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This is a survey on the recent theory on minimizing the normalized volume function attached to any klt singularities.
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We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a klt singularity is unique up to rescaling. This is achieved by defining stability thresholds for valuations, and then showing that a valuation is a minimizer if and only if it is K-semistable, and that K-semistable valuation is unique up to rescaling. As applications, we prove a finite degree formula for volumes of klt singularities and an effective bound of the local fundamental group of a klt singularity.
Dilation theory is a paradigm for studying operators by way of exhibiting an operator as a compression of another operator which is in some sense well behaved. For example, every contraction can be dilated to (i.e., is a compression of) a unitary operator, and on this simple fact a penetrating theory of non-normal operators has been developed. In the first part of this survey, I will leisurely review key classical results on dilation theory for a single operator or for several commuting operators, and sample applications of dilation theory in operator theory and in function theory. Then, in the second part, I will give a rapid account of a plethora of variants of dilation theory and their applications. In particular, I will discuss dilation theory of completely positive maps and semigroups, as well as the operator algebraic approach to dilation theory. In the last part, I will present relatively new dilation problems in the noncommutative setting which are related to the study of matrix convex sets and operator systems, and are motivated by applications in control theory. These problems include dilating tuples of noncommuting operators to tuples of commuting normal operators with a specified joint spectrum. I will also describe the recently studied problem of determining the optimal constant $c = c_{theta,theta}$, such that every pair of unitaries $U,V$ satisfying $VU = e^{itheta} UV$ can be dilated to a pair of $cU, cV$, where $U,V$ are unitaries that satisfy the commutation relation $VU = e^{itheta} UV$. The solution of this problem gives rise to a new and surprising application of dilation theory to the continuity of the spectrum of the almost Mathieu operator from mathematical physics.
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $xin (X,Delta)$ satisfies the ACC if the coefficients of $Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.
Convolutional networks are large linear systems divided into layers and connected by non-linear units. These units are the articulations that allow the network to adapt to the input. To understand how a network manages to solve a problem we must look at the articulated decisions in entirety. If we could capture the actions of non-linear units for a particular input, we would be able to replay the whole system back and forth as if it was always linear. It would also reveal the actions of non-linearities because the resulting linear system, a Linear Interpreter, depends on the input image. We introduce a hooking layer, called a LinearScope, which allows us to run the network and the linear interpreter in parallel. Its implementation is simple, flexible and efficient. From here we can make many curious inquiries: how do these linear systems look like? When the rows and columns of the transformation matrix are images, how do they look like? What type of basis do these linear transformations rely on? The answers depend on the problems presented, through which we take a tour to some popular architectures used for classification, super-resolution (SR) and image-to-image translation (I2I). For classification we observe that popular networks use a pixel-wise vote per class strategy and heavily rely on bias parameters. For SR and I2I we find that CNNs use wavelet-type basis similar to the human visual system. For I2I we reveal copy-move and template-creation strategies to generate outputs.
We review a gravitational model based on noncommutative geometry and the spectral action principle. The space-time geometry is described by the tensor product of a four-dimensional Riemanian manifold by a discrete noncommutative space consisting of only two points. With a specific choice of the finite dimensional involutive algebra, the noncommutative spectral action leads to the standard model of electroweak and strong interactions minimally coupled to Einstein and Weyl gravity. We present the main mathematical ingredients of this model and discuss their physical implications. We argue that the doubling of the algebra is intimately related to dissipation and the gauge field structure. We then show how this noncommutative spectral geometry model, a purely classical construction, carries implicit in the doubling of the algebra the seeds of quantization. After a short review on the phenomenological consequences of this geometric model as an approach to unification, we discuss some of its cosmological consequences. In particular, we study deviations of the Friedmann equation, propagation of gravitational waves, and investigate whether any of the scalar fields in this model could play the role of the inflaton.
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