Friedrich Bauermeister
Physical Link Theory and Gordian Links
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Knot Theory studies the continuous transformation of curves in R^3 with no intersections. This insufficiently models how real rope behaves: real rope is not an idealized curve of zero thickness. Instead, it has a fixed thickness and a fixed length. Accounting for this, one can start studying "physical" knot and link theory.
Many of the same questions of classical knot and link theory can also be asked in physical knot and link theory. There is a notion of physical link homotopy and physical link isotopy. Additionally, there is the question: For an abstract link, what is the shortest representative with thickness 1?
It turns out that classical link theory and physical link theory are quite different. One way to see this is by exhibiting links that link isotopic (or homotopic), but which fail to be "physically" link isotopic (or homotopic).
In this talk we will show a few examples, including a construction from my still-in-progress Master's thesis of a two-component link that is link-homotopic to the unlink, but not physically link homotopic to the unlink. We will also talk about (locally) length-minimizing link configurations.
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Brian Mintz
Why (almost) all puzzles, and many games, are really just mazes (and how to solve them)
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Puzzles come in many shapes and forms, but one approach can solve an astounding number of them. By representing each state of the puzzle as a point in an abstract space, with connections when one can transition between states with a valid move, one can often find solutions easily. This talk explains this connection in detail, with several fun examples and puzzles for participants to interact with! If time allows, we'll also discuss an amazing result known as the ''Sprague-Grundy Theorem'' that uses this technique to solve a broad class of games. If you like puzzles, or get frustrated when you can't solve them, this talk is for you!
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Ryan Maguire
Double, double, toil and trouble: The mathematics of double-double arithmetic.
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The standard IEEE-754 double precision floating point arithmetic
allows for 52 significant bits, 53 if one includes the implicit
leading bit past the binary point. Translating this to decimal
one can expect between 15 and 16 decimals of accuracy in their
computations. Mathematical computations may need more than
this, say, in the rendering of the Mandelbrot set. Arbitrary
precision arithmetic is dreadfully slow, and quadruple precision
isn't readily available on most common architectures. The solution
is double-double arithmetic. The idea dates back to at least
the 1970s but is still actively researched and experimented with.
In this talk we'll go over the original Dekker algorithms and
discuss the implementation of double-double.
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Ryan Maguire
How to Compute Sine of 10^100
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libm, the standard C library for mathematical functions,
requires implementations for the trigonometric functions.
The difficulty of computing the remainder after division
by a transcendental number, like pi, led to early
implementations having poor or completely gibberish results
for large inputs. One infamous example threw an error for
values |x| > 26. Modern implementations can readily handle
values up to 10^308, the maximum for 64-bit floating point
numbers. We'll talk about how these implementations work
and how to correctly compute the sine of large numbers.
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Jonathan Lindbloom
The conjugate gradient method: A top 10 algorithm of the 20th century
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The computational primitive of matrix inversion appears ubiquitously throughout the pure and applied sciences. Unfortunately, direct inversion methods relying on matrix factorizations -- those taught in introductory linear algebra courses -- are computationally infeasible for large-scale matrix inversion tasks arising in applications. In this talk, will provide an overview of the conjugate gradient (CG) iterative method for solving linear systems involving SPD matrices. The CG method falls under the umbrella of Krylov subspace iterative methods, which were deemed a so-called ''top 10 algorithm of the 20th century''. We will begin by outlining the key ideas and properties of the algorithm, and then proceed to discuss related topics such as preconditioning and least squares problems. In the remaining time, we will provide a sketch of several areas of active research.
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