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We survey progress from the past five years on the distribution of mass in high-dimensional convex bodies and in probability distributions with convexity properties. The concentration of measure phenomenon has traditionally been studied in highly regular or structured settings, such as spheres, Hamming cubes, Gaussian measures, Markov chains, and martingales. It turns out that convexity assumptions provide an alternative source of regularity in high dimensions with remarkably similar features: Lipschitz functions are highly concentrated, the isoperimetric problem is nearly saturated by half-spaces (up to logarithmic factors), and the central limit theorem is nearly as strong as in the setting of independent random variables. The main developments discussed include the resolution of Bourgain’s slicing problem and the Variance Conjecture, as well as recent progress on the isoperimetric problem for high-dimensional convex bodies. Based on joint work with P. Bizeul and J. Lehec.

We will survey basic tools of project scheduling including Gannt charts, CPM and PERT. We will then consider a new point of view that takes into account the different resources that different potential contractors may have when scheduling the same project. We will also consider the aspects of policies for many similar projects. We will do so taking into account only operational considerations. Nonetheless, we will show that this purely application driven approach can lead to a lot of interesting and diverse mathematics and physics including enumerative combinatorics, Lorentzian geometry, Kardar-Parisi-Zhang processes (Integrable probability) and wave propagation in hyperbolic media. The talk will be self contained.

Model theory of Abelian groups is extensively studied in the literature also in recent years. An identical inclusion is a formula that can be expressed as a (possibly infinitary) disjunctive identity u = v1 ∨ u = v2 ∨ u = v3 ∨ . . . , or, equivalently, as a universally closed identical equality of subsets of words (terms). For groups and rings, the classes defined by identical inclusions and by infinitary disjunctive identities are coincide, for semigroups they do not coincide. A class of algebras defined by a set of identical inclusions is called an inclusive variety. An inclusive variety that can not be defined by first order formulas is called a nonelementary inclusive variety. An inclusive variety defined by a system of identical inclusions - each depending on a finite set of variables - is called a quasielementary inclusive variety.

We describe elementary, nonelementary and quasielementary inclusive varieties of Abelian groups. There exist continuum many inclusive varieties of each of these kinds. We also determine Abelian groups defined by identical inclusions up to isomorphism and classify Abelian groups up to inclusive equivalence.

The theory of characters is central to the representation theory of finite groups. Harish-Chandra introduced the theory of characters as a fundamental tool for studying infinite-dimensional representations of real and p-adic groups G. In this setting, the character is a conjugation-invariant distribution on the group. For p-adic reductive groups, Harish-Chandra obtained a beautiful formula for the characters of cuspidal representations in terms of orbital integrals.

One of Harish-Chandra’s most remarkable results—valid for real reductive groups and for reductive p-adic groups over fields of characteristic zero—is the regularity theorem, which establishes that the characters of irreducible representations of such groups are represented by functions that are locally in L1(G). He achieved this by obtaining a bound on the character in terms of a power of a rather elementary function, namely the discriminant function on the group (which measures the distances between eigenvalues). In the positive-characteristic case, this approach to proving local integrability fails, and the problem remains open.

In this talk, we will describe our approach and results concerning the regularity of characters of representations of p-adic groups over fields of positive characteristic. In the case of GLn, this leads to a proof of the Harish-Chandra regularity theorem for cuspidal representations. Our key tool is the so-called Chevalley map (for G=GLn, this is the map sending a matrix to the coefficients of its characteristic polynomial): we show that it sends compactly supported smooth measures to essentially bounded measures, and from this we deduce the regularity theorem.

No prior familiarity with p-adic groups or representation theory will be assumed.

This talk is based on a series of joint works with Aizenbud, Gourevitch, and Kazhdan (see arXiv:2602.16389), as well as on very recent joint work by the same authors with Avni.