The classical Ramsey numbers r(s,t) denote the minimum n such that every red-blue coloring of the edges of the complete graph K_n contains either a red clique of order s or a blue clique of order t. These quantities are the centerpiece of graph Ramsey Theory, and have been studied for almost a century. The Erdős-Szekeres Theorem (1935) shows that for each s ≥ 2, r(s,t) = O(t^s - 1) as t →∞. We introduce a new approach using pseudorandom graphs which shows r(4,t) = Ω(t^3/(log t)^4) as t →∞, answering an old conjecture of Erdős, and we illustrate how to apply this approach to many other Ramsey and related combinatorial problems.
A graph $G$ with vertex set $\{v_1,v_2,\ldots,v_n\}$ is an intersection graph of segments if there are segments $s_1,\ldots,s_n$ in the plane such that $s_i$ and $s_j$ have a common point if and only if $\{v_i,v_j\}$ is an edge of~$G$. In this expository paper, we consider the algorithmic problem of testing whether a given abstract graph is an intersection graph of segments. It turned out that this problem is complete for an interesting recently introduced class of computational problems, denoted by $\exists\mathbb{R}$. This class consists of problems that can be reduced, in polynomial time, to solvability of a system of polynomial inequalities in several variables over the reals. We discuss some subtleties in the definition of $\exists\mathbb{R}$, and we provide a complete and streamlined account of a proof of the $\exists\mathbb{R}$-completeness of the recognition problem for segment intersection graphs. Along the way, we establish $\exists\mathbb{R}$-completeness of several other problems. We also present a decision algorithm, due to Muchnik, for the first-order theory of the reals.
We survey the complexity class ∃ℝ, which captures the complexity of deciding the existential theory of the reals. The class ∃ℝ has roots in two different traditions, one based on the Blum-Shub-Smale model of real computation, and the other following work by Mnëv and Shor on the universality of realization spaces of oriented matroids. Over the years the number of problems for which ∃ℝ rather than NP has turned out to be the proper way of measuring their complexity has grown, particularly in the fields of computational geometry, graph drawing, game theory, and some areas in logic and algebra. ∃ℝ has also started appearing in the context of machine learning, Markov decision processes, and probabilistic reasoning. We have aimed at collecting a comprehensive compendium of problems complete and hard for ∃ℝ, as well as a long list of open problems. The compendium is presented in the third part of our survey; a tour through the compendium and the areas it touches on makes up the second part. The first part introduces the reader to the existential theory of the reals as a complexity class, discussing its history, motivation and prospects as well as some technical aspects.
These notes follow a lecture series at the "Singularities and low dimensional topology" winter school at the Rényi Institute in January 2023, with a target audience of graduate students in singularity theory and low-dimensional topology. The lectures discuss the basics of four-dimensional manifold topology, connecting this rich subject to knot theory on one side and to contact, symplectic, and complex geometry (through Stein surfaces) on the other side of the spectrum.
These notes provide an introduction to the stable homotopy types in Khovanov theory (due to Lipshitz-Sarkar) and in knot Floer theory (due to Manolescu-Sarkar). They were written following a lecture series given by Sucharit Sarkar at the Renyi Institute during a special semester on "Singularities and low-dimensional topology", organised by the Erdos Center.
This survey has been written in occasion of the School and Workshop about Optimal Transport on Quantum Structures at Erd\"os Center in September 2022. We discuss some recent results on noncommutative entropic optimal transport problems and their relation to the study of the ground-state energy of a finite-dimensional composite quantum system at positive temperature, following the work [FGP23]. In the first part, we review some of the classical primal-dual formulations of optimal transport in the commutative setting, including extensions to multimarginal problems and entropic regularisation. We discuss the main features of the entropic problem and show how optimisers can be efficiently computed via the so-called Sinkhorn algorithm. In the second part, we discuss how to apply these ideas to a noncommutative setting, in particular on the space of density matrices over finite dimensional Hilbert spaces. In this framework, we present equivalences between primal and dual formulations, and use them to characterise the optimisers. Despite the lack of explicit formulas due to the noncommutative nature of the problem, one can also show that a suitable quantum version of the Sinkhorn algorithm converges to the minimiser of the entropic problem. In the final part of this work, we discuss similar results for bosonic and fermionic systems.
This text is a set of lecture notes for a 4.5-hour course given at the Erd\"os Center (R\'enyi Institute, Budapest) during the Summer School "Optimal Transport on Quantum Structures" (September 19th-23rd, 2023). Lecture I introduces the quantum analogue of the Wasserstein distance of exponent $2$ defined in [F. Golse, C. Mouhot, T. Paul: Comm. Math. Phys. 343 (2016), 165-205], and in [F. Golse, T. Paul: Arch. Ration. Mech. Anal. 223 (2017) 57-94]. Lecture II discusses various applications of this quantum analogue of the Wasserstein distance of exponent $2$, while Lecture III discusses several of its most important properties, such as the triangle inequality, and the Kantorovich duality in the quantum setting, together with some of their implications.
These notes are based on the lectures given by the second author at the School on Optimal Transport on Quantum Structures at Erdös Center in September 2022. The focus of the exposition is on two recently introduced approaches on quantum optimal transport: one based on quantum channels as generalized transport plans, the other based on the notion of Hamming-Wasserstein distance of order 1 on multiple-qubit systems. The material is presented in an elementary manner with a focus on the finite-dimensional setting.
This document presents the contents of three lectures delivered by the author at the Erd\H{o}s Center School ``Optimal Transport on Quantum Structures'', Septemer 19-23, 2022 in Budapest, Hungary. It presents a fairly self contained account of an active topic of current research, and this account should be accessible to most graduate students, as befits lectures for a school. The main results are known, but there a number of new proofs and some new results.
First, we define the lattice cohomology associated with the link of a normal surface singularity, whenever this link is a rational homology sphere. This is done via the lattice of a resolution and well-chosen (Riemann-Roch type) weight functions of the lattice points. We prove that it is independent of all the choices and depends only on the link. The author conjectured that it is isomorphic as a graded Z[U]-module with the Heegaard Floer homology of the link (this fact was verified recently by Zemke). In parallel we define the graded roots as well as improvements of the 0-homology group. We compute this topological lattice cohomology and the graded root in many examples (star shaped graphs, surgery 3-manifolds). Then we discuss its path-version, the path lattice cohomology and its relationship with the geometric genus (of any analytic structure). In the final part we discuss the 'analytic pair': the analytic lattice cohomology. We compare the two theories and we test their behavior with respect to certain analytic deformations.
Sequential importance sampling offers an alternative way to approximately evaluate the permanent. It is a stochastic algorithm which seems to work in practice but has eluded analysis. This paper offers examples where the analysis can be carried out and the first general bounds for the sample size required. This uses a novel importance sampling proof of Brégman’s inequality due to Lovász.
We consider the problem of minimizing a convex function over a convex set given access only to an evaluation oracle for the function and a membership oracle for the set. We give a simple algorithm which solves this problem with $\tilde{O}(n^2)$ oracle calls and $\tilde{O}(n^3)$ additional arithmetic operations. Using this result, we obtain more efficient reductions among the five basic oracles for convex sets and functions defined by Grotschel, Lovasz and Schrijver.
