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The Kechris-Pestov-Todor\v{c}evi\'c correspondence (KPT-correspondence for short) is a surprising correspondence between model theory, combinatorics and topological dynamics. In this paper we present a categorical re-interpretation of (a…

Martin and Osswald \cite{Martin07} have recently proposed many generalizations of combination rules on quantitative beliefs in order to manage the conflict and to consider the specificity of the responses of the experts. Since the experts…

We point out that the recursive formula that appears in Erickson's presentation "Fusible Numbers" is incorrect, and pose an alternate conjecture about the structure of fusible numbers. Although we are unable to solve the conjecture, we…

A theory of matchings for finite subsets of an abelian group, introduced in connection with a conjecture of Wakeford on canonical forms for homogeneous polynomials, has since been extended to the setting of field extensions and to that of…

Several results about the union-closed sets conjecture are presented.

Controlling multiple wave properties simultaneously poses a key challenge in coherent control of wave transport. We present a theory for joint coherent control of transmission, reflection, and absorption in linear systems. We prove that the…

In this paper, we use the theory of symmetric Dirichlet forms to give a probabilistic interpretation of Calder\'{o}n's inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes.

We show that, consistently, there is a Borel set which has uncountably many pairwise very non-disjoint translations, but does not allow a perfect set of such translations.

Hilbert and Ackermann asked for a method to consistently extend incomplete theories to complete theories. G\"odel essentially proved that any theory capable of encoding its own statements and their proofs contains statements that are true…

We introduce the first method for translating text embeddings from one vector space to another without any paired data, encoders, or predefined sets of matches. Our unsupervised approach translates any embedding to and from a universal…

We deduce decompositions of natural representations of general linear groups and symmetric groups from combinatorial bijections involving tableaux. These include some of Howe's dualities, Gelfand models, the Schur-Weyl decomposition of…

Preservation theorems provide a direct correspondence between the syntactic structure of first-order sentences and the closure properties of their respective classes of models. A line of work has explored preservation theorems relativised…

In the encoder-decoder architecture for neural machine translation (NMT), the hidden states of the recurrent structures in the encoder and decoder carry the crucial information about the sentence.These vectors are generated by parameters…

The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this…

Complemented lattices and uniquely complemented lattices are very important, not only in mathematics, but also in physics, biology, and even in social sciences. They have been investigated for a long time, especially by Huntington,…

We extend the Duffin--Schaeffer conjecture to the setting of systems of $m$ linear forms in $n$ variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no $n$-by-$m$ systems…

General potential theories concern the study of functions which are subharmonic with respect to a suitable constraint set (called a subequation) in the space of 2-jets. While interesting in their own right, general potential theories are…

This paper treats the variation of sets. We attempt to formulate convergence and continuity of set-valued functions in a different way from the theories on sequences of sets and correspondence. In the final section, we also attempt to…

This is a survey of recent advances in commutative algebra, especially in mixed characteristic, obtained by using the theory of perfectoid spaces. An explanation of these techniques and a short account of the author's proof of the direct…

There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem…