Abstract: Increasingly, workplaces are requesting Diversity Statements from future employees as they become more intentional with their efforts to foster diversity, equity, and inclusion. Graduate school applications are also another group requesting diversity statements for admission. This talk will give an overview of what diversity statements are and things to consider when crafting your own.

In this talk we prove that certain Near-field Ptychographic measurements are robustly invertible for specific Point Spread Functions (PSFs) and physical masks which lead to well-conditioned lifted linear systems. We then apply a block phase retrieval algorithm using weighted angular synchronization and prove that the proposed approach accurately recovers the measured sample for these specific PSF and mask pairs. We then numerically demonstrate an approach for solving Blind far-field Ptychography under reasonable assumptions. Finally, we discuss an extension of the Johnson-Lindenstrauss lemma for finite sets and demonstrate accurate compressive nearest neighbor classification in comparison to J-L embeddings. We then generalize this approach to infinite sets/manifolds.
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This will be a hybrid seminar and take place in C329 Wells Hall and via Zoom at https://msu.zoom.us/j/99426648081?pwd=ZEljM3BPUXg2MjVUMVM5TnlzK2NQZz09 .

Congruences between Hecke eigenforms have a long history. Most of these have a simple modular representation theoretic interpretation. Serre developed a systematic approach to predicting the exceptional congruences that arise when the corresponding Galois representation is tamely ramified at p. We will discuss a generalization of this phenomenon in higher rank. This is joint work with Le Hung, Levin, and Morra.

We will start by introducing a real space model of a scalar electromagnetic field coupled to a continuum collection of two level atoms. From this we will obtain a pair of nonlocal partial differential equations describing the energy eigenstates that have at most one photon present in the field. The rest of the talk will discuss spectral results in two different types of atomic distributions.
1. Compactly supported densities: In this setting the atoms are contained in a finite region in space. We will state necessary and sufficient conditions for the existence of eigenstates, as well as an upper bound on the number of such states.
2. Periodic densities: In this setting the atoms exhibit the symmetries of a lattice. We will present a decomposition of the continuous spectrum into spectral bands and state a corresponding structure theorem.
This work is joint with Erik Hiltunen, John Schotland, and Michael Weinstein.

The algebraic K theory of a manifold M is a space whose homotopy groups record interesting invariants of the manifold. Some of these invariants, like the Wall finiteness obstruction and Whitehead torsion, depend only on the fundamental group G of M and can also be defined in terms of the algebraic K theory of the group ring Z[G]. The relationship between the algebraic K theory of M and the algebraic K theory of Z[G] is made precise by a comparison map called the linearization. In this talk we will review these ideas and discuss some recent work on a refinement of the linearization map to the genuine equivariant algebraic K theory of spaces. This is joint work with Maxine Calle and Andres Mejia.

The Stanley symmetric function F_w(x1,x2,…) associated to a permutation w can be obtained via a stable limit of Schubert polynomials. These symmetric functions F_w are known to be Schur-positive for any w. In 2018, the backstable double Schubert polynomial was introduced by Lam, Lee, and Shimozono and used to define the double Stanley symmetric function F_w(X,Y) in two independent sets of commuting variables X and Y. Using geometric techniques, they showed that the expansion of F_w(X,Y) into backstable double Schur functions enjoy an analogous positivity. In this talk, we present a combinatorial proof of this positivity for vexillary permutations having positive support. If time permits, we will discuss progress towards the general vexillary case and to the K-theoretic setting. This is joint work with Zachary Hamaker and Tianyi Yu.

Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions?
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.

Games are widely used to test reinforcement learning paradigms, to study competitive systems in economics and biology, and to model decision tasks. Empirical game theory studies games through observation of populations of interacting agents. We introduce a generic low-dimensional embedding scheme that maps agents into a latent space which enables visualization, interpolation, and strategic feature extraction. The embedding can be used for feature extraction since it represents a generic game as a combination of simpler low dimensional games. Through examples, we illustrate that these components may correspond to basic strategic trade-offs. We then show that the embedding scheme can represent all games with bounded payout, or whose payout has finite variance when two agents are sampled at random. We develop a formal approximation theory for the representation, study the stability of the embedding, provide sufficient sampling guidelines, and suggest regularizers which promote independence in the identified features.

The understanding of invariant foliations is very important in the theory of uniformly and partially hyperbolic dynamics. The main theme of this talk is to study transitive Anosov (or uniformly hyperbolic) systems having a decomposition of the form $E^s + E^c + E^u$, where $E^c$ expands uniformly. There are two foliations that we will consider, the (center) unstable foliation $W^{cs}$ and the strong unstable foliation $W^u$, tangent to $E^c + E^u$ and $E^u$, respectively.
The foliation $W^{cu}$ is very well understood. It is known that the foliation is minimal, i.e. every leaf is dense, and that there is only one ergodic invariant measure "compatible" with that foliation, the so-called SRB measure. However, the strong unstable foliation is not well understood. In this talk, I will survey some recent progress in the direction of understanding topological and ergodic properties of the strong unstable foliation and how this is related to measure rigidity for u-Gibbs measures.

The Birkhoff-von Neumann theorem is a fundamental result in matrix analysis. It describes the structure of the set of n by n bistochastic matrices as the compact convex set whose extreme points are the permutation matrices. If one interprets bistochastic matrices as noisy classical information channels, it becomes very natural to ask whether a version of the Birkhoff-von Neumann theorem holds for bistochastic quantum channels? (That is, for unital completely positive and trace preserving maps between matrix algebras). The ongoing search for a non-commutative version of the Birkhoff-von Neumann theorem has a long history, and this search has led to some fascinating interactions between operator algebras and quantum information theory over the last two decades. I'll give a light survey on some of these ideas, and (time permitting) I'll explain how tools from quantum group theory shed new light on this problem.

In this talk, I first review some rigidity results for smooth dynamical systems and then I will talk about the symmetric rigidity result for continuous dynamics with bounded geometry. I will explain why the symmetric rigidity result is important in the study of the Teichmuller theory for one-dimensional dynamical systems. After that, I will show how to use the geometric method we have developed in studying the Furstenburg conjecture.

I will give a motivated introduction to skein theory on surfaces and 3-manifolds with coefficients in an arbitrary ribbon braided tensor category, following the approach laid out by Kevin Walker (after Turaev, Przytycki, and others). This is a very general formalism for constructing manifold invariants from algebraic or diagrammatic data. Together, these invariants assemble to form a topological quantum field theory (that is to say, they satisfy a natural gluing formula). We recover the familiar skein theories (like the Kauffman bracket) by making particular choices of the coefficient category. No prior familiarity with skein modules, tensor categories, or TQFT is expected or required!

In this essentially topological talk, I will classify all principal actions of a general locally compact quantum group on a topological graph. We will glance at the applications to the operator algebraic setting, but content ourselves to describe particular classes of actions coming from a suitably defined “covering space” of the topological graph, which generalizes covering spaces of ordinary topological spaces.

In 1997, H. Hennion used a non-standard metric to show a kind of multiplicative ergodic theorem for the convergence of an infinite product of positive random matrices. Recently Movassagh and Schenker proved a quantum-channel version of Hennion's ergodic theorem. We will discuss the necessary background to understand the generalization of Hennion's metric to the state space of a tracial von Neumann algebra $(M,\tau)$, and a characterization of contraction mappings in this metric. We will discuss generalizations of the theorems of Movassagh and Schenker to everywhere-defined, positive maps on the noncommutative $L^1$ space. Time permitting, we will sketch the proof of our characterization of contraction mappings and discuss its applications to locally normal states on the spin chain. This is part of work in progress in collaboration with Brent Nelson.

The bigraded (Kauffman bracket) skein module of a 3-manifold is a vector space with two gradings indexed by H_1(M;Z/2). One recovers the usual Kauffman bracket skein (with its usual grading) by setting one of the gradings to zero; it turns out that setting the other grading to zero recovers the skein module associated to the adjoint group PGL_2. More generally, there is a bigraded skein module associated with any simple Lie algebra g which encodes the skein modules associated with all the connected Lie groups with Lie algebra g. Our motivation for studying such objects comes from the (geometric - or rather, topological) Langlands program, which predicts a surprising duality relating the dimensions of different skein modules associated to the same closed 3-manifold M. In the Kauffman bracket case, this prediction boils down to an unexpected symmetry between the two gradings. This is joint work in progress with David Ben-Zvi, David Jordan and Pavel Safronov.

In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. The analysis of these Markov chains takes advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands (LRBs).
This talk will explore a generalization called LRBs of Groups (LRBGs) that delightfully mixes LRBs and group theory. The principal example throughout will be a LRBG defined by S. Hsiao, which links the braid arrangement with the Mantaci-Reutenauer algebra. The talk is based on joint work with Jose Bastidas (LACIM/UQAM) and Sarah Brauner (U Minnesota).

Understanding the behavior of solutions to the compressible Euler equations for large times necessitates a sharp analysis of possible singularities that can form. Our understanding of shock singularities in three space dimensions has enjoyed a dramatic surge in progress in the past two decades due in part to the mathematical techniques that were developed to study Einstein’s equations. In this talk, I will discuss my recent work which provides a sharp localized description of a shock singularity as part of the boundary of maximal development of smooth data. The set of Cartesian spacetime points on which a singularity occurs, which we call the singular boundary $\mathcal{B}$, has the structure of an embedded hypersurface with very degenerate causal properties. I will give an overview of the difficulties that occur in the construction of the singular boundary, and if time permits, also discuss the construction of the Cauchy horizon which emanates from the past boundary of $\mathcal{B}$.

A manifold is called Poincaré-Einstein if it has constant negative Ric curvature and admits a suitable compactification. As the framework for AdS/CFT correspondence, it has been intensively studied over the last thirty years. One guiding principle in this area is to understand the connection between the manifold and its conformal boundary, and in my talk I will show an inequality between Yamabe invariants in this direction and then generalize this to manifolds with a lower Ric curvature bound.

Manifold learning algorithms can recover the underlying low-dimensional parametrization of high-dimensional point clouds. This talk will extend this paradigm in two directions. First, we ask if it is possible, in the case of scientific data where quantitative prior knowledge is abundant, to explain a data manifold by new coordinates, chosen from a set of scientifically meaningful functions. Second, I will show how some popular manifold learning tools and applications can be recreated in the space of vector fields and flows on a manifold. Central to this approach is the order 1-Laplacian of a manifold, $\Delta_1$, whose eigen-decomposition, known as the Helmholtz-Hodge Decomposition, provides a basis for all vector fields on a manifold. We present a consistent estimator for $\Delta_1$, which enables visualization, analysis, smoothing and semi-supervised learning of vector fields on a manifold. In topological data analysis, we describe the 1st-order analogue of spectral clustering, which amounts to prime manifold decomposition. Furthermore, from this decomposition a new algorithm for finding shortest independent loops follows.
Joint work with Yu-Chia Chen, Weicheng Wu, Samson Koelle, Hanyu Zhang and Ioannis Kevrekidis

Abstract: How can we use tools from abstract algebra to quantify the difference between the surface of a basketball and the surface of a donut? The overarching goal of an area of mathematics called algebraic topology is to use algebra to study and distinguish topological spaces. In this talk, we’ll explore one answer to this question by using algebraic objects called homotopy groups to tame the wily, squishy geometry of topological spaces. Then, we’ll flip our guiding question around to see what topology can tell us about tricky problems in algebra. In doing so, we’ll investigate some topological connections to an interesting invariant of rings called algebraic K-theory and I'll talk about my work bridging topology and algebra.

An algebraic variety is called rational if it is birational to projective space. Determining which varieties are rational is a difficult problem which has been at the heart of algebraic geometry since the Italian school. In recent years, a new set of questions borne out of these rationality problems has gained considerable attention: given that a variety is not rational, how can we measure how far it is from being rational? Much of the progress in this budding field of measures of irrationality has been obtained by adapting tools used in the study of rationality problems. However, so far these adaptations have been limited to the simplest tools. I will describe a new cycle-theoretic method to give lower bounds on measures of irrationality which is an adaptation of the decomposition of the diagonal. Just as the study of the decomposition of the diagonal has led to breakthroughs in the field of rationality problems, I hope that modified diagonals can be used to great effect to study the degree of irrationality. As a proof of concept, I will describe how this perspective unifies seemingly unrelated results about the degree of irrationality.

Schrödinger operators with random potentials are very important models in quantum mechanics, in the study of transport properties of electrons in solids. In this talk, we study the approximation of eigenvalues via landscape theory for some random Schrödinger operators. The localization landscape theory, introduced in 2012 by Filoche and Mayboroda, considers the landscape function u solving $Hu=1$ for an operator $H$. Landscape theory has remarkable power in studying the eigenvalue problems for a large class of operators and has led to numerous “landscape baked” results in mathematics. We first give a brief review of the localization landscape theory. Then we focus on some recent progress of the landscape-eigenvalue approximation for operators on general graphs. We show that the maximum of the landscape function is comparable to the reciprocal of the ground state eigenvalue, for Anderson or random hopping models on certain graphs with growth and heat kernel conditions, as well as on some fractal-like graphs such as the Sierpinski gasket graph. There will be precise asymptotic behavior of the ground state energy for some 1D chain models, as well as numerical stimulations for excited states energies. The talk is based on recent joint work with L. Shou and W. Wang.

A divisible convex set O is a convex, bounded, and open subset of an affine chart of the real projective space, on which acts cocompactly a discrete group G of projective transformations. The quotient M=O/G is a closed manifold with a real projective structure. These objects have a rich theory, which involves ideas from dynamical systems, geometric group theory and (G,X)-structures. Moreover, they are an important source of examples of discrete subgroups of Lie groups, and have links with Anosov representations. In dimension 3, there is a strong link between the projective geometry of M and its JSJ and geometric decompositions: M is cut by projectively totally geodesic embedded disjoint tori into pieces admitting a finite-volume hyperbolic structure.
In this talk, we will survey known examples of divisible convex sets, and then describe new examples obtained in collaboration with Gabriele Viaggi, of irreducible, non-symmetric, and non-strictly convex divisible convex sets in arbitrary dimensions (at least 3). Our examples are also new in dimension 3.

Reading cut the hyperplanes in a real central arrangement H into pieces called shards, which reflect order-theoretic properties of the arrangement. We show that shards have a natural interpretation as certain generators of the fundamental group of the complement of the complexification of H. Taking only positive expressions in these generators yields a new poset that we call the pure shard monoid.
When H is simplicial, its poset of regions is a lattice, so it comes equipped with a pop-stack sorting operator Pop. In this case, we use Pop to define an embedding Crackle of Reading's shard intersection order into the pure shard monoid. When H is the reflection arrangement of a finite Coxeter group, we also define a poset embedding Snap of the shard intersection order into the positive braid monoid; in this case, our three maps are related by Snap = Crackle Pop. This is joint work with Colin Defant.

In this talk, I'll describe a deterministic particle method for the weighted porous medium equation. The key idea behind the method is to approximate the PDE via certain highly nonlocal continuity equations. The formulation of the method and the proof of its convergence rely on the Wasserstein gradient flow formulation of the aforementioned PDEs. This is based on joint work with Katy Craig, Karthik Elamvazhuthi, and Matt Haberland.

Beckman and Quarles published the following theorem in 1953. If $n \geq 2$ and if $f$ is a function from Euclidean $n$-space to itself satisfying $||f(x)-f(y)||=1$ whenever $||x-y||=1$, then $||f(x)-f(y)||=||x-y||$ for every pair of points $x$ and $y$.
I'll discuss a conjectured generalization for complete Riemannian manifolds, some supporting theorems, and the apparent role of convexity in the 'preserve one (distance), preserve all (distances)' phenomenon.

Modular forms are holomorphic functions with a wealth of symmetries. Even though these functions are borne out of complex analysis, their Fourier coefficients contain a wealth of arithmetic information. Even bounding the sizes of these coefficients involve very deep mathematics -- the best bounds follow from Deligne's proof of the Weil conjectures, for which he was awarded the Fields medal.
In this talk, rather than looking at complex absolute values, we will instead focus on the p-adic size of p-th Fourier coefficient for a prime number p. We will state a conjecture (the ghost conjecture) which gives an exact description of these sizes for all modular forms. This funnily named conjecture converts difficult automorphic questions into more accessible combinatorial ones. We will discuss the state of this conjecture and its applications to several open questions on slopes of modular forms.

Fix a weight 2 CM modular form f with trivial character and level $l^2$ for some prime $l$. How many eigenforms of the same weight and level are congruent to f modulo a prime p? We will sketch a proof that when $l$ is -1 mod p, this number is always divisible by p. Such an f is an example of a modular form that is “vexing at $l$ mod p”, and the p-divisibility phenomenon is true for all such vexing forms (at least if p is at least 5). These vexing forms are so named due to the difficulties they pose in modularity lifting theorems. The divisibility result is deduced from a structure result on the corresponding Hecke algebra: we show that it is free over the group ring of a cyclic p-group. Our techniques use both modular representation theory as well as geometric/cohomological methods. This is joint work in progress with Robert Pollack and Preston Wake.

Title: High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems

Date: 04/21/2023

Time: 4:00 PM - 5:00 PM

Place: C304 Wells Hall

Contact: Mark A Iwen ()

We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. We first review some examples of gradient flows in generalized optimal transport spaces from the Onsager principle. We then use a one-step time relaxation optimization problem for time-implicit schemes, namely generalized Jordan-Kinderlehrer-Otto schemes. Their minimizing systems satisfy implicit-in-time schemes for initial value gradient flows with first-order time accuracy. We adopt the first-order optimization scheme ALG2 (Augmented Lagrangian method) and high-order finite element methods in spatial discretization to compute the one-step optimization problem. This allows us to derive the implicit-in-time update of initial value gradient flows iteratively. We remark that the iteration in ALG2 has a simple-to-implement point-wise update based on optimal transport and Onsager's activation functions. The proposed method is unconditionally stable for convex cases. Numerical examples are presented to demonstrate the effectiveness of the methods in two-dimensional PDEs, including Wasserstein gradient flows, Fisher--Kolmogorov-Petrovskii-Piskunov equation, and two and four species reversible reaction-diffusion systems.
This is a joint work with Stanley Osher from UCLA and Wuchen Li from U. South Carolina.

This talk will begin with an introduction to higher homotopy groups, focusing on a particularly fundamental invariant, the homotopy groups of spheres. Concrete examples will lead us to define stable homotopy groups and spectra. From there, we will enter the land of G-equivariant homotopy theory, where everything in sight has an action by the group G. We'll see how keeping track of rotation and symmetry information adds rigidity to calculations, making computations more difficult, and in turn, answers more rich in information than the classical non-equivariant case.

Free probabilistic invariants tend to come in two flavors: microstates, which determine regularity through approximation by matrices; and non-microstates, which determine regularity through behavior under a non-commutative differential calculus. In this talk, I will focus on the former, which are named after the notion of microstates in Boltzmann's entropy theory—the inspiration for Voiculescu's free analogues. I will provide an introduction to free entropy, free entropy dimension, and 1-bounded entropy, and discuss why they are useful in the study of operator algebras.

Abstract : Lattice gauge theories in two spatial dimensions provide many examples of topological order. An important class of examples are Kitaev's quantum double models, which are lattice gauge theories with finite gauge group G, and have topological order described by the quantum double of G. A more general class of models may be obtained by gauging a given state of the lattice spin system that has a global G-symmetry. In this setting the quantum double models are obtained by gauging a trivial product state. If we gauge a non-trivial SPT state however, we expect to get a different topological order. G-symmetric SPT states in two spatial dimensions have an index [a] that takes values in the third cohomology group $H^3(G, U(1))$. It is conjectured that the state obtained by gauging such an SPT state has topological order given the a-twisted quantum double of G, i.e. by the Dijkgraaf-Witten theory given by G and the cocycle a. We prove this conjecture in the case $G=Z_2$.
This is joint work with Alvin Moon and Boris Kjær.

Brown-Gitler spectra have many uses in homotopy theory, largely due to the fact that they realize finite pieces of the dual Steenrod algebra, which is a key input for computing stable homotopy. However, Brown-Gitler spectra were not originally constructed with homotopical applications in mind. Instead, Brown and Gitler first constructed these spectra to aid in studying the immersion conjecture for differentiable manifolds. This talk will provide an overview of homotopical and geometric motivations for constructing Brown-Gitler spectra. We will describe four different constructions due to a range of authors, and discuss how these Brown-Gitler spectra are used in homotopy computations. These descriptions will motivate a sketch of current research directions in equivariant and motivic homotopy theory, and culminate in a statement of new results from work in progress joint with Guchuan Li and Elizabeth Tatum.

This expository talk will consist of two parts. The first part will be an introduction to Hessenberg varieties and their relation to many other topics such as Schubert calculus, representation theory, the theory of (quasi)symmetric functions, and combinatorics. There is much interesting work in this area, so this first part will be a "survey" talk, and I will try to give an overall sense of the history and some of the big themes, instead of spending a lot of time on details. In the second portion of the talk, I will recount what is known about Newton-Okounkov bodies of Hessenberg varieties. In fact, not very much is known, so this second part will be shorter, and more speculative. We will close with some open questions which I hope that members of the audience would be interested to answer!

We will start with an introduction to the weak cosmic censorship conjecture and the problem of constructing naked singularities for the Einstein vacuum equations. Then we will explain our discovery of a new type of self-similarity and explain how this allows us to construct the desired naked singularity solutions.

I will discuss the Hardy-Littlewood integral inequality with sharp constant on the Heisenberg goup proved by Frank and Lieb. I will outline a simpler proof which bypasses the sophisticated argument for existence of a minimizer and is based on the study of the 2nd variation of subcritical functionals. This is joint work with Fengbo Hang.

The analysis of tensor data, i.e., arrays with multiple directions, has become an active research topic in the era of big data. Datasets in the form of tensors arise from a wide range of scientific applications. Tensor methods also provide unique perspectives to many high-dimensional problems, where the observations are not necessarily tensors. Problems in high-dimensional tensors generally possess distinct characteristics that pose great challenges to the data science community.
In this talk, we discuss several recent advances in tensor learning and their applications in computational imaging and microbiome. We also illustrate how we develop statistically optimal methods and computationally efficient algorithms that interact with the modern theories of computation, high-dimensional statistics, and non-convex optimization.

THIS IS AN ONLINE SEMINAR
In this talk we will discuss recent progress on singular star flows. A star vector is a smooth vector field $X$ such that for all vector fields $Y$ sufficiently close to $X$, all periodic orbits and singularities of $Y$ are hyperbolic. It has long been conjectured that every (generic) star flow has only finitely many chain recurrence classes. In this talk, we will consider this problem from an ergodic theory point of view and show that $C^1$ open and densely, every singular star flow has only finitely many measures of maximal entropy (MME); furthermore, the support of each MME is a homoclinic class.

Title: Structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations and stochastic Maxwell equations

Date: 04/28/2023

Time: 4:00 PM - 5:00 PM

Place: C304 Wells Hall

Contact: Mark A Iwen ()

Many partial differential equations (PDEs) can be written as a multi-symplectic Hamiltonian system, which obeys the multi-symplectic conservation law. In this presentation, we present and study semi-discrete discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian PDEs, and show that the proposed DG methods can simultaneously preserve the multi-symplectic structure and energy conservation with a general class of numerical fluxes, which includes the well-known central and alternating fluxes. Applications to the wave equation, the Benjamin--Bona--Mahony equation, the Camassa--Holm equation, the Korteweg--de Vries equation and the nonlinear Schrodinger equation are discussed. In the second part, we consider discontinuous Galerkin methods for the stochastic Maxwell equations with additive (or multiplicative) noise. It is shown that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations.