The inverse Galois problem over the rationals has inspired several variants over time.
In this talk, following an introduction to the classical problem and the basic notions involved, we will briefly review the variant which asks for the fewest possible ramified primes in a Galois realization of a finite group over the rationals Q, and then spend most of the remaining talk on a recent variant that asks for Galois realizations of a finite group G over Q in which all the nontrivial inertia subgroups have order two. The only groups for which this can happen are those generated by elements of order two, for example finite nonabelian simple groups. If such a group G has a "regular" realization over the rational function field Q(t), as the splitting field of a polynomial f(t,x), then there are computable conditions on the polynomial which guarantee the existence of infinitely many specializations of t into Q which yield realizations over Q with all inertia groups of order two. As examples, three finite simple groups will be given, together with corresponding polynomials. One application of this result is to unramified realizations of these groups over quadratic fields. Another application is to the problem which motivated this result (time permitting). Joint work with Joachim Koenig and Daniel Rabayev.
Given a countable group G and a probability measure m on G, a function from G to the reals is said to be m-harmonic if it satisfies the mean-value property with respect to averages taken using m. It has long been known that all commutative groups are Choquet-Deny groups: namely, they admit no non-constant bounded harmonic functions. More generally, it has been shown that all virtually nilpotent groups are Choquet-Deny. We show that in the class of finitely generated groups, only virtually nilpotent groups are Choquet-Deny. This proves a conjecture of Kaimanovich and Vershik (1983), who suggested that groups of exponential growth are not Choquet-Deny. Our proof does not use the superpolynomial growth property of non-virtually nilpotent groups, but rather that they have quotients with the infinite conjugacy class property (ICC). Indeed, we show that a countable discrete group is Choquet-Deny if and only if it has no ICC quotients.
Joint work with Joshua Frisch, Yair Hartman and Pooya Vahidi Ferdowsi
The talk will focus on optimization on the high-dimensional sphere when the objective function is a polynomial with independent Gaussian coefficients. Such random processes are called spherical spin glasses in physics, and have been extensively studied since the 80s. I will describe certain geometric properties of spherical spin glasses unique to the full-RSB case, and explain how they can be exploited to design a polynomial time algorithm that finds points within vanishing error from the global minimum.
This will be the second of several lectures in which we will study the paper:
R. Clouatre and M. Hartz, Multiplier algebras of complete Nevanlinna-Pick spaces: dilations, boundary representations and hyperrigidity, J. Funct. Anal. 274 (2018), no. 6, 1690–1738.
Arxiv link to the paper: https://arxiv.org/pdf/1612.03936.pdf
A graph polynomial is weakly distinguishing (on a graph property A) if for almost all finite graphs G (in A) there is a finite graph H with P(G;X)=P(H;X). We show that the clique polynomial and the independence polynomial are weakly distinguishing, and give sufficient conditions on graph properties C for the generating functions of induced subgraphs with property C to be weakly distinguishing. In addition, we give sufficient conditions on graph properties A for the Tutte, the Domination and the Characteristic polynomials to be weakly distinguishing on A.
Advisor: Makowsky Johann
Abstract: A graph polynomial is weakly distinguishing (on a graph property A) if for almost all finite graphs G (in A) there is a finite graph H with P(G;X)=P(H;X). We show that the clique polynomial and the independence polynomial are weakly distinguishing, and give sufficient conditions on graph properties C for the generating functions of induced subgraphs with property C to be weakly distinguishing.
In addition, we give sufficient conditions on graph properties A for the Tutte, the Domination and the Characteristic polynomials to be weakly distinguishing on A.
Estimating a manifold from (possibly noisy) samples appears to be a difficult problem. Indeed, even after decades of research, there are no (computationally tractable) manifold learning methods that actually "learn" the manifold. Most of the methods try, instead, to embed the manifold into a low-dimensional Euclidean space. This process inevitably introduces distortions and cannot guarantee a robust estimate of the manifold.
In this talk, we will discuss a new method to estimate a manifold in the ambient space, which is efficient even in the case of an ambient space of high dimension. The method gives a robust estimate to the manifold and its tangent, without introducing distortions. Moreover, we will show statistical convergence guarantees.
It is on a work in progress, joint with Barak Sober.