fundamental within mathematics to our ability to understand three-dimensional spaces. In this talk we use linear algebra to generate polynomials, which help decide whether two given knots are different. This is a surprisingly hard question! The polynomials can also help us know when to look for hidden symmetries in the knots. The first knot polynomial we introduce was found by James Waddell Alexander II in 1923. We then discuss new polynomials arising from research with undergraduates Vincent Longo, Alyssa Springstead, and Hoang Cao at The College of New Jersey.

**Generalizing composition of functions and Operads**

**Peter Bonventre, Union College Class of 2011**

Friday, February 8th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Given two single-variable functions, we are allowed to take their composite to produce a new function again of a single variable. In this talk, we will ask: “In what other contexts does ‘composition of functions’ make sense?”. We will slowly broaden our definitions of “function” and “composition”, starting with the types of functions that appear in the Calculus sequence, and moving to include well-behaved geometric figures. This will lead us to the abstract concept of an

operad. We will give several examples, as well as an interpretation of what these new objects can do for us.

**The Joy of Abstraction**

**Kimmo Rosenthal, Union College**

Thursday, January 31st, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

The imagination is the only genius. It is intrepid and eager and the extreme of its achievement lies in abstractionWallace Stevens.It may seem incongruous for the epigraph to a math talk to be from one of the great American poets. However, while the ubiquity and utility of mathematics is widely acknowledged, its burnish of aestheticism is much less so. Can the old dictum “art for art’s sake” be replaced by “math for math’s sake”? In this day and age, when relevance, applicability, and connections with other disciplines are touted as paramount, is there still a place for purely abstract mathematics viewed more as an intellectual art form? Abstraction has always appealed to me and indeed guided me. Why does it often provoke outright hostility? We shall follow the path of abstraction from the set theory of Cantor (called a “corrupter of youth”) to point-set topology, followed by the mysterious emergence of Bourbaki (the mathematician who never existed), and finally category theory, which earned the epithet of “abstract nonsense”. Of course, there will be some mathematics along the way, reasonably modest in scope.

**Hall’s Marriage Theorem**

**Alan Taylor, Union College**

Thursday, January 24th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Suppose we have a collection of women and a collection of men, and each woman finds some of the men acceptable (and the rest not). When is it possible to match each woman with a man she considers acceptable, subject to the obvious constraint that the matching be one to one? The answer to this metaphorical question is a beautiful result in finite combinatorics known as Hall’s marriage theorem. We will discuss Hall’s theorem, sketch a proof of it, and consider a couple of natural questions it suggests, all with the hope of providing an illustration of how research gets done in mathematics.

**Summer Opportunities for Math Students**

Thursday, January 17th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

This week’s seminar will focus on ways in which you can put your mathematical skills to use over the summer.

Julia Greene ’19 will speak about the

Teaching Experiences for UndergraduatesProgram, Professor Jeff Hatley will speak aboutResearch Experiences for Undergraduates, and Keri Willis of the Becker Career Center will speak about summer internships.

**Turning the lights out, mathematician style**

**Leila Khatami, Union College**

Thursday, November 1st, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

“LIGHTS OUT!” is a single player game played on a 5 by 5 grid where each cell has a button that can be turned on or off. Pressing a button toggles the light in the cell and its neighboring cells. The game starts with some cells turned on and some turned off. The goal of the game is to turn all cells off. The game was originally introduced in 1995 as a handheld electronic came. Nowadays, the original game, as well as many of its variants, are readily accessible in app stores and elsewhere. It is not obvious (or even true!) that all starting configurations of the game are “solvable”. In this talk, we use mathematical tools to see if a game is “solvable”. We also briefly discuss ways to find the most efficient solutions for solvable games.

**Solving the General Cubic Equation**

ax3ax3+bx2bx2+cx+d=0cx+d=0

**Paul Friedman, Union College**

Thursday, October 25th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

The solution to the general quadratic equation,

ax2+bx+c=0ax2+bx+c=0,

is well-known to most high school students:

It was also known to many “ancient” cultures … some dating back to 2000 BC! However, the solution to the general cubic equation, ax3+bx2+cx+d=0ax3+bx2+cx+d=0, is not as well known, and it was not found until the 1500s.In this talk, we will look at how the Renaissance mathematicians Scipione del Ferro, Tartaglia, and Cardano, solved the cubic equation, though we will do so using modern language and notation. As a cute consequence, we will be able to derive some remarkable identities, such as:

**Fair Division of a Graph: Envy Freeness up to one Good, or Two**

**William Zwicker****, Union College**

Thursday, Oct 18th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Countries A and B are dividing up a disputed island with several cities, linked by roads:

- Each city must go entirely to A, or entirely to B
- A city may be worth more to one country than to the other
- You must be able to drive among A’s cities without going through B’s
Can cities be allocated in a way that leaves neither country jealous of the other’s share?

No – not in general. But with certain road networks one can always get within one city of this ideal. Which networks are these? With more than 2 countries, the question gets harder . . . in interesting ways.

In its “classical” setting Fair Division concerns sharing a single, continuously divisible resource. Some solutions can be adapted to this new setting, but a lot needs to change.

**Forms of Remigration – Émigré Jewish Mathematicians and Germany in the Immediate Post-War Period**

**Volker Remmert, ****, of the Bergische Universitaet Wuppertal**

Tuesday, Oct 9th, 4:45pm, Bailey 207, with reception at 4:15pm in Bailey 204

Over the last twenty years or so there has been a steady flow of historical studies on remigration into Germany in the immediate post-war period. These studies have described three main forms of academic remigration to Germany after World War II:

1) returning to universities in Germany on a permanent basis as university professors;

2) returning as visiting professors, assessing Germany without any obligation to stay;

3) returning for guest lectures and academic visits.

In this context my interest is in Jewish émigré mathematicians and their stance to Germany in the immediate post-war period.

**Math, Music, and Health Science**

**Danielle Gregg ’19 and Robert Righi ’19, ****Union College Undergraduates**

Thursday, Oct 4th, 1:00pm, Bailey 207

Much recent research has focused on discerning topological and geometric features of data. For example, by observing the “birth” and “death” of holes via an algebraic method known as persistent homology, we can distinguish noise from significant features in data. In analyzing the “shape” of data our research diverges into two separate fields: music and health science. How can one use geometric and topological methods to classify a variety of degenerative diseases of the eye or compare songs within an artist’s discography? Come learn about what two Union students researched over the past summer as well as the often non-linear research process.

**Action Graphs and Catalan Numbers**

**Julie Bergner, ****University of Virginia and Cornell University**

Thursday, Sept 27th, 1:00pm, Bailey 207

The Catalan numbers are given by a recursively-defined sequence and arise from over 200 different kinds of combinatorial objects. In 2013, two of my undergraduate research students, Gerardo Alvarez and Ruben Lopez, showed that a family of directed graphs called action graphs gives a new way to obtain this sequence. Since these graphs are defined inductively, one might ask what sequences we can get by using a different initial graph but the same induction process. Last year, three more students, Cedric Harper, Ryan Keller, and Mathilde Rosi-Marshall, looked into this question. They found new families, called kk-initial action graphs, which produce self-convolutions of the Catalan sequence. In this talk we’ll introduce the sequences and graphs involved and talk about how these comparisons were made.

**Cutting Up Space: Hilbert’s Third Problem and the Dehn Invariant**

**Jonathan Campbell, ****Vanderbilt University**

Friday, Sept 21st, 1:00pm, Bailey 207

Give two polyhedra of equal volume, can you cut up one into a finite number of pieces, and reassemble it into the other? This was a problem posed by Hilbert in a famous address. I’ll go through the two dimensional analogue of this problem, and present Dehn’s beautiful solution to Hilbert’s question. Time permitting, I’ll give some hint of how this easily stated problem shows up in my own research.

**Counting sudokus**

**Professor Brenda Johnson, ****Union College**

Thursday, Sept 13th, 1:00pm, Bailey 207

Sudoku is a popular puzzle involving a 9×9 grid in which one has to arrange the numbers 1 through 9 so that each row, column, and block contains all nine numbers. There are many interesting mathematical questions involving sudoku puzzles. In this talk, we’ll focus on a couple of questions related to counting sudokus. After discussing how many possible solutions there are for 9×9 and 4×4 sudokus, we’ll look at ways in which one can generate new sudokus from old ones, and whether or not these techniques can be used to generate all sudokus of a given size.

Please view a list of seminars from previous years here.