First talk at 4:00 by by Ananth Shankar (MIT http://math.mit.edu/~ananths/)
Exceptional splitting of abelian surfaces over global function fields.
Let A denote a non-constant ordinary abelian surface over a global function field (of characteristic p > 2) with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. Then we prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves. If time permits, I will also talk about applications of our results to the p-adic monodromy of such abelian surfaces. This is joint work with Davesh Maulik and Yunqing Tang.
Second talk at 5:15 Jordan Ellenberg (University of Wisconsin http://www.math.wisc.edu/~ellenber/)
What is the tropical Ceresa class and what should it be?
This is a highly preliminary talk about joint work with Daniel Corey and Wanlin Li. The Ceresa cycle is an algebraic cycle canonically attached to a curve C, which appears in an intriguing variety of contexts; its height can sometimes be interpreted as a special value, the vanishing of its cycle class is related to the Galois action on the nilpotent fundamental group, it vanishes on hyperelliptic curves, etc. In practice it is not easy to compute, and we do not in fact know an explicit non-hyperelliptic curve whose Ceresa class vanishes. We will discuss a definition of the Ceresa class for a tropical curve, explain how to compute it in certain simple cases, and describe progress towards understanding whether it is possible for the Ceresa class of a non-hyperelliptic tropical curve to vanish. (The answer is: "sort of”.) The tropical Ceresa class sits at the interface of algebraic geometry, graph theory (because a tropical curve is more or less a metric graph), and topology: for we can also frame the tropical Ceresa class as an entity governed by the mapping class group, and in particular by the question of when a product of commuting Dehn twists can commute with a hyperelliptic involution in the quotient of the mapping class group by the Johnson kernel.
We study whether all stationary solutions of 2D Euler equation must be radially symmetric, if the vorticity is compactly supported or has some decay at infinity. Our main results are the following:
(1) On the one hand, we are able to show that for any non-negative smooth stationary vorticity that is compactly supported (or has certain decay as |x|->infty), it must be radially symmetric up to a translation.
(2) On the other hand, if we allow vorticity to change sign, then by applying bifurcation arguments to sign-changing radial patches, we are able to show that there exists a compactly-supported, sign-changing smooth stationary vorticity that is non-radial.
We have also obtained some symmetry results for uniformly-rotating solutions for 2D Euler equation, as well as stationary/rotating solutions for the SQG equation. The symmetry results are mainly obtained by calculus of variations and elliptic equation techniques. This is a joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao.
One can regard a (trained) feedforward neural network as a particular type of function , where is a (typically high-dimensional) Euclidean space parameterizing some data set, and the value of the function on a data point is the probability that the answer to a particular yes/no question is "yes." It is a classical result in the subject that a sufficiently complex neural network can approximate any function on a bounded set. Last year, J. Johnson proved that universality results of this kind depend on the architecture of the neural network (the number and dimensions of its hidden layers). His argument was novel in that it provided an explicit topological obstruction to representability of a function by a neural network, subject to certain simple constraints on its architecture. I will tell you just enough about neural networks to understand how Johnson's result follows from some very simple ideas in piecewise linear geometry. Time permitting, I will also describe some joint work in progress with K. Lindsey aimed at developing a general theory of how the architecture of a neural network constrains its topological expressiveness.
Given an m-dimensional manifold M that is homotopy equivalent to an n-dimensional manifold N (where n<m), a spine of M is a piecewise-linear embedding of N into M (not necessarily locally flat) realizing the homotopy equivalence. When m-n=2 and m>4, Cappell and Shaneson showed that if M is simply-connected or if m is odd, then it contains a spine. In contrast, I will show that there exist smooth, compact, simply-connected 4-manifolds which are homotopy equivalent to the 2-sphere but do not contain a spine (joint work with Tye Lidman). I will also discuss some related results about PL concordance of knots in homology spheres (joint with Lidman and Jen Hom).
The talk presents an extension for high dimensions of an idea from a recent result concerning near optimal adaptive finite element methods (AFEM). The usual adaptive strategy for finding conforming partitions in AFEM is ”mark → subdivide → complete”. In this strategy any element can be marked for subdivision but since the resulting partition often contains hanging nodes, additional elements have to be subdivided in the completion step to get a conforming partition. This process is very well understood for triangulations received via newest vertex bisection procedure. In particular, it is proven that the number of elements in the final partition is limited by constant times the number of marked cells. This motivated us [B., Fierro, Veeser, in preparation] to design a marking procedure that is limited only to cells of the partition whose subdivision will result in a conforming partition and therefore no completion step is necessary. We also proved that this procedure is near best in terms of both error of approximation and complexity. This result is formulated in terms of tree approximations and opens the possibility to design similar algorithms in high dimensions using sparse occupancy trees introduced in [B., Dahmen, Lamby, 2011]. The talk describes the framework of approximating high dimensional data using conforming sparse occupancy trees.
This talk will be about polynomial decompositions that are relevant in machine learning. I will start with the well-known low-rank symmetric tensor decomposition, and present a simple new algorithm with local convergence guarantees, which seems to handily outperform the state-of-the-art in experiments. Next I will consider a particular generalization of symmetric tensor decomposition, and apply this to estimate subspace arrangements from very many, very noisy samples (a regime in which current subspace clustering algorithms break down). Finally I will switch gears and discuss representability of polynomials by deep neural networks with polynomial activations. The various polynomial decompositions in this talk motivate questions in commutative algebra, computational algebraic geometry and optimization. The first part of this talk is joint with Emmanuel Abbe, Tamir Bendory, Joao Pereira and Amit Singer, while the latter part is joint with Matthew Trager.
I will talk about a conjecture that in Gibbs states of one-dimensional spin chains with short-ranged gapped Hamiltonians the quantum conditional mutual information (QCMI) between the parts of the chain decays exponentially with the length of separation between said parts. The smallness of QCMI enables efficient representation of these states as tensor networks, which allows their efficient construction and fast computation of global quantities, such as entropy. I will present the known partial results on the way of proving of the conjecture and discuss the probable approaches to the proof and the obstacles that are encountered.
We formulate a geometric nonlinear theory of the mechanics of accretion. In this theory the material manifold of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material isadded to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body. Balance laws are discussed and the initial-boundary value problem of accretion is formulated. The particular cases where the growth surface is either fixed or traction-free are studied and some analytical results are provided. We numerically solve several accretion problems and calculate the residual stresses in nonlinear elastic bodies induced from accretion.