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The foundations of classical Algebraic Geometry and Real Algebraic Geometry are the Nullstellensatz and Positivstellensatz. Over the last two decades the basic analogous theorems for matrix and operator theory (noncommutative variables)…
In this paper, we establish the existence of team-optimal policies for static teams and a class of sequential dynamic teams. We first consider the static team problems and show the existence of optimal policies under certain regularity…
We consider a hypergraph (I,C), with possible multiple (hyper)edges and loops, in which the vertices $i\in I$ are interpreted as agents, and the edges $c\in C$ as contracts that can be concluded between agents. The preferences of each agent…
Colloquially, there are two groups, $n$ men and $n$ women, each man (woman) ranking women (men) as potential marriage partners. A complete matching is called stable if no unmatched pair prefer each other to their partners in the matching.…
The many-to-one stable matching problem provides the fundamental abstraction of several real-world matching markets such as school choice and hospital-resident allocation. The agents on both sides are often referred to as residents and…
A central problem in analyzing networks is partitioning them into modules or communities. One of the best tools for this is the stochastic block model, which clusters vertices into blocks with statistically homogeneous pattern of links.…
We begin a systematic development of structure theory for a first order theory, which is stable over a monadic predicate. We show that stability over a predicate implies quantifier free definability of types over stable sets, introduce an…
The work lays the foundations of the theory of changeable sets. In author opinion, this theory, in the process of it's development and improvement, can become one of the tools of solving the sixth Hilbert problem least for physics of…
We develop a framework to give upper bounds on the "practical" computational complexity of stability problems for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems using…
In this paper we provide a first-ever epistemic formulation of stabilizing agreement, defined as the non-terminating variant of the well established consensus problem. In stabilizing agreements, agents are given (possibly different) initial…
We study a matching problem between agents and public goods, in settings without monetary transfers. Since goods are public, they have no capacity constraints. There is no exogenously defined budget of goods to be provided. Rather, each…
In this paper, we introduce the notions of motivic representation stability that is an algebraic counterpart of the notion of representation stability. In the process, we also introduce the notion of motivic decomposition for varieties…
We use algorithmic methods from online learning to explore some important objects at the intersection of model theory and combinatorics, and find natural ways that algorithmic methods can detect and explain (and improve our understanding…
A group is said to be stable if it is isomorphic to its automorphism group. We investigate how we can extend centerless groups to construct finite stable groups with nontrivial centers. To this end, we classify all finite stable groups…
Given a set U of alternatives, a choice (correspondence) on U is a contractive map c defined on a family Omega of nonempty subsets of U. Semantically, a choice c associates to each menu A in Omega a nonempty subset c(A) of A comprising all…
Asking which sets are fixed-parameter tractable for a given parameterization constitutes much of the current research in parameterized complexity theory. This approach faces some of the core difficulties in complexity theory. By focussing…
In this survey article we summarize the current state of research in representation stability theory. We look at three different, yet related, approaches, using (1) the category of FI-modules, (2) Schur-Weyl duality, and (3)…
Optimal solutions of combinatorial optimization problems can be sensitive to changes in the cost of one or more elements of the ground set E. Single and set tolerances measure the supremum / infimum possible change such that the current…
Many-to-many matching with contracts is studied in the framework of revealed preferences. All preferences are described by choice functions that satisfy natural conditions. Under a no-externality assumption individual preferences can be…
We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between…