## Student Seminars

- Speaker: Samantha Youmans (Canisius College)
- Title: Non Linear Classifications of Ultrametric Spaces

**Abstract:**

The following conditions are shown to be equivalent for a metric space *(X,d)*: (1) *X* is ultrametric, (2) *X* has infinite roundness, (3) *X* has infinite generalized roundness, and (4) *X* has strict *p*-negative type for all *p ≥ 0*. In order to apply these equivalences, we show that an *n +1* point metric space *(X,d)*, where *n > 1*, has strict 2-negative type if and only if *(X,d)* can be isometrically embedded in the Euclidean space *R ^{n}* of dimension

*n*but it may not be isometrically embedded in any Euclidean space

*R*of dimension

^{r}*r < n*. This theorem is a variant of Schoenberg's finite metric space theorem. As all ultrametric spaces are seen to have strict 2-negative type by (4), we thus obtain a new proof of Lemin's isometric embedding theorem.

- Speaker: Regina Pollack (Canisius College)
- Title: Connections between Mathematics and Music: The Auditory Aesthetics of the Baroque Era

** Abstract: **The Baroque era signifies a new beginning of music theory and a transition from early musical compositions to an era of strong musical pattern and technique. It is known that the most basic musical substances such as beat, rhythm, and pitch, all have an obviously mathematical and scientific foundation. Although these theoretical aspects of music are important to analyze in such a mathematical style, it is especially interesting to consider the aesthetic portion of music in a mathematical sense. Within newer branches of mathematics such as Information Theory, one can analyze the predictability of musical tones and pattern and attempt to discover a musical composition's aesthetic potential. The baroque era introduced musical theory that has been a continuous source of inspiration for musical composers and its connection to mathematics is extremely significant to the development of later works.

- Speaker: Danielle Mallare (Canisius College)
- Title: New Techniques for Computing Generalized Roundness

** Abstract:**

Computing the exact generalized roundness of a metric space depends on one being able to solve complicated families of non linear inequalities. The notion arose in relation to the study of uniformly continuous maps between metric spaces that have uniformly continuous inverses. Even in the case of a finite metric space of small cardinality, exact computation of the generalized roundness of the space can easily remain elusive. Recently some new techniques for enabling this type of computation have been isolated. The purpose of this talk will be to discuss some of these developments, particularly as they pertain to computing the generalized roundness of finite and countable metric trees, but also to graph metrics in general.

- Speaker: Mary Russell (Canisius College)
- Title: Analyzing Circular Decks and Probability Transition Matrices

**Abstract:**

We consider shuffling strategies for a circular deck of cards. This is a deck that does not have either a top or bottom, but rather treats each card as having a card above and below it. It is interesting to examine decks with this different structure as well as varying size of decks. We will analyze two methods of shuffling on a circular deck and prove the surprising fact that they are equivalent. Additionally, we will analyze the probabilities of transitioning between decks of cards that differ in size by a single card.

- Speaker: Samantha Youmans (Canisius College)
- Title: Non-linear Geometry of Ultrametric Spaces

**Abstract:**

An ultrametric space is a metric space that satisfies a strong triangle inequality. In practice, this means that any three distinct points in an ultrametric space form an isosceles triangle in which the length of the base does not exceed the length of the legs. Ultrametric spaces arise in a variety of fields including number theory, physics, theoretical biology, and so on. It is possible to visualize ultrametric spaces in terms of certain trees. The purpose of this talk will be to examine this process and to describe some new non linear characterizations of ultrametric spaces.

- Speaker: Jason Tata (Canisius College)
- Title: The Supremal
*p*-negative Type of a Finite Metric Space

** Abstract:**

We will define generalized roundness exponent *p* and *p*-negative type of metric space *(X,d)*. Then we'll define the supremal *p*-negative type of a finite metric space *(X,d)* and give an explicit expression for calculating it in terms of the associated distance matrix for *(X,d)*. The method is used to compute the supremal *p*-negative type of complete bipartite graphs with the usual path metric.

- Speaker: Christina Ruggeri (Canisius College)
- Title: The Negative Binomial

** Abstract:**

The negative binomial will be explained and it will be shown how it relates to the Poisson distribution. Its use in the "capture, recapture" process will also be explained.

- Speaker: Dr. Kelly Delp (Buffalo State College)
- Title: Playing with surfaces: spheres, monkey pants, and zippergons

**Abstract:**We describe a process, inspired by clothing design, of smoothing an octahedron to form round sphere. This process can be adapted to construct many different surfaces out of paper and craft foam. (This work is joint with Bill Thurston of Cornell.)

- Speaker: Kristen May, Patrick O'Loughlin, Mary Russell (Canisius College)
- Title: On QR-decomposition

- Speaker: Jerod Sikorskyj (Canisius College)
- Title: Dark Matter: A basic physical introduction and numerical model

- Speaker: Christina Ruggeri (Canisius College)
- Title: Convex Polygons formed by Tangrams

** Abstract:**

It will be proven that exactly thirteen convex polygons can be formed from the seven tangram pieces. It will be shown that the tangram can be broken up into sixteen congruent triangles and a table of cases will show the possible arrangements of these sixteen triangles to prove the main theorem.

- Speaker: Mary Russell (Canisius College)
- Title: Sensitivity of Initial Conditions Seen in Dynamical Systems

** Abstract:**

Dynamical systems is a relatively new field of mathematics that tries to predict future states based on past states. This field studies the behavior of certain functions under iteration. One essential feature of chaotic dynamic systems is sensitive dependence on the initial condition. In a chaotic system, even the smallest difference in the initial conditions can cause the outputs to diverge eventually from each other exponentially. Such a system does not have regularity in its behavior, but we are able to predict the eventual separation between certain points through Lyapunov Exponents. Lyapunov Exponents are able to predict behavior even in non-periodic orbits by evaluating the product of the derivatives of the iterations of the function. This talk will analyze different initial conditions and their eventual outcome. Understanding the patterns of the initial conditions of these systems is extremely useful to predicting patterns in areas such as meteorology and biology.

- Speaker: Mary Russell (Canisius College)
- Title: Matrix Groups as Manifolds

Abstract:

Throughout this talk we will be looking at matrices that we are familiar with from any linear algebra course. As we know these matrices can represent linear transformations. We now want to look at sets of matrices that form a group. That is, we look at a set of matrices that satisfy the necessary axioms defined in an abstract algebra course. We define the term matrix group with the language of topology. We can learn more about the group including tangent spaces to the group. An especially interesting tangent space is located at the identity. This space is called the Lie algebra and is a subspace of the Euclidean space. This fact shows matrix groups behave nicely. We later show that all matrix groups do in fact have a Lie algebra by creating a one-parameter group through matrix exponentiation. This matrix exponentiation parametrizes all points in the matrix group, thus proving any matrix group is a manifold. The Lie algebra encodes information regarding the matrix group, including the dimension of the group. The goal of this talk is to show how a matrix group of a certain dimension is a manifold of the same dimension.

- Speaker: Regina Pollack (Canisius College)
- Title: Fermat's Last Theorem when
*n=4*

**Abstract**:

Pierre de Fermat brought about much consternation over the past couple of centuries because his infamous Last Theorem was left without a proof. Stemming from several aspects of Number Theory, Fermat's Last Theorem states that there are no positive integer solutions of the equation *x ^{n}+y^{n}=z^{n}* for every integer

*n>2*. Countless mathematicians, adamant about proving this theorem, greatly contributed to a final answer, but none were able to fully succeed until Andrew Wiles' proof appeared in the late 1990s. One such important contribution was the proof of this Theorem for the case when

*n=4*which was indirectly described by Fermat himself. This talk will discuss the proof of the case when

*n=4*and will illustrate Fermat's strategy of infinite descent leading to a contradiction.

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