Colloquium Ical Atom Mailing List

The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing. In particular, we will outline an approach towards proving the conjectures posed by Fox and Kuttler back in 1983 on the asymptotics of sloshing frequencies in two dimensions. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.

The main goal of the talk is to show how some classical constructions in Geometry and Analysis appear (and in a unique way) from elementary and very simple properties. For example, the polarity relation and support functions are very important and well known constructions in Convex Geometry, but some elementary properties uniquely imply these constructions, and lead to their functional versions, say, in the class of log-concave functions. It turns out that they are uniquely defined also for this class, as well as for many other classes of functions. In this talk we will use these Geometric results as an introduction to the main topic which involves the analogous results in Analysis. We will start the Analysis part by characterizing the Fourier transform (on the Schwartz class in R^n) as, essentially, the only map which transforms the product to the convolution, and discuss a very surprising rigidity of the Chain Rule Operator equation (which characterizes the derivation operation). There will be more examples pointing to an exciting continuation of this direction in Analysis.

The results of the geometric part are mostly joint work with Shiri Artstein-Avidan, and of the second, Analysis part, are mostly joint work with Hermann Koenig.

The talk will be easily accessible for graduate students.

In the finite-dimensional representation theory of the symmetric groups $S_n$ over the base field $\mathbb{C}$, there is an an interesting phenomena of “stabilization” as $n \to \infty$: some representations of $S_n$ appear in sequences $(V_n)_{n \geq 0}$, where each $V_n$ is a finite-dimensional representation of $S_n$, where $V_n$ become “the same” in a certain sense for $n >> 0$.

One manifestation of this phenomena are sequences $(V_n)_{n \geq 0}$ such that the characters of $S_n$ on $V_n$ are “polynomial in $n$”. More precisely, these sequences satisfy the condition: for $n>>0$, the trace (character) of the automorphism $\sigma \in S_n$ of $V_n$ is given by a polynomial in the variables $x_i$, where $x_i(\sigma)$ is the number of cycles of length $i$ in the permutation $\sigma$.

In particular, such sequences $(V_n)_{n \geq 0}$ satisfy the agreeable property that $\dim(V_n)$ is polynomial in $n$.

Such “polynomial sequences” are encountered in many contexts: cohomologies of configuration spaces of $n$ distinct ordered points on a connected oriented manifold, spaces of polynomials on rank varieties of $n \times n$ matrices, and more. These sequences are called $FI$-modules, and have been studied extensively by Church, Ellenberg, Farb and others, yielding many interesting results on polynomiality in $n$ of dimensions of these spaces.

A stronger version of the stability phenomena is described by the following two settings:

• The algebraic representations of the infinite symmetric group $$S_{\infty} = \bigcup_{n} S_n,$$ where each representation of $$S_{\infty}$$ corresponds to a ``polynomial sequence'' $$(V_n)_{n \geq 0}$$.
• The "polynomial" family of Deligne categories $$Rep(S_t), ~t \in \mathbb{C}$$, where the objects of the category $$Rep(S_t)$$ can be thought of as "continuations of sequences $$(V_n)_{n \geq 0}$$" to complex values of $$t=n$$.

I will describe both settings, show that they are connected, and explain some applications in the representation theory of the symmetric groups.

Many interesting problems in additive combinatorics have a translation to geometric questions. A classical example to this is when Elekes used point-line incidence bounds on the sum-product problem of Erdos and Szemeredi. In this talk we will see more examples and will list several open problems in additive combinatorics.

In infinite ergodic theory, absolutely normalized pointwise convergence of ergodic sums fails. Sometimes, this is replacable by weaker modes of convergence. Namely distributional limits and/or weak limits (as in e.g. “rational ergodicity”). We’ll review the subject exhibiting “natural examples” and then see the “latest news” that every random variable on the positive reals occurs as the distributional limit of some infinite ergodic transformation. As a corollary, we obtain a complete exhibition of the possible “$A$-rational ergodicity properties” ($0 < A \le \infty$) for infinite ergodic transformations. The main construction follows by “inversion” from a related construction showing that every random variable on the positive reals occurs as the distributional limit of the partial sums some positive, ergodic stationary process normalized by a 1-regularly varying normalizing sequence.

The “latest news” is joint work with Benjamin Weiss. For further info. see arXiv:1604.03218.

Over the course of the last 15 years or so, Minhyong Kim has developed a method for making effective use of the fundamental group to bound sets of solutions to hyperbolic equations; his method opens a new avenue in the quest for an effective version of the Mordell conjecture. But although Kim’s approach has led to the construction of explicit bounds in special cases, the problem of realizing the potential effectivity of his methods remains a difficult and beautiful open problem. In the case of the unit equation, this problem may be approached via ``motivic’’ methods. Using these methods we are able to describe an algorithm; its output upon halting is provably the set of integral points, while its halting depends on conjectures. This will be a colloquium-version of a talk that I gave at the algebraic geometry seminar here in November of 2015.

We describe the Calabi-Yau problem on complex manifolds and its analog in non-archimedean geometry. On complex manifolds the Calabi-Yau problem asks locally for solutions of a PDE of Monge-Ampère type. The complex Calabi-Yau problem was posed by Calabi in 1954 and solved by Yau in 1978. After a presentation of the complex case we give a brief introduction to non-archimedean geometry and report on the recent progress in the non-archimedean case.

Motivated by the formula, due to Bourgain, Brezis and Mironescu, $\lim_{\varepsilon\to 0^+} \int_\Omega\int_\Omega \frac{|u(x)-u(y)|^q}{|x-y|^q}\,\rho_{\varepsilon}(x-y)\,dx\,dy=K_{q,N}\|\nabla u\|_{L^{q}}^q\,,$ that characterizes the functions in $L^q$ that belong to $W^{1,q}$ (for $q>1$) and $BV$ (for $q=1$), respectively, we study what happens when one replaces the denominator in the expression above by $|x-y|$. It turns out that, for $q>1$ the corresponding functionals “see’’ only the jumps of the $BV$ function. We further identify the function space relevant to the study of these functionals, the space $BV^q$, as the Besov space $B^{1/q}_{q,\infty}$. We show, among other things, that $BV^q(\Omega)$ contains both the spaces $BV(\Omega)\cap L^\infty(\Omega)$ and $W^{1/q,q}(\Omega)$. We also present applications to the study of singular perturbation problems of Aviles-Giga type.

Let G be a locally compact group. For example it could be a discrete group or a Lie group. A random closed subgroup of G, whose distribution is invariant under conjugation by elements of G is called an “invariant random subgroup of G” or IRS for short.

IRS turn out to be very useful in a surprisingly wide array of applications even outside of group theory. Yielding significant contributions to a-priori unrelated subjects such as these mentioned in the title.

I will survey some of these developments by stating one theorem in each of these subjects explaining exactly how IRS come into the picture.