The generalization error of a classifier is related to the complexity of the set of functions among which the classifier is chosen. We study a family of low-complexity classifiers consisting of thresholding a random one-dimensional feature. The feature is obtained by projecting the data on a random line after embedding it into a higher-dimensional space parametrized by monomials of order up to k. More specifically, the extended data is projected n-times and the best classifier among those n, based on its performance on training data, is chosen. We show that this type of classifier is extremely flexible, as it is likely to approximate, to an arbitrary precision, any continuous function on a compact set as well as any Boolean function on a compact set that splits the support into measurable subsets. In particular, given full knowledge of the class conditional densities, the error of these low-complexity classifiers would converge to the optimal (Bayes) error as k and n go to infinity. On the other hand, if only a training dataset is given, we show that the classifiers will perfectly classify all the training points as k and n go to infinity. We also bound the generalization error of our random classifiers. In general, our bounds are better than those for any classifier with VC dimension greater than O (ln n) . In particular, our bounds imply that, unless the number of projections n is extremely large, there is a significant advantageous gap between the generalization error of the random projection approach and that of a linear classifier in the extended space. Asymptotically, as the number of samples approaches infinity, the gap persists for any such n. Thus, there is a potentially large gain in generalization properties by selecting parameters at random, rather than optimization.
$\\$
A preprint of this work can be found here: https://arxiv.org/abs/2108.06339
$\\$
This will be a hybrid seminar and take place in C117 Wells Hall and via Zoom at https://msu.zoom.us/j/99426648081?pwd=ZEljM3BPUXg2MjVUMVM5TnlzK2NQZz09 .

I will talk about Ogg's conjecture and its generalization. This is joint work with Ken Ribet (and should serve as an introduction to Ken's talk on Friday).

A discrete parameter quantum process is represented by a sequence of quantum operations, which are completely positive maps that are trace non-increasing. Given a stationary and ergodic sequences of such maps, an ergodic theorem describing convergence to equilibrium for a general class of such processes was recently obtained by Movassagh and Schenker. Under irreducibility conditions we obtain a law of large numbers that describes the asymptotic behavior of the processes involving the Lyapunov exponent. Furthermore, a central limit type theorem is obtained under mixing conditions. In the continuous time parameter, a quantum process is represented by a double-indexed family of positive map valued random variables. For a stationary and ergodic family of such maps, we extend the results by Movassagh and Schenker to the continuous case.

Nakajima's graded quiver varieties are complex algebraic varieties associated with quivers. They are introduced by Nakajima in the study of representations of universal enveloping algebras of Kac-Moody Lie algebras, and can be used to study cluster algebras. In the talk, I will explain how to precisely locate the supports of the triangular basis of skew-symmetric rank 2 quantum cluster algebras by applying the decomposition theorem to various morphisms related to quiver varieties, thus prove a conjecture proposed by Lee-Li-Rupel-Zelevinsky in 2014.

The contact invariant, defined by Kronheimer and Mrowka, is
an element in the monopole Floer homology of a 3-manifold canonically
attached to a contact structure. I will discuss how the contact
invariant places constraints on the topology of families of contact
structures, and how it can be used to detect non-trivial
contactomorphisms given by "Dehn twists" on spheres. The main new tool
is a generalisation of the contact invariant to an invariant of
families of contact structures.

The KPZ universality class contains one-dimensional random growth models, which under quite general assumptions exhibit similar (non-Gaussian) scaling behavior. For special initial states, the limiting distributions surprisingly coincide with those from the random matrix theory. The physical explanation is that in the space of Markov processes, these models are all being rescaled to a universal fixed point. This scaling invariant fixed point was first characterized in joint work with Jeremy Quastel and Daniel Remenik. In our work, we found a surprising relation between random growing interfaces and the solutions of the classical integrable systems.

We present results on the stability of equilibria (time-independent solutions) of the Vlasov-Maxwell equation. In particular, linear stability criteria for certain classes of equilibria are discussed. We also give a result on the nonlinear stability of an initial-boundary value problem for the Vlasov-Poisson equation.
**Note: speaker will present Virtually. Participants can join in person to view the presentation in C304, or through the Zoom link.**

If a and b are integers that satisfy a simple nonvanishing
condition, the cubic equation y^2 = x^3 + ax + b defines an elliptic
curve over the field of rational numbers. Elliptic curves have been
studied for millennia and seem to occur all over the place in
mathematics, physics and other sciences. In my talk, I'll explain how a
specific elliptic curve provides the solution to a surprisingly hard "brain
teaser" that had a big run on social media a few years ago.

Over 40 years ago, I proved the finiteness of the group of cyclotomic torsion points on an abelian variety over a number field. (A torsion point is cyclotomic if its coordinates lie in the field obtained by
adjoining all roots of unity to the base field.) If the abelian variety is one that we know well, and if the number field is the field of rational numbers, we can hope to determine explicitly the group of
its cyclotomic torsion points. I will illustrate this theme in the situation studied by Barry Mazur in his landmark "Eisenstein ideal" article, i.e., that where the abelian variety is the Jacobian of the
modular curve $X_0(p)$.