Recent progress in Mathematics and Statistics

Newton polyhedra relate algebraic geometry and singularity theory with the geometry of convex polyhedra within the framework of toric geometry. This connection is useful in both directions. On the one hand, it provides explicit answers to problems in algebra and singularity theory in terms of convex polyhedra. For instance, according to the Bernstein-Khovanskii-Koushnirenko (BKK) theorem, the number of solutions of a generic system of n equations in (C^*)^n with  fixed Newton polyhedra is equal to the mixed volume of the Newton polyhedra multiplied by n!. This suggests that there should be an analog of the famous Alexandrov- Fenchel inequalities from the theory of mixed volumes in algebraic geometry. (These inequalities can be considered as a broad generalization of the classical isoperimetric inequality.) On the other hand, algebraic theorems of a general nature (such as the Hirzebruch-Riemann-Roch theo- rem) suggest unexpected results in the geometry of convex polyhedra. The theory of Newton-Okounkov bodies connects algebra and geometry in the broad framework of general algebraic varieties. This relationship is useful in many directions. It suggests the existence of birationally invariant theory of intersection of divisors and provides elementary proofs of Alexandrov-Fenchel inequalities in the theory of intersections and their local versions for the multiplicities of intersections of ideals in local rings. Alexandrov-Fenchel geometric inequalities easily follow from their algebraic analogs. In the theory of invariants, this connection gives analogues of the BKK theorem for horospherical varieties and some other varieties with the action of a reductive group. In abstract algebra, this relationship allows us to introduce a broad class of graded algebras, the Hilbert functions of which are not necessarily polynomials for large argument values, but have polynomial asymptotics. In my presentation, I will introduce these results in a way that is accessible to a general mathematical audience.

This is a combinatorics talk hiding in geometric clothing. The clothing consists of the local motivic monodromy conjecture, an analogue of the Weil conjectures. Given a polynomial f with integer coefficients, it predicts a remarkable relationship between arithmetic properties (number of solutions to f = 0 modulo an integer) and topological properties (eigenvalues of monodromy of the Milnor fibre of the complex hypersurface defined by f). The conjecture remains wide open in general. Surprisingly, it remains open for "generic" choices of f, even though there are well known and relatively simple combinatorial formulas for all the relevant quantities. This brings us to the heart of the talk: for "generic" f, we relate the local motivic monodromy conjecture to a long-standing open question of Stanley on triangulations of simplices. Progress towards the latter question then helps resolve the local motivic monodromy conjecture. This is joint work with Matt Larson and Sam Payne.

Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots in R^3 of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the 4D Poincare conjecture using Rasmussen’s invariant from Khovanov homology. It is yet unclear whether their strategy can work. I will explain several recent results in this direction and some of the challenges that appear. A key problem is to certify when a knot is slice (bounds a disk in four-dimensional half-space), which can be tackled with machine learning. The talk is based on joint work with Sergei Gukov, Jim Halverson, Marco Marengon, Lisa Piccirillo, Fabian Ruehle, Mike Willis, and Sucharit Sarkar.

Topological solitons occur in nonlinear field theories from various areas of mathematical physics. Originally, they served as particle models in certain quantum field theories. Recently, there has been a strong trend towards topological solitons in variational models of condensed matter. The recent experimental discovery of so-called "chiral magnetic skyrmions" has raised hopes to use topology as a concept for future information technologies. In this talk I will present some fundamental mathematical questions, advances and open problems in this exciting field at the interface of analysis, geometry and physics.

The pentagon relation is an interesting relation in mathematics. I will discuss how to get the super OSP (1|2) generalization of this relation and motivate it from different topic in Mathematical Physics such as, AGT correspondence, Teichmueller TQFT, and Integrability.

There are several topological invariants that one may associate to a finitely generated group G -- the characteristic varieties, the resonance varieties, and the Bieri–Neumann–Strebel invariants -- which keep track of various finiteness properties of certain subgroups of G. These invariants are interconnected in ways that makes them both more computable and more informative. I will describe one such connection, made possible by tropical geometry, and I will provide examples and applications pertaining to complex geometry and low-dimensional topology.

Deep learning (the training of deep neural nets) is a very simple idea. Yet it has led to many striking applications throughout science and industry over the last decade. It has also become a major tool for applied mathematicians. In pure mathematics the impact has so-far been modest. I will discuss a few instances where it has proved useful, and led to interesting results in pure mathematics. I will also reflect on my experience as a pure mathematician interacting with deep learning. Finally, I will discuss what can be learned from the successful examples that I understand, and try to guess an answer to the question in the title. (Deep learning also raises interesting mathematical questions, but this talk won't be about this.)

Anomalous diffusion and Levy flights, which are characterized by the occurrence of random discrete jumps of all scales, have been observed in a plethora of natural and engineered systems, ranging from the motion of molecules to climate signals.

Mathematicians have recently unveiled mechanisms to generate anomalous diffusion, both stochastically and deterministically. However, there exists to the best of our knowledge no explicit example of a spatially extended system which exhibits anomalous diffusion without being explicitly driven by Levy noise.

We provide the first explicit example of a stochastic partial differential equation which albeit only driven by normal Gaussian noise supports anomalously diffusive propagating front solutions. This is an entirely emergent phenomenon without explicitly built-in mechanisms for anomalous diffusion. This is joint work with Chunxi Jiao.

Since the 1920's, it has been known that the spatially homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes generically develop curvature singularities in the contracting time direction along spacelike hypersurfaces, known as big bang singularities, both in vacuum and for a wide range of matter models. For many years, it remained unclear if the big bang singularities were physically relevant. It was thought by some that big bang singularities were due to the unphysical assumption of spatial homogeneity and that they would disappear in non-homogenous spacetimes, or in other words, big bang singularities were unstable under nonlinear perturbations as solutions to the Einstein field equations. A partial resolution to this situation came in 1967 when Hawking established his singularity theorem that guarantees a cosmological spacetime will be geodesically incomplete for a large class of matter models and initial data sets, including highly anisotropic ones.

While Hawking's singularity theorem guarantees that cosmological spacetimes are geodesically incomplete (i.e. at least one observer will experience something pathological at a finite time in the past) for a large class of initial data sets, it is silent on the cause of the geodesic incompleteness. It has been widely anticipated that the geodesics incompleteness is due to the formation of curvature singularities, and it is an outstanding problem in mathematical cosmology to rigorously establish the conditions under which this expectation is true and to understand the dynamical behaviour of cosmological solutions near singularities.

In this talk, I will begin by introducing the FLRW and Kasner solutions of the Einstein-scalar field equations, which are exact, spatially homogeneous solutions that play a distinguished role in the analysis of big bang singularities. After briefly providing context for the FLRW and Kasner solutions in the historical development of the field of cosmology, I will define what it means for a FLRW/Kasner big bang singularity to be stable. With this notion in hand, I will then discuss the recent influential FLRW and Kasner big bang stability proofs of Rodnianski-Speck and Fournodavlos-Rodnianski-Speck. One aspect of these stability results that I will pay particular attention to is their global nature. To conclude the talk, I will discuss some recent work done in collaboration with Florian Beyer where we improve the Rodnianski-Speck FLRW big bang stability result by establishing that the FRLW big bang is locally stable, which is a significantly stronger notion of stability with important physical consequences that I will briefly discuss. Time permitting, I will also briefly discuss open questions and future directions for research.