Undergraduate Colloquium Winter 2011

January 12, 2011

Games Night

Dr. Frank Bauerle, Continuing Lecturer, UCSC Mathematics Department 

To start off the new year (the new decade really), we will be hosting Games Night, where we will be bringing a variety of Games to play. A few examples we may choose from are: Settler's of Catan, Ricochet Robots, Quoridor, and Set. 

Hope to see you there for a little friendly competition!

January 19, 2011

Derivatives of Complex Maps

Victor Dods, Department of Mathematics 

We'll talk about a way to represent complex numbers as real 2x2 matrices, and then show how to easily compute tangent maps (i.e. derivatives in the 23A sense) of functions of complex variables (not necessarily holomorphic). 

Knowledge of elementary linear algebra, vector calculus, and the complex numbers is all that is necessary. We will also discuss the Cauchy-Riemann equations.

Find more information at the accompanying article: http://www.thedods.com/victor/papers/derivatives_of_complex_maps.pdf

January 26, 2011

The Gauss-Bonnet Theorem: From Curvature to Topology

Professor Viktor Ginzburg, UCSC Mathematics Department

The sum of the exterior angles of a planar polygon is 360 degrees regardless of the number of vertices of the polygon. This simple fact can be thought of as the first instance of a connection between geometry and topology: every exterior angle is a geometrical quantity that can be changed by a slight alteration of the polygon. However, the total sum of the angles remains constant, i.e., it represents a topological quantity completely determined by the facts that the plane is flat (zero curvature) and that the polygon has no self-intersections. 

What if a polygon is allowed to have self-intersections or replaced by a closed curve? What if the polygon is located on the sphere rather than a plane? Finally, what happens if we go one dimension up and consider a surface in the three-dimensional space instead of a curve on the plane? Answers to all these questions involve the notion of curvature and lead to various versions of the Gauss-Bonnet theorem (the 19th century) which was the starting point of one of the central themes in the 20th century geometry: connections between geometrical and topological quantities.

February 2, 2011

Games Night

Dr. Frank Bauerle, Continuing Lecturer, UCSC Mathematics Department 

We will be having another Games Night, where as usual we will be playing a variety of Games (feel free to bring your own). A few examples we may choose from are: Settler's of Catan, Ricochet Robots, Quoridor, and Set. 

Hope to see you there for a little friendly competition!

February 9, 2011

Two Paradoxes and Their Influence on the Development of Mathematics and Logic

Dr. Frank Bauerle, Continuing Lecturer, UCSC Mathematics Department 

In the first half of this talk we will discuss the "Liar's Paradox" and Goedel's creative use of it in his first incompleteness theorem. Secondly we will look at "Galileo's Paradox" about infinity, and discuss Cantor's solution to this dilemma. Then we will conclude by connecting both topics through Cantor's "Continuum Hypothesis".

February 16, 2011

Games Night

Dr. Frank Bauerle, Continuing Lecturer, UCSC Mathematics Department 

We will be having another Games Night, where as usual we will be playing a variety of Games (feel free to bring your own). A few examples we may choose from are: Settler's of Catan, Ricochet Robots, Quoridor, and Set.

Hope to see you there for a little friendly competition!

February 23, 2011

Cancelled - Representations of the Symmetric Group

Visiting Professor Scott Crofts, UC Santa Cruz Mathematics Department 

Representation Theory is the study of realizing abstract mathematical objects as collections of linear transformations on a fixed vector space. Often times these mathematical objects describe some type of symmetry and their representations thus provide concrete descriptions for this. In the theory of finite groups, the symmetric group is a sort of universal object, making its representation theory particularly important. 

In this talk I will define the symmetric group and describe the beautiful classification and construction of its irreducible representations. The only prerequisite is some basic linear algebra.

March 2, 2011

Two or three perception-filters in mathematics: logical algebra, ratios and, possibly, category theory.

Professor Laurence Barker, Department of Mathematics, Bilkent University 

Are we missing some easier ways of perceiving mathematical concepts? Of course, one cannot exhibit an example of something that one is overlooking. As with any question in methodology of mathematics, the relevant data can come only from the past: we must do some history. We shall give a brief indication of some problems that motivated: the logical approach to algebra developed during the 19th century (which resisted the notion of a mathematical object); the theory of ratios developed during the 4th century BC (which resisted the notion of a fractional number). If we can understand why those dogmas seemed attractive at those times, then that may, perhaps, give us some perspective on some present-day outlooks.

March 9, 2011

Games Night: The Game of Pigs

Dr. Frank Bauerle, Continuing Lecturer, UCSC Mathematics Department 

This week we will play and analyze "The Game of Pigs", a game that involves a combination of strategy and luck. This game is one in a class of games called jeopardy dice games. We will have fun playing it and then look at the mathematics of it. Bring your lucky (but not loaded) dice or take your chance with ours! Time permitting, we can also play some other related dice games. Visit this site for an on-line version of the game: http://cs.gettysburg.edu/projects/pig/piggame.html