Mathematics

Weighted Diophantine Approximation on Manifolds

Demi Allen (University of Bristol)

Abstract: Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. Consequently, studying the size of sets of numbers that can be approximated to within a given precision is very important. In the most classical setting, a $\psi$-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function $\psi$. From a measure-theoretic point of view, this classical set of $\psi$-well-approximable numbers in Euclidean space is well understood. However, it is far less clear what happens when this set, or its various generalisations or higher dimensional analogues, is intersected with other natural sets such as curves, manifolds, or fractals. Studying such questions has become a hugely popular topic of interest in Diophantine Approximation in recent years. In this talk I will discuss recent joint work with Baowei Wang (HUST, China) concerning the particular topic of weighted Diophantine Approximation on manifolds.

Decidability bounds for Presburger arithmetic extended by sine

Eion Blanchard (Illinois)

Abstract: Since 1929, when Presburger formalized his eponymous arithmetic $(\mathbb{Z}, <, +)$—integers with order and addition, but not multiplication—various extensions of this theory have been studied. Presburger arithmetic (PA) is decidable, as are some of its extensions, such as that by base-2 exponentiation. In joint work with Philipp Hieronymi, we extend PA by the sine function. Let sine-Presburger arithmetic ($\sin$-PA) be the theory of $(\mathbb{R}, <, +, \mathbb{Z}, \sin)$ with quantification only over integers. We show that the $\sin$-PA fragment with 4 alternating quantifier blocks is undecidable by encoding the halting problem. On the other hand, we present a decision procedure for the existential fragment of $\sin$-PA, under assumption of Schanuel’s Conjecture from transcendental number theory.

An improvement on Łuczak's connected matching method

Shoham Letzter (University College London)

Abstract: A connected matching in a graph $G$ is a matching contained in a connected component of $G$. A well-known method due to Łuczak reduces Ramsey-type problems about paths and cycles in complete graphs to Ramsey-type problems about connected matchings in almost complete graphs. We show that these can be further reduced to Ramsey-type problems about connected matchings in complete graphs.

Wave front propagation for FKPP reaction-diffusion equation on a class of infinite random trees

Wenqing Hu (Missouri University of Science and Technology)

Abstract: The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees is considered in this talk. We show that a travelling wave front emerges, provided that the reaction rate is large enough. The wave travels at a speed that can be quantified via a variational formula involving the random branching degrees and the random branch lengths of the tree. This speed is slower than that of the same equation on the real line, and we can estimate this slow-down in terms of the structure of the tree. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. This idea, combined with the Feynman-Kac formula, connect our analysis of the wave front propagation to the Large Deviations Principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis for this multi-skewed Brownian motion involves delicate estimates for an infinite product of 2 by 2 random matrices parametrized by the structure of the tree and for hitting times of a random walk in random environment. Joint work with Wai-Tong (Louis) Fan (Indiana University) and Grigory Terlov (UIUC).

Compositions as Trans-Scalar Identity

Gualtiero Piccinini (University of Missouri, St. Louis)

Abstract: We define mereologically invariant composition as the relation between a whole object and its parts when the object retains the same parts during a time interval. We argue that mereologically invariant composition is identity between a whole and its parts taken collectively. Our reason is that parts and wholes are equivalent measurements of a portion of reality at different scales in the precise sense employed by measurement theory. The purpose of these scales is the numerical representation of primitive relations between quantities of objects. To show this, we prove representation and uniqueness theorems for composition. Thus, mereologically invariant composition is trans-scalar identity.

Zoom Info: Please email Kay Kirkpatrick (kkirkpat@illinois.edu).