Krieger’s generator theorem says that all free ergodic measure preserving actions (under natural entropy constraints) can be modelled by a full shift. Recently, in a sequence of two papers Mike Hochman noticed that this theorem can be strengthened: He showed that all free homeomorphisms of a Polish space (under entropy constraints) can be Borel embedded into the full shift. In this talk we will discuss some results along this line from a recent paper with Tom Meyerovitch and ongoing work with Spencer Unger.
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With Meyerovitch, we established a condition called flexibility under which a large class of systems are almost Borel universal, meaning that such systems can model any free Z\^{}d action on a Polish space up to a null set. The condition of flexibility covered a large class of examples including those of domino tilings and the space of proper 3-colourings and answered questions by Robinson and Sahin. However extending the embedding to include the null set is a daunting task and there are many partial results towards this. Using tools developed by Gao, Jackson, Krohne and Seward, along with Spencer Unger we were able to get Borel embedding of symbolic systems (as opposed to all Borel systems) under some very similar assumptions which still covered all the examples that we were interested in. This answered questions by Gao and Jackson and recovered results announced by Gao, Jackson, Krohne and Seward.

I'll discuss joint with Jekel-Nelson-Sinclair. We give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using 1-bounded entropy. The main techniques are concentration of measure on unitary groups as well as Voiculescu's asymptotic freeness theorem.

Given a smooth Jordan curve and a cyclic quadrilateral (a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle) we show that there exist four points on the Jordan curve forming the vertices of a quadrilateral similar to the one given. The smoothness condition cannot be dropped (since not all cyclic quadrilaterals can be inscribed in all triangles), while the cyclicity is necessary (since the circle is itself a smooth Jordan curve). The proof involves some results in symplectic topology. No prior knowledge assumed.
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Joint work with Josh Greene.

Consider the following statement: If $G$ is a finitely generated group, and all elements of $G$ have finite order, then $G$ is a finite group. Is it true? Nope. We'll construct a counterexample (the Grigorchuk group), and then talk a little about the properties and representations of any such counterexample.

In ergodic theory, one often studies measure-preserving actions of countable groups on probability spaces. Bernoulli shifts are a class of such actions that are particularly simple to define, but despite several decades of study some elementary questions about them still remain open, such as how they are classified up to isomorphism. Progress in understanding Bernoulli shifts has historically gone hand-in-hand with the development of a tool known as entropy. In this talk, I will review classical concepts and results, which apply in the case where the acting group is amenable, and then I will discuss recent developments that are beginning to illuminate the case of non-amenable groups.

I will show how to construct a very large family of new examples of convex ancient and translating solutions to mean curvature flow in all dimensions. At $t=-\infty$, these examples resemble a family of standard Grim hyperplanes of certain prescribed orientations. The existence of such examples has been suggested by Hamilton and Huisken—Sinestrari. Our examples include solutions with symmetry group $D\times \mathbb Z_2$, where $D$ is the symmetry group of any given regular polytope, and, surprisingly, many examples which admit only a single reflection symmetry. We also exhibit a family of eternal solutions which do not evolve by translation, settling a conjecture of Brian White in the negative. Time permitting, I will present further structure and partial classification results for this class of solutions, as well as some open questions and conjectures.
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Joint with T. Bourni and G. Tinaglia.

The success of deep learning is due, to a large extent, to the remarkable effectiveness of gradient-based optimization methods applied to large neural networks. In this talk I will discuss some general mathematical principles allowing for efficient optimization in over-parameterized non-linear systems, a setting that includes deep neural networks. Remarkably, it seems that optimization of such systems is "easy". In particular, optimization problems corresponding to these systems are not convex, even locally, but instead satisfy locally the Polyak-Lojasiewicz (PL) condition allowing for efficient optimization by gradient descent or SGD. We connect the PL condition of these systems to the condition number associated to the tangent kernel and develop a non-linear theory parallel to classical analyses of over-parameterized linear equations.
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In a related but conceptually separate development, I will discuss a new perspective on the remarkable recently discovered phenomenon of transition to linearity (constancy of NTK) in certain classes of large neural networks. I will show how this transition to linearity results from the scaling of the Hessian with the size of the network.
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Joint work with Chaoyue Liu and Libin Zhu.

To a cuspidal automorphic representation of GSp(4) over
$\mathbb Q$, one can associate a compatible system of Galois
representations $\{\rho_p\}_{p \; \mathrm{prime}}$. For $p$ sufficiently
large, the deformation theory of the mod-$p$ reduction $\overline
\rho_p$ is expected to be unobstructed -- meaning the universal
deformation ring is a power series ring. The global obstructions to
deforming $\overline \rho_p$ is controlled by certain adjoint Bloch-Kato
Selmer groups, which are expected to be trivial for $p$ large enough.
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I will talk about some recent results showing that there are no local
obstructions to the deformation theory of $\overline \rho_p$ for almost
all $p$.
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This is joint work with M. Broshi, C. Sorensen, and T. Weston.

In complex algebraic geometry, the decomposition
theorem asserts that semisimple geometric objects remain semisimple
after taking direct images under proper algebraic maps. This was
conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a
series of long papers via harmonic analysis and D-modules.
In this talk, I would like to explain a simpler proof in the case of
semisimple local systems using a more geometric approach. As a
byproduct, we recover a weak form of Saito's decomposition theorem for
variations of Hodge structures.
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Joint work in progress with Chuanhao Wei.

This talk, the first on the third paper https://arxiv.org/abs/2009.03243 of a series, is partly a continuation of talks given in the fall. See: http://www.math.ucsd.edu/\~{}benchow/cc-seminar 20.html