Calendar

I discuss the impact of the positive cosmological constant on the interplay between the equivalence principle in general relativity, and the rules of quantum mechanics. There is an ambiguity in the definition of a phase of a wave function measured by inertial and accelerating observes which is a non-relativistic analogue of the Unruh effect. This will be put in the framework of a non-relativistic limit of twistor space.

We prove a conjecture by Cameron and Aydinian in three dimensions, showing that the density of the largest sum-free subset of [n]^3 approaches (10 + √15)/20 as n approaches infinity. The resolution of the two-dimensional problem was accomplished by Elsholtz and Rackham in 2017. In this work, we introduce a different approach that involves considering a relaxed linear program on the projection [n]^2 and then constructing a suitable dual solution. We also explore potential approaches for addressing the higher-dimensional version of this problem.

For information about the Combinatorics Seminar, please visit http://math.mit.edu/seminars/combin/

In the setting of lattice gauge theories with finite (possibly non-Abelian) gauge groups at weak coupling, we prove exponential decay of correlations for a wide class of gauge invariant functions, which in particular includes arbitrary functions of Wilson loop observables. Based on joint work with Sky Cao.

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability.

We will first review these motivations, then present the ”mean-field” derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature.

Talk will be followed by Tea in the Math Common Room – Science Center, 4th Floor

The flag variety of rank r=(r_1,…,r_k) has points corresponding to collections of subspaces (V_1,…, V_k) with V_i of dimension r_i such that V_i is contained in V_{i+1}. It can be embedded into a multi-projective space, where it is cut out by the incidence Plücker relations. We explore two extensions of this variety: First, we study the nonnegative flag variety, which corresponds to a subset of the flag variety consisting of flags that can be represented by totally positive matrices. Second, we study the tropicalization of the flag variety and, more specifically, its nonnegative part. In both cases, we provide equivalent descriptions of these spaces for flag varieties of rank r=(a,a+1,…,b), where r consists of consecutive integers. We also explore descriptions of the nonnegative tropical flag variety in terms of polytopal subdivisions. Finally, based on explicit computations, we highlight a number of possible fan structures on the totally nonnegative tropical flag variety. This talk is based on joint work with Chris Eur and Lauren Williams.

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For more info, see https://math.mit.edu/combin/