This was a long day. There were a lot of short talks to attend, so this will be a pretty lengthy report (probably why I've been putting it off). As such, I am splitting it into two parts.
Talk 1: "Ligeti's Combinatorial Tonality" by Clifton Callendar
I decided to attend some math and music talks, and I'm glad I did, as they were a fun way to start the morning. This first one, as suggested by the title, was a study of some of Liegti's later works, particularly the piece "Disorde." The pieces are neither harmonious or atonal, but a blend of the two. To describe this phenomena, Callendar uses the notions of inter harmonies, intra harmonies, and then interval content. In particular, he performed a statistical analysis of the standard deviation of the interval content for Disorde. The piece corresponds well to statistically expected interval content, but with more emphasis on tritones. Interestingly enough, however, the piece corresponds better with the expected intervals and notes as it goes on. There are two proposed reasons for this. First, the left and right hands of the piano become more separated as the piece goes on, so the increased dissonance is less noticeable to human ears. Second, the piece has something called Iso-rhythmic structure, and as the piece becomes denser towards the end, this becomes compressed, allowing for fewer possible notes and enforcing the randomness. Both reasons seem to be partially correct, based off of a statistical analysis. This was a really fun talk, particularly because we got to listen to the piece and to the speaker play on a keyboard.
Talk 2: "Voicing Transformations and Linear Representation of Uniform Triadic Transformations" by Thomas M. Fiore
Thomas' motivation for this work was Webern concerto no. 9, because it's rich pattern. Thomas is proposing a more robust system for understanding triadic transformations, based on global reflections, contextual reflections, and voicing reflections. In terms of linear algebra, these are matrices which act on the note space. Thomas takes the semi-direct product of the symmetric group on 3 elements with the matrix group generated by these reflections to create what he calls J. J ends up having very nice structure. It also can be used to described a wide variety of classical progressions. For instance, there are elements of J whose orbit produces the diatonic fifths sequence and a subgroup of J is isomorphic to the Hook group. The Hook group generates the uniform triadic transformations, and was known before, but this is the first time a linear description of this group and its action has been realized. The major downside of this work is that it ignores inversions. So the progression is only represented up to inversion. Still pretty cool. I like how this material is essentially clever applications of undergraduate mathematics.
Talk 3: "Hypergraph Regularity, Ultraproducts, and a Game Semantics" by Henry Towsner
This was another talk that was related to that fateful blog post by Terry Tao on non-standard analysis. Henry is studying the ultraproducts of finite (or finitely generated/finite dimensional) objects. There is a recurring phenomena where properties of the ultraproduct translate into uniform statements about the class of finite objects forming the ultraproduct. An example of this is meta-stability. For finite graphs, this is the Szeneredi Regularity Lemma. A corresponding statement can be proved for hypergraphs, but is horribly complicated to state and prove. Henry, proceeding philosphically, was wondering about the role of the ultraproduct in these statements, and whether a better semantics was possible which could describe this phenomena. Building on the game-theoretic form of first order logic semantics, Henry described a game setup that can describe these uniform statements, called uniform semantics. This avoids ultraproducts, and can actually be used to obtain specific bounds, which the ultraproduct method could not. This is honestly super cool, and I'm really intrigued by the further applications of this that Henry mentioned.
Talk 4: "Measurable Chromatic Numbers" by Clinton Conley
This is essentially what it sounds like (if you know the terminology). The objects of study here are graph structures on Polish spaces, where the edge relation is Borel. The chromatic number is the minimum number of colors needed to color the graph in such a way that if two nodes share an edge, they are different colors. This however splits into three numbers. The raw number, where the axiom of choice can be used to select starting points in components of the graph, The Borel number, where the colorings under consideration have to be Borel, and the measure number, where the colorings under consideration have to be (let's say Lebesgue) measurable and you only have to color the graph up to a set of full measure. For countable graphs, there is a neat result called Brooke's theorem, which states that if the degree of the graph is less than d, the chromatic number is at most d+1. If the graph contains no finite cliques and no odd cycles, then the chromatic number is at most d. This extends to the Borel chromatic number, except that Marks proved that there are acyclic Borel graphs of degree d (for any d) whose Borel chromatic number is d+1. The example produced by Marks are of high complexity, however, which is maybe the only way one could ask for this result to be improved. For the measure chromatic number, there are actually two variants: the first listed above, where you color up to a set of full measure, and another where for all epsilon, you color the graph up to set of measure 1 - epsilon. These two number can be different, as seen in the example of the graph generated by the irrational rotation of the circle (3 for the full measure version, 2 for the epsilon version). In fact, any graph generated by the Borel action of an amenable group has epsilon measure chromatic number 2. So for measure chromatic number, the irrational rotation of the circle, is essentially the only new criteria that needs to be added to Brooks theorem. With this in hand, and the fact that the Borel actions of amenable groups generate hyperfinite graphs (up to measure), one might hope that Marks' examples somehow rely on being of high complexity. Indeed, it's not possible to use smooth graphs for such a class of examples. However, by tweaking Marks' example in a very non-trivial way, Clinton was able to create (for all d) hyperfinite acyclic graphs of degree d whose Borel chromatic number is d+1.
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