## Past Seminars

### Spring 2018

#### Quadratic Orbits Modulo n: A Familiar Object Viewed from a Different Vantage Point

Rob Sulman, Mathematics, SUNY Oneonta

**Abstract: **We considered repeated iterations of quadratic functions f acting on the ring (**Z**/n**Z**, +_{n} , •_{n}). The result is a rich variety of structures, the orbits consisting of cycles as well as whiskers when a given f is not one-to-one. In addition, some of the structure of the group of units of the ring are seen in these graphs. We began with moduli a power of 2, where inverse-connections (between units) are included in the orbit graphs. We saw that as the powers increased (for a fixed f ), the corresponding orbit graphs grew in a consistent manner. For the modulus not a power of two, there is a different kind of variety, with a tendency to produce cycles containing various whisker structures. This talk is accessible to students with good calculating ability. The knowledge of the notion of a group would be helpful.

#### Computation in Algebra

Eran Crockett, Mathematics - Binghamton University

**Abstract:** An example of an algebraic computation you may have done is to use Gaussian elimination to solve a system of linear equations. Assuming basic arithmetic operations take 1 second of time, Gaussian elimination can be completed in roughly n^3 seconds (here n is the maximum of the number of variables and number of equations). After working through this carefully, we showed there is another efficient algorithm that answers a similar question for finite groups. Finally, we generalized this question to arbitrary finite algebraic structures and asked when are there efficient solutions.

#### Quantum Theoretic Reality: the EPR Experiment

David Clark, Mathematics, SUNY New Paltz

**Abstract:** Since its inception a century ago, quantum theory has served to accurately predict behavior at the atomic and subatomic levels. In his famous statement, “God does not play dice!”, Albert Einstein objected to its probabilistic formulation, maintaining that there must be a more fundamental theory that would supply the missing information. This lecture presented a famous thought experiment which he and two colleagues devised to establish their point of view. Years after his passing the technology was developed to carry out this experiment and — contrary to Einstein’s expectations — it showed that there can be no more complete theory. This forced us to accept a reality at the subatomic level that is quite different from our human scale experience.

### Fall 2017

#### A Bit of Metric Geometry

By Ulysses Alvarez, Binghamton University

** Abstract: **We defined a notion of curvature in abstract spaces using triangles.

#### These are a few of my favorite groups

By Matt Zaremsky, SUNY Albany

** Abstract: **Group theory can be viewed as the study of symmetries of geometric objects, via abstract algebraic tools called “groups.” In the other direction, when handed an abstract group, one might wonder if there is a nice geometric object whose symmetries it describes. This talk was an introduction to the interplay between geometry and algebra, via some of my favorite groups. Along the way, I discussed a colorful cast of characters including groups of permutations, groups of braids, matrix groups, a fascinating class of groups with the unfortunately non-descriptive name “Thompson’s groups,” and more. As time permitted, I specifically discussed some of my results on topological finiteness properties of groups.

#### Investigating Calculus Students’ Conception of Continuity

By Jayleen Wangle, SUNY Oneonta

**Abstract:**

Continuity is a central yet subtle concept in Calculus I. Yet very few students seem to grasp the nature of continuity. This study uses a mixed methods model to investigate collegiate calculus students' understanding of continuity. I discussed participant displayed depth of understanding of function, limit, and continuity in terms of the constructs defined by Dubinsky’s (1991) Action-Process-Object-Schema (APOS) theory. A prominent finding was that participants who demonstrated a stronger conception of the function displayed a more in-depth understanding of continuity.