The colloquium is preceded by tea and cookies at 3:30 in MONT 201 and usually followed by dinner with the speaker.

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Title: The universal triangle-free graph has finite big Ramsey degrees

Speaker: Natasha Dobrinen (University of Denver)

Time: Thursday, September 21, 2017 at 4:00 pm

Place: MONT 214Abstract: It is a central question in the theory of homogeneous relational structures as to which structures have finite big Ramsey degrees. This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorcevic in 2005. An infinite structure $\mathbf{S}$ is homogeneous if any isomorphism between two finitely generated substructures of $\mathbf{S}$ can be extended to an automorphism of $\mathbf{S}$. A homogeneous structure $\mathbf{S}$ is said to have finite big Ramsey degrees if for each finite substructure $\mathbf{A}$ of $\mathbf{S}$, there is a number $n$, depending on $\mathbf{A}$, such that any coloring of the copies of $\mathbf{A}$ in $\mathbf{S}$ into finitely many colors can be reduced down to no more than $n$ colors on some substructure $\mathbf{S}'$ isomorphic to $\mathbf{S}$. This is interesting not only as a Ramsey property for infinite structures, but also because of its implications for topological dynamics.

Prior to work in this paper, finite big Ramsey degrees had been proved for a handful of homogeneous structures: the rationals (Devlin 1979) the Rado graph (Sauer 2006), ultrametric spaces (Nguyen Van Thé 2008), and enriched versions of the rationals and related circular directed graphs (Laflamme, Nguyen Van Thé, and Sauer 2010). According to Nguyen Van Thé , "so far, the lack of tools to represent ultrahomogeneous structures is the major obstacle towards a better understanding of their infinite partition properties." We address this obstacle by providing new tools to represent the homogeneous triangle-free graph and developing the necessary Ramsey theory to deduce finite big Ramsey degrees. The methods developed seem robust enough that correct modifications should likely apply to a large class of homogeneous structures omitting some finite substructure.

Title: How hard is it to compute an isomorphism?

Speaker: Barbara Csima (University of Waterloo)

Time: Thursday, October 5, 2017 at 4:00 pm

Place: MONT 214Abstract: We say a structure is computable (or computably presented) if it has domain the natural numbers, and if all functions, relations and constants are uniformly computable. This definition is not isomorphism invariant. We can ask: between two computable copies of a given structure, what is this simplest isomorphism that must exist? For example, between any two computable copies of the rationals, there always exists a computable isomorphism, since the usual back-and-forth isomorphism can be found effectively. On the other hand, the difficulty of computing an isomorphism between two computable copies of $(\mathbb{N}, <)$ is exactly that of the halting set, since we need to answer the single-quantifier questions of which element is least, next, etc. In this talk, we present what is known about Turing degrees that arise as degrees of isomorphisms for computable structures.

Title: Algorithms, Equations, and Logic

Speaker: Martin Davis (NYU-Courant and UC Berkeley)

Time: Tuesday, October 10, 2017 at 4:00 pm

Place: Schenker 151Abstract: Comparison of different ways of specifying collections of numbers will lead to conclusions about limitations in what can be computed and what can be proved.

Title: Unsolvability and Undecidability in the Diophantine Realm

Speaker: Martin Davis (NYU-Courant and UC Berkeley)

Time: Thursday, October 12, 2017 at 4:00 pm

Place: Schenker 151Abstract: The results leading to the unsolvability of Hilbert's 10th problem will be discussed with applications and extensions.

Title: Complexity in fields of generalized power series

Speaker: Julia Knight (University of Notre Dame)

Time: Thursday, October 19, 2017 at 4:00 pm

Place: MONT 214Abstract: Newton (in 1676) and Puiseux (in 1850) showed that if $K$ is real closed, or algebraically closed of characteristic $0$, then the field of Puiseux series with coefficients in $K$ is also real closed, or algebraically closed. Mac Lane extended the Newton-Puiseux Theorem to ``Hahn fields'' We are interested in measuring complexity of the root-taking process. We want to know whether it is effective? If not, we want to say how far it is from being effective.

Title: Vaught's conjecture in computability theory

Speaker: Antonio Montalbán (UC Berkeley)

Time: Thursday, October 26, 2017 at 4:00 pm

Place: MONT 214Abstract: We'll describe Vaught's conjecture, which is one of the most well-known and longest-standing open questions in logic. The conjecture essentially says that the continuum hypothesis holds when restricted to counting the number of models of a theory. We'll mention the author's result that this conjecture is equivalent to a computability-theoretic statement.

Title: Mathematical Aspects of Arbitrage

Speaker: Ioannis Karatzas (Columbia University)

Time: Thursday, November 2, 2017 at 4:00 pm

Place: MONT 214Abstract: We introduce simple models for financial markets and, in their context, the notions of "portfolio rules" and of "arbitrage". The normative assumption of "absence of arbitrage" is central in the modern theories of mathematical economics and finance. We relate it to concepts iii the theory of probability, such as "fair game", "martingale", "coherence" in the sense of deFinetti, and "equivalent martingale measure". We also survey recent work in the context of the Stochastic Portfolio Theory pioneered by E.R. Fernholz. This theory provides descriptive conditions under which arbitrage, or "outperformance", opportunities do exist, then constructs simple portfolios that implement them. Finally we explain how, even in the presence of such arbitrage, most of the standard mathematical theory of finance still functions, though in somewhat modified form.

Title: Computability Theory and Diophantine Approximation

Speaker: Theodore Slaman (UC Berkeley)

Time: Thursday, November 9, 2017 at 4:00 pm

Place: SCHN 151Abstract: We will discuss similarly motivated themes of study in Diophantine Approximation and Computability Theory: normality/algorithmic randomness, approximation by algebraic numbers/Kolmogorov complexity. We will describe recent work on topics common to both areas and suggest questions for the future.

Title: On Euclidean Yang-Mills Random Fields

Speaker: Bruce Driver ( University of California, San Diego )

Faculty Sponsor:Maria Gordina

Time: Tuesday, November 14, 2017 at 4:00 pm

Place: MONT 214Abstract: The aim of this talk is to discuss some aspects of the Clay-Mathematics millennium prize problem involving the "quantization" of Yang-Mills fields. We will begin with a gentle introduction to one possible method towards rigorously constructing this model which involves integration over certain infinite dimensional spaces. We will then specialize to the case d=2 (one space and one time dimension) where known probabilistic methods may be applied to define the theory. In the context of the d=2 theory (d=4 is the relevant physical dimension), we will then discuss some intriguing identities which were first discovered and heuristically "proved" by the physicists Makeenko and Migdal (1979) and Kazakov and Kostov (1980). The final goal of this talk is to explain how the heuristic Makeenko and Migdal machinery leads to another "natural" rigorous proof T. Lévy's (2011) extended version of the physicist's original identities.

Organizer: Ralf Schiffler