Curvature pinching theorems restrict the topology of smooth manifolds satisfying suitable curvature assumptions. In some situations the Ricci flow can transform initial integral curvature bounds into later pointwise bounds and thereby extend pointwise to integral pinching results. I will first review $L^p$ integral pinching theorems of Gursky, Hebey--Vaugon, Dai--Wei--Ye, and others, which all rely on supercritical powers p greater than n/2 or on Chern--Gauss--Bonnet in dimension four. Then I will discuss how stronger control of the Sobolev inequality obtained using Perelman's mu-functional can be used to address the critical case $p=n/2$, leading both to a generalization of previous results as well as to a separate pinching result in the asymptotically flat setting. Some of the work presented is joint with Guofang Wei and Rugang Ye.

The variational principle of Lagrangian PDE is encoded geometrically in a multisymplectic structure. This multisymplectic structure gives rise to a covariant formulation of Noether's theorem and conservation of multisymplecticity, a spacetime generalization of the symplecticity of Lagrangian mechanics. Multisymplecticity has many important physical implications, such as reciprocity in electromagnetism and conservation of wave action, a key ingredient in the stability analysis of wave propagation problems. Furthermore, upon introducing a foliation of spacetime, multisymplecticity can be reformulated as symplecticity on an infinite-dimensional phase space. Consequently, understanding how these multisymplectic structures are affected under discretization is an important aspect to the numerical integration of this class of PDE.
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In this talk, after discussing the multisymplectic formulation of Lagrangian PDE, we discuss how the multisymplectic structure is affected under discretization of the variational principle via the finite element method. We show how choices of discretization which preserve functional and geometric relationships of the underlying function spaces naturally give rise to discrete analogs of the continuum multisymplectic structures. In particular, we discuss how cochain projections and group-equivariant projections from the continuum function spaces to the finite element function spaces induce discrete analogs of the variational, multisymplectic, and Noether-theoretic structures.
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This is joint work with Melvin Leok.

In the 70s, Fred Cohen and Peter May gave a description of the mod $p$ homology of a free $E_n$-algebra in terms of certain homology operations, known as Dyer--Lashof operations, and the Browder bracket. These operations capture the failure of the $E_n$ multiplication to be strictly commutative, and they prove useful for computations. After reviewing the main ideas from May and Cohen's work, I will discuss a framework to generalize these operations to homology with certain twisted coefficient systems and give a complete classification of twisted operations for $E_{\infty}$-algebras. I will also explain computational results that show the existence of new operations for $E_2$-algebras.

Of all the polynomials over $\mathbb{Q}_p$, those of degree divisible by $p$ are most complicated. Similarly, the $p$-local subgroup structure of $\operatorname{GL}_n(\mathbb{C})$ is more complicated if $p$ divides $n$. The talk will be an example based introduction to these ideas, which fall under the ``wild aspect'' of the local Langlands correspondence. Along the way we will define $\mathbb{Q}_p$, some reductive groups of exceptional type, and exhibit wild $p$-structures in both of these examples.

We give a brief introduction to moduli of shtukas, with an eye towards its application in the Langlands correspondence for global function fields.

Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form C x R, then this tangent cone is unique for a large class of minimal cones C. One of the hypotheses in this result is that C x R is integrable and this excludes the case when C is the Simons cone over $S^3 x S^3$. The main result in this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in this non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the setting of tangent cones with non-isolated singularities.

In this talk, I will talk about my construction of relative moduli
spaces of stable sheaves over the stack of quasipolarized surfaces.
For this, I first retrace some of the classical results in the theory
of moduli spaces of sheaves on surfaces to make them work over the
nonample locus. Then I will recall the theory of good moduli spaces,
whose study was initiated by Alper and concerns an intrinsic (stacky)
reformulation of the notion of good quotients from GIT. Finally, I use
a criterion by Alper-Heinloth-Halpern-Leistner to prove existence of
the good moduli space. The application of the construction that I have
in mind is extending the Strange Duality results to degree two K3
surfaces - this part is still work in progress.

In this talk, we discuss a few simple and optimal methods for solving stochastic variational inequalities (VI). A prominent application of our algorithmic developments is the stochastic policy evaluation problem in reinforcement learning. Prior investigations in the literature focused on temporal difference (TD) learning by employing nonsmooth finite time analysis motivated by stochastic subgradient descent leading to certain limitations. These encompass the requirement of analyzing a modified TD algorithm that involves projection to an a-priori defined Euclidean ball, achieving a non-optimal convergence rate and no clear way of deriving the beneficial effects of parallel implementation. Our approach remedies these shortcomings in the broader context of stochastic VIs and in particular when it comes to stochastic policy evaluation. We developed a variety of simple TD learning type algorithms motivated by its original version that maintain its simplicity, while offering distinct advantages from a non-asymptotic analysis point of view. We first provide an improved analysis of the standard TD algorithm that can benefit from parallel implementation. Then we present versions of a conditional TD algorithm (CTD), that involves periodic updates of the stochastic iterates, which reduce the bias and therefore exhibit improved iteration complexity. This brings us to the fast TD (FTD) algorithm which combines elements of CTD and our newly developed stochastic operator extrapolation method. For a novel index resetting stepsize policy FTD exhibits the best known convergence rate. We also devised a robust version of the algorithm that is particularly suitable for discounting factors close to 1.

The theory of Drinfeld modules is remarkably similar to the
theory of abelian varieties, but their local monodromy behaves
differently and is poorly understood. In this talk I will present a
research program which aims to fully describe this monodromy. The
cornerstone of this program is a ``z-adic'' variant of Grothendieck's
l-adic monodromy theorem.
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The talk is aimed at a general audience of number theorists and
arithmetic geometers. No special knowledge of monodromy theory or
Drinfeld modules is assumed.

In an SRPT queue, the job with the shortest remaining processing time is served first, with preemption. The SRPT scheduling rule is of interest due to its optimality properties; it minimizes queue length (number of jobs in system). However, even with Markovian distributional assumptions on the processing times, an exact analysis is not possible. Hence, approximations in the form of a fluid (functional law of large numbers) limit or a diffusion (functional central limit theorem) limit can provide insights into system performance. It was shown by Gromoll, Kruk and Puha (2011) that, if the processing time distribution has unbounded support, then, under standard heavy traffic conditions, the diffusion limit of the queue length process is identically equal to zero. This exhibits the queue length minimization property of SRPT in sharpest relief. It also demonstrates that the SRPT queue length process is orders of magnitude smaller than the workload process in the diffusion limit. (The workload process tracks the time it will take the server to process the work associated with each job in system.) In this talk, we report on progress in characterizing this order of magnitude difference. We find that distribution dependent scaling must be used to obtain a nontrivial limit for the queue length and the associated measure valued state descriptor. The scaling captures the order of magnitude difference, and the nature of the limit is dependent on the tail decay of the processing time distribution.
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This work is joint with Sayan Banerjee (UNC) and Amarjit Budhiraja (UNC).

The spreading behavior of new or invasive species is a central topic in ecology. The modeling of free boundary problems is widely studied to better understand the nature of spreading behaviors of new species. From mathematical modeling point of view, it is a challenge to perform numerical simulations of the free boundary problems, due to the moving boundaries, the topological changes, and the stiffness of the system.
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Our work is concerned with numerical simulations of the long-term dynamical behavior of invasive species modeled by reaction-diffusion equations with free boundaries. We develop a front-tracking method to track the locations of the moving boundary explicitly in one dimension and higher dimensions with spherical symmetry. In two dimensional cases, we employ the level set method to handle topological bifurcations. For single invasive species, we numerically analyze the spreading-vanishing dichotomy in the diffusive logistic model. Various numerical experiments are presented in the two-dimensional spaces to show that the population range tend to be more and more spherical as time increases no matter what geometrical shape the initial population range has if the invasive species spreads successfully. For two invasive species in a weak-strong competition case, we examine how the long-time dynamics of the model changes as the initial functions are varies. Specifically, we simulate the ``chase-and-run coexistence'' phenomenon by choosing the initial function properly. The spreading behavior under time-periodic perturbation of the environment is also considered in our work.

Most of us learn about determinants in an algebra course, but they have important geometrical meanings. I'll explain a few instances of these in this talk.