in this talk, i will discuss some models and algorithms from
two different fields: (1) machine learning, including logistic
regression and deep learning, and (2) numerical pdes, including
multigrid methods. i will explore mathematical relationships between
these models and algorithms and demonstrate how such relationships can
be used to understand, study and improve the model structures,
mathematical properties and relevant training algorithms for deep
neural networks. in particular, i will demonstrate how a new
convolutional neural network known as mgnet, can be derived by making
very minor modifications of a classic geometric multigrid method for
the poisson equation and then explore the theoretical and practical
potentials of mgnet.

the talk presents a quick review on landau damping for vlasov-poisson system near penrose stable data, followed by a joint work with d. han-kwan and f. rousset, where the damping, with screening potential, is proved for data with (essentially) $c^{1}$ regularity on the whole space.

a surface has geometric free-boundary in a barrier hypersurface if its boundary meets the barrier orthogonally, like a bubble on a bathtub. we extend brakke's weak notion of mean curvature flow to have a free-boundary condition, and using toy examples we show why this extension is necessary. contrary to the classical flow, for which the barrier is ``invisible,'' the weak flow allows for the surfaces to ``pop'' upon tangential contact with the barrier. when the initial surface is mean-convex, we generalize white's partial regularity and structure theory to the free-boundary setting. this is in part joint work with robert haslhofer, mohammad ivaki, and jonathan zhu.

khovanov homology, as defined by mikhail khovanov based on the representation theory of quantum groups, is a powerful invariant for knots. it categorifies the famous jones polynomial, meaning that jones polynomial can be recovered as its euler characteristic. despite its simple combinatorial definition, khovanov homology has deep relation with quantum topology, gauge theory and representation theory. in this talk, i will recall the definition of khovanov homology and introduce some of its important properties and applications (including rassmussen's combinatorial proof of the milnor conjecture). no previous knowledge on knot theory will be assumed.

low rank approximation has been extensively studied in the past. it is most suitable to reproduce rectangular like structures in the data. in this talk i introduce a generalization using ``shifted'' rank-1 matrices to approximate $a\in\mathbb{c}^{m\times n}$. these matrices are of the form $s_{\lambda}(uv^*)$ where $u\in\mathbb{c}^m$, $v\in\mathbb{c}^n$ and $\lambda\in\mathbb{z}^n$. the operator $s_{\lambda}$ circularly shifts the $k$-th column of $uv^*$ by $\lambda_k$.

these kind of shifts naturally appear in applications, where an object $u$ is observed in $n$ measurements at different positions indicated by the shift $\lambda$. the vector $v$ gives the observation intensity. this model holds e.g., for seismic waves that are recorded at $n$ sensors at different times $\lambda$.

the main difficulty of the above stated problem lies in finding a suitable shift vector $\lambda$. once the shift is known, a simple singular value decomposition can be applied to reconstruct $u$ and $v$. we propose a greedy method to reconstruct $\lambda$ and validate our approach in numerical examples.

reference:

f. boss mann, j. ma, enhanced image approximation using shifted rank-1 reconstruction, inverse problems and imaging, accepted 2019, https://arxiv.org/abs/1810.01681

the kpz equation, a model for the random growth of rough interfaces, has been the subject of great physical and mathematical interest since its introduction in 1986. by simple changes of variables, it is closely related to the stochastic heat equation, which models the partition function of a random walk in a random environment, and the stochastic burgers equation, a simple model for turbulence. i will explain several recent results about the existence, classification, and properties of spacetime-stationary solutions to these equations on ${\bf r}^d$ for various values of $d$. these solutions thus represent the behavior of the models in large domains on long time scales. most of the results are joint work with various combinations of c. graham, y. gu, l. ryzhik, and o. zeitouni.

symmetric functions are highly ubiquitous in algebraic combinatorics, with connections to the representation theory of $gl_n$ and $s_n$, the geometry of grassmannians, and more. there is a classical way to 'compose' symmetric functions called plethysm which has a nice representation-theoretic interpretation. we will describe a related operation called chern plethysm which has inputs given by a symmetric function f and a vector bundle e and outputs a symmetric polynomial. chern plethysm provides numerous schur-positivity results, indicating a representation-theoretic connection, but finding this connection remains an open problem. joint with sara billey and vasu tewari.

the moduli space m of complex curves of fixed topology is an orbifold classifying space for surface bundles. as such the cohomology rings of m and its various orbifold covers give characteristic classes for surface bundles. i will discuss the steinberg module which is central to the duality present in these cohomology rings. i will then explain current joint work with t. brendle and a. putman on surfaces with marked points which expands on results of n. fullarton and a. putman for surfaces without marked points. we show that certain finite-sheeted orbifold covers m[l] of m have large nontrivial q-cohomology in their cohomological dimension.

given a graph $g$, one can compute the eigenvalues of its adjacency matrix $a_g$. remarkably, these eigenvalues can tell us quite a bit about $g$. more generally, spectral graph theory consists of taking a graph $g$, associating to it a matrix $m_g$, and then using algebraic properties of $m_g$ to recover combinatorial information about $g$. this talk is the first in a series of introductory talks to the subject of spectral graph theory. in particular, we'll be discussing the adjacency matrix and the information encoded by its eigenvalues.

first stated in 1973, the cycle double cover conjecture states that every bridgeless graph has a collection of cycles which cover every edge exactly twice. the problem remains wide open to this day despite much work. this talk is to present my partial results on the problem, explain why the tutte polynomial is awesome, and to draw some of the beautiful proofs by picture that arose in my work.

nonlocal models are experiencing a firm upswing recently as more realistic alternatives to the conventional local models for studying various phenomena from physics and biology to materials and social sciences. in this talk, i will describe our recent effort in taming the computational challenges for nonlocal models. i will first highlight a family of numerical schemes - the asymptotically compatible schemes - for nonlocal models that are robust with the modeling parameter approaching an asymptotic limit. second, i will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. this also motivates the development of new mathematical results - for instance, a new trace theorem that extends the classical results.

although new nonlocal models have been gaining popularity in various applications, they often appear as phenomenological models, such as the peridynamics model in fracture mechanics. here i will illustrate how to characterize the origin of nonlocality through homogenization of wave propagation in periodic media.

rational varieties are some of the simplest examples of varieties, e.g. most of their points can be parametrized by affine space. it is natural to ask (1) how can we determine when a variety is rational? and (2) if a variety is not rational, can we measure how far it is from being rational? a famous particular case of this problem is when the variety is a smooth hypersurface in projective space. this problem has attracted a great deal of attention both classically and recently. the interesting case is when the degree of the hypersurface is at most the dimension of the projective space (the ``fano''range), these hypersurfaces share many of the properties of projective space. in this talk, we present recent work with nathan chen which says that smooth fano hypersurfaces of large dimension can have arbitrarily large degrees of irrationality, i.e. they can be arbitrarily far from being rational.