Description: This article discusses an abstract index theorem on non-compact Riemannian manifolds. Abstract: We prove an abstract index theorem for essentially self-adjoint Fredholm supersymmetric first-order elliptic differential operators on Hermitian vector bundles over complete oriented Riemannian manifolds. According to our main result the supersymmetric L2-index of such an operator can be expressed as the sum of a "local contribution" (the familiar Atiyah-Singer index form, suitably restricted to and integrated over a finite region) and a "boundary contribution" (which depends only on the restriction of the operator at large distances). This is done by splicing together local parametrices and Green's operators defined "at infinity". The result yields (in fact is equivalent to) a generalisation of the relative index theorem of Gromov and Lawson.

Description: This article addresses, from a historical perspective, Newton's sums of like powers of natural numbers.

Description: This paper discusses autonomous robot localization using WiFi fingerprinting. Abstract: We are using widely available 802.11 wireless networks to determine the location of autonomous robots. Before a robot can accomplish a simple task such as moving to a specific coordinate, it must accurately know its current location with-in a certain degree of accuracy. Humans often take their eye sight and spatial awareness for granted. For a robot, the computational difficulty of solving the same problem becomes apparent. Our implementation creates a database of wireless signal strengths of a given area and uses the current signal strength reading within the area to find a weighted signal space distance. The "closest" point in the database should also correlate with the current position of the robot. Given the robots correct location, the authors can successfully navigate around any area with sufficient Wi-Fi coverage.

Description: This presentation discusses research on predictable patterns in continued fractions and decimal expansions. Abstract: A continued fraction is a representation of a number by series of fractions inside fractions, such that the numerator of every fraction is a one. Decimal expansion is another way to express a number. Many times in mathematics when we have two different ways to write the same expression, we look for connections between the two notations. When trying to express a number, we encounter an interesting anomaly between continued fractions and decimal expansions. A randomly chosen number, with a predictable decimal pattern, will have an unpredictable continued fraction. Furthermore, a randomly chosen continued fraction, with predictable partial quotients, will have an unpredictable decimal expansion. However, there are those few exceptions, one of which the author is studying in depth. It is a binary sequence closely tied with the Fibonacci numbers, a series of numbers that often occur in nature. Through the author's research, the author hopes to find an understanding to this aforementioned anomaly and bridge continued fractions and decimals. In this presentation, the author will show where predictability lies and furthermore where there is chaos still to be settled.

Description: In this presentation, the author will briefly introduce the subject of Descriptive Set Theory and the motivation for its study. The author will discuss the idea of a projective set and also define the mathematical notion of a "tree" as an example of a projective set. The author will conclude with a brief mention of a significant result that can be proved using the notion of a tree.