In these two talks, we will first discuss Cheeger and Naber's quantitative estimates of singular sets on manifolds with lower Ricci curvature and also review some recent developments. As consequences, we will discuss the hessian estimates of harmonic functions on Einstein manifolds.

We introduce notions of finite presentation which serve as analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we then introduce analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules, we show that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan. We pursue applications to abstract harmonic analysis, where, for a locally compact group G, we show that A(G)-nuclearity of the inclusion $C*_r(G) \to C^*_r(G)**$ and $A(G)$-semi-discreteness of $VN(G)$ are both equivalent to amenability of $G$. We also present the equivalence between $A(G)$-injectivity of the crossed product $G\bar{\ltimes}M$, $A(G)$-semi-discreteness of $G\bar{\ltimes} M$, and amenability of W*-dynamical systems $(M,G,\alpha)$ with $M$ injective.

The multisymplectic structure of Lagrangian and Hamiltonian PDEs is a covariant generalization of the field-theoretic symplectic structure and encodes many important physical conservation laws. Multisymplectic integrators are a class of numerical methods which, at the discrete level, preserve the multisymplectic structure of a Lagrangian or Hamiltonian field theory. By preserving this structure at the discrete level, a multisymplectic integrator admits discrete analogs of the conservation laws encoded by multisymplecticity. Such methods have been used, for example, for structure-preserving modeling of nonlinear wave phenomena and for stable discretizations of plasma physics problems. There have also been recent investigations into the application of discrete multisymplectic structures for lattice quantum field theory.
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Traditionally, such multisymplectic integrators have been constructed from the variational perspective in the Lagrangian framework, or from directly discretizing the equations of motion in the Hamiltonian framework and subsequently determining the conditions for the discretization method to be multisymplectic in an ad hoc manner. In this talk, after discussing the necessary background material, I will discuss a systematic framework for constructing multisymplectic Hamiltonian integrators variationally utilizing the notion of a Type II generating functional. This framework only requires a choice of finite-dimensional function space and quadrature, so it is applicable to unstructured meshes, whereas traditional Hamiltonian multisymplectic integrators require rectangular meshes. As an application of this framework, I will derive the class of multisymplectic partitioned Runge--Kutta methods and show that, in this framework, discretizing via a tensor product partitioned Runge--Kutta expansion in spacetime is well-defined if and only if the partitioned Runge--Kutta methods are symplectic in space and time. This is joint work with Prof. Melvin Leok.
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Time permitting, I will discuss future research directions, such as applications to lattice quantum field theory.

Consider a Conway two-sphere S intersecting a knot K in 4 points, and thus decomposing the knot into two 4-ended tangles, T and T’. We will first interpret Khovanov homology Kh(K) as Lagrangian Floer homology of a pair of specifically constructed immersed curves C(T) and C'(T’) on the dividing 4-punctured sphere S. Next, motivated by several tangle-replacement questions in knot theory, we will describe a recently obtained structural result concerning the curve invariant C(T), which severely restricts the types of curves that may appear as tangle invariants. The proof relies on the matrix factorization framework of Khovanov-Rozansky, as well as the homological mirror symmetry statement for the 3-punctured sphere.
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This is joint work with Liam Watson and Claudius Zibrowius.

Point-free measure theory is an approach to measure theory in a more abstract viewpoint. Specifically, it forgets the notion of ``points" and tries to recover the whole measure theory only with the notion of measurable sets. We will briefly see why this viewpoint has a potential to liberate us from the agony that we feel all the time when an uncountable collection pops up in measure theory. We will also talk about how to define ``measurable functions" in the point-free way.

Let $u$ be a harmonic function in $\Omega \subset \mathbb{R}^d.$ It is known that in the interior, the singular set $\mathcal{S}(u) = \{u=|\nabla u|=0 \}$ is $(d-2)$-dimensional, and moreover $\mathcal{S}(u)$ is $(d-2)$-rectifiable and its Minkowski content is bounded (depending on the frequency of $u$). We prove the analogue near the boundary for $C^1$-Dini domains: If the harmonic function $u$ vanishes on an open subset $E$ of the boundary, then near $E$ the singular set $\mathcal{S}(u) \cap \overline{\Omega}$ is $(d-2)$-rectifiable and has bounded Minkowski content. Dini domain is the optimal domain for which $\nabla$ u is continuous towards the boundary, and in particular every $C^{1,\alpha}$ domain is Dini. The main difficulty is the lack of monotonicity formula near the boundary of a Dini domain.
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This is joint work with Carlos Kenig.

We discuss some recent results on model-based methods for online reinforcement learning (RL). The goal of online RL is to adaptively explore an unknown environment and learn to act with provable regret bounds. In particular, we focus on finite-horizon episodic RL where the unknown transition law belongs to a generic family of models. We propose a model based `value-targeted regression' RL algorithm that is based on optimism principle: In each episode, the set of models that are `consistent' with the data collected is constructed. The criterion of consistency is based on the total squared error of that the model incurs on the task of predicting values as determined by the last value estimate along the transitions. The next value function is then chosen by solving the optimistic planning problem with the constructed set of models. We derive a bound on the regret, for arbitrary family of transition models, using the notion of the so-called Eluder dimension proposed by Russo \& Van Roy (2014).

In these two talks, we will first discuss Cheeger and Naber's quantitative estimates of singular sets on manifolds with lower Ricci curvature and also review some recent developments. As consequences, we will discuss the hessian estimates of harmonic functions on Einstein manifolds.

In a seminal paper, Chatterjee and Varadhan derived an LDP for the dense Erd\H{o}s-R\'{e}nyi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is \emph{inhomogeneous} or \emph{constrained}.
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In this talk, we will explore large deviations for dense random graphs, beyond the ``mean-field'' setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erd\H{o}s-R\'{e}nyi random graphs.
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Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.

Toeplitz operators are fundamental and ubiquitous in signal processing and information theory as models for convolutional (filtering) systems. Due to the fact that any practical system can access only signals of finite duration, however, time-limited restrictions of Toeplitz operators are also of interest. In the discrete-time case, time-limited Toeplitz operators are simply Toeplitz matrices. In this talk we survey existing and present new bounds on the eigenvalues (spectra) of time-limited Toeplitz operators, and we discuss applications of these results in various signal processing contexts. As a special case, we discuss time-frequency limiting operators, which alternatingly limit a signal in the time and frequency domains. Slepian functions arise as eigenfunctions of these operators, and we describe applications of Slepian functions in spectral analysis of multiband signals, super-resolution SAR imaging, and blind beamforming in antenna arrays.
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This talk draws from joint work with numerous collaborators including Zhihui Zhu from the University of Denver.

Fix a prime integer $p$. Set $R$ the completed valuation ring
of the maximal unramified extension of $\mathbb{Q}_p$. For $X :=
X_1(N)$ the modular curve with $N$ at least 4 and coprime to $p$,
Buium-Poonen in 2009 showed that the locus of canonical lifts enjoys
finite intersection with preimages of finite rank subgroups of $E(R)$
when $E$ is an elliptic curve with a surjection from $X$. This is done
using Buium's theory of arithmetic ODEs, in particular interesting
homomorphisms $E(R) \to R$ which are arithmetic analogues of Manin maps.
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In this talk, I will review the general idea behind this result and
other applications of arithmetic jet spaces to Diophantine geometry and
discuss a recent analogous result for quasi-canonical lifts, i.e., those
curves with Serre-Tate parameter a root of unity. Here the ODE Manin
maps are insufficient, so we introduce a PDE version of Buium's theory
to provide the needed maps. All of this is joint work with A. Buium.

I will state the question of uniqueness for centers of
blow ups and blow downs of birational maps, explain what is currently
known and give two applications. The first is to the structure of
Cremona groups, namely their nongeneration by involutions in dimension
$>$= 3. The second application is for the Grothendieck ring of
varieties, of dimension $<$= 2, over perfect fields.
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Based on joint work with H.-Y. Lin, and with H.-Y. Lin and S. Zimmermann.