PhD Candidate
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
322 SEO (M/C 249)
851 S Morgan St
Chicago, IL 60607-7045
Office: 710 SEO
Email: rwalke20@uic.edu
Phone: (312) 413-8323
My research interests lie in logic and model theory. I have been studying applications of the Shelah 2-rank in the context of stable theories and applications of VC-dimension in the context of NIP theories. I have also been working to develop more robust understanding of and applications for distality. To this end, I introduced and developed distality rank as a property of first-order theories. My most recent research shows that in the stable context, distality rank has geometric implications. I am working to determine whether or not these implications extend to NIP theories. Distality rank is also relevant in the context of IP theories, where I am working to develop a more homogeneous version of the hypergraph regularity lemma for n-dependent structures introduced by Artem Chernikov and Henry Towsner. My advisor is David Marker.
Abstract: We introduce tree dimension and its leveled variant in order to measure the complexity of leaf sets in binary trees...
. We then provide a tight upper bound on the size of such sets using leveled tree dimension. This, in turn, implies both the famous Sauer-Shelah Lemma for VC dimension and Bhaskar's version for Littlestone dimension, giving clearer insight into why these results place the exact same upper bound on their respective shatter functions. We also classify the isomorphism types of maximal leaf sets by tree dimension. Finally, we generalize this analysis to higher-arity trees.
Abstract: Building on Pierre Simon's notion of distality, we introduce distality rank as a property of first-order theories...
and give examples for each rank m such that 1 ≤ m ≤ ω. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called m-determinacy and show that theories of distality rank m require certain products to be m-determined. Furthermore, for NIP theories, this behavior characterizes m-distality. If we narrow the scope to stable theories, we observe that m-distality can be characterized by the maximum cycle size found in the forking geometry,
so it coincides with (m − 1)-triviality. On a broader scale, we see that m-distality is a strengthening of Saharon Shelah's notion of m-dependence.
April 6, 2022 -- Joint Mathematics Meetings (ASL Special Session: Model-Theoretic Classification)
March 26, 2022 -- AMS Spring Central Sectional Meeting (ASL Special Session: Model Theory and Its Applications)
March 4, 2021 -- Maryland Model Theory Seminar
April 7, 2020 -- Notre Dame Model Theory Seminar
March 27, 2020 -- ASL North American Annual Meeting (Special Session: Model Theory)
December 6, 2019 -- Caltech-UCLA Logic Seminar
October 29, 2019 -- UIC Logic Seminar
UIC Model Theory seminar
Abstract: Vapnik-Chervonenkis dimension and density are two measures of combinatorial complexity which arose from the study of probability theory. During this two-part talk...
, we will define these measures, give some examples, and discuss the consequences of the famous Sauer-Shelah Lemma. We will move into the model-theoretic context to define the VC density of a theory and discuss recent work by Aschenbrenner, Dolich, Haskell, Macpherson, and Starchenko which finds uniform bounds on VC density for RCVF and ACVF.
Abstract: The Szemerédi Regularity Lemma (1976) has proven to be a very important tool in extremal graph theory with many applications to number theory and computer science as well...
. It basically says that the vertices of any finite graph can be partitioned in such a way that the edges between any pair of sets from the partition are uniformly (or randomly) distributed up to a requested nonzero margin of error ε. Furthermore, the size of partition needed to obtain such regularity depends only on ε, not on the size or complexity of the graph. However, in 1997, Timothy Gowers showed that the size of partition needed in the general case grows faster than an exponential tower of height polynomial in 1/ε. Recently, many subcategories of hypergraphs, such as those with bounded VC dimension and those defined by semialgebraic sets of bounded complexity, have been shown to require only polynomial growth in terms of 1/ε. We will be discussing the results of a paper by Artem Chernikov and Sergei Starchenko in which they develop and prove a model-theoretic analog of the regularity lemma for NIP hypergraphs, both finite and infinite, using finitely approximated Keisler measures. They also show that regular partitions are definable and, when VC dimension is bounded, their size can be bounded by a polynomial in 1/ε. In addition, if the hypergraph is stable, all defective pairs can be eliminated. Alternatively, if the hypergraph is defined in a distal structure, there is a definable partition for which all pairs are homogenous in terms of the edge relation.
Abstract: So far in the seminar we have been discussing o-minimal theories. For the final weeks, we will broaden our scope to include all NIP theories...
and study the behavior of invariant types. My source will be Pierre Simon's paper of the same name (see link). We will be covering Section 2.
UIC Louise Hay Logic seminar
Abstract: Recursive saturation and S-saturation are interesting concepts at the intersection of model theory and recursion theory...
. In this talk, we will define and discuss recursively saturated models, Scott sets, effectively perfect theories, standard systems, and S-saturated models. We will prove that any recursively saturated model of an effectively perfect theory is S-saturated for exactly one Scott set S. It is natural to ask if all Scott sets arise in this fashion. In 1982, Knight and Nadel showed that all Scott sets of size at most ℵ_{1} arise as the standard system of some nonstandard model of Peano Arithmetic. We will prove a more general version of this result which applies to any computable theory. This closes the question under CH. We will discuss recent progress which closes the question for specific theories when CH is not assumed.
Abstract: A major focus of model theory is the study of definable sets. In this talk we will discuss externally definable sets and Shelah expansions...
. Our goal will be to prove a theorem of Saharon Shelah stating that in the NIP setting, Shelah expansions have quantifier elimination. While proving the theorem, we will review and use several basic techniques involving quantifier-free types, heirs, coheirs, and coheir sequences. We will also discuss some areas of open research concerning externally definable sets.
Abstract: Vapnik-Chervonenkis dimension and density are two measures of combinatorial complexity which arose from the study of probability theory. During this two-part talk...
, we will discuss these measures and their duals both in the classical and model-theoretic contexts, prove the famous Sauer-Shelah Lemma, discuss the relationship between VC dimension and NIP, and time permitting discuss some recent applications and open questions.