A solution to a geometric flow is called ancient if it has a backhistory going back infinitely far in time. Ancient solutions of parabolic PDE are analogous to entire solutions of elliptic PDE. In particular, they play a fundamental role in understanding singularity formation.
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Perelman studied ancient solutions to the Ricci flow in dimension 3 which are kappa-noncollapsed, and proved a crucial structure theorem for these ancient kappa-solutions. Moreover, Perelman conjectured that, up to scaling, every noncompact ancient kappa-solution in dimension 3 is isometric to either the Bryant soliton or the standard cylinder (or a quotient thereof). In these lectures, I will discuss the proof of this conjecture.

For every infinite set X we define S(X) as the group of all permutations of X. On its subgroup consisting of all finitely supported permutations there exists a natural group homomorphism signature. However, thanks to an observation of Vitali in 1915, we know that this group homomorphism does not extend to S(X). In the talk we extend the signature on the subgroup of S(X) consisting of all piecewise isometric elements (strongly related to the Interval Exchange Transformation group). This allows us to list all of its normal subgroups and gives also informations about an element of the second cohomology group of some groups.

In recent years, topological and geometric data analysis (TGDA) has emerged as a new and promising field for processing, analyzing and understanding complex data. Indeed, geometry and topology form natural platforms for data analysis, with geometry describing the ``shape'' behind data; and topology characterizing / summarizing both the domain where data are sampled from, as well as functions and maps associated to them.
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In this talk, I will show how topological (and geometric ideas) can be used to analyze graph data, which occurs ubiquitously across science and engineering. Graphs could be geometric in nature, such as road networks in GIS, or relational and abstract. I will particularly focus on the reconstruction of hidden geometric graphs from noisy data, as well as graph matching and classification. I will discuss the motivating applications, algorithm development, and theoretical guarantees for these methods. Through these topics, I aim to illustrate the important role that topological and geometric ideas can play in data analysis.

Modeling high dimensional data by latent tree graphical models is a common approach in multiple
machine learning applications. In these models, the key task is to infer the structure of the tree, given
only observations on its leaves. A canonical example of this setting is the tree of life, where the
evolutionary history of a set of organisms is inferred by their nucleotide or protein sequences.
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In this talk, we will show that the tree structure is strongly related to the spectral properties of a fully
connected graph, defined over the terminal nodes of the tree. This relation forms the theoretical basis
of two new methods to recover latent tree models. Comparing our approach to several competing
methods, we show that in many settings, spectral methods have stronger theoretical guarantees and
work better in practice.

Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic surfaces with a common dual has reinvigorated the study of concordance invariants defined by Freedman and Quinn in the 90's, along with homotopy theoretic invariants of Dax from the 70's obstructing isotopy of disks. Using the Dax invariant, we will give conditions under which pairs of homotopic properly embedded disks in a smooth 4-manifold with boundary with a common dual are isotopic.

I will discuss topics in Natural Language Processing

It has been proved by R. E. Greene and H. Wu that a simply-connected complete Kahler manifold negatively pinched sectional curvature possesses a complete Bergman metric. I will briefly review the history and present the estimates of the Bergman metric using the bounded geometry. This talk is based on the joint work with Yau.

A solution to a geometric flow is called ancient if it has a backhistory going back infinitely far in time. Ancient solutions of parabolic PDE are analogous to entire solutions of elliptic PDE. In particular, they play a fundamental role in understanding singularity formation.
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Perelman studied ancient solutions to the Ricci flow in dimension 3 which are kappa-noncollapsed, and proved a crucial structure theorem for these ancient kappa-solutions. Moreover, Perelman conjectured that, up to scaling, every noncompact ancient kappa-solution in dimension 3 is isometric to either the Bryant soliton or the standard cylinder (or a quotient thereof). In these lectures, I will discuss the proof of this conjecture.

A central problem of machine learning is the following. Given data of the form $\{(y_i, f(y_i)+\epsilon_i)_{i=1}^M\}$, where $y_i$'s are drawn randomly from an unknown (marginal) distribution $\mu^*$ and $\epsilon_i$ are random noise variables from another unknown distribution, find an approximation to the unknown function $f$, and estimate the error in terms of $M$.
The approximation is accomplished typically by neural/rbf/kernel networks, where the number of nonlinear units is determined on the basis of an estimate on the degree of approximation, but the actual approximation is computed using an optimization algorithm.
Although this paradigm is obviously extremely successful, we point out a number of perceived theoretical shortcomings of this paradigm, the perception reinforced by some recent observations about deep learning.
We describe our efforts to overcome these shortcomings and develop a more direct and elegant approach based on the principles of approximation theory and harmonic analysis.\\
%We demonstrate a duality between certain problems of function approximation and probability estimation in machine learning and problems of super-resolution in signal separation. In particular, we will explain how the same tools from harmonic analysis can be used for both purposes, leading to a unified theory. We will demonstrate our ideas with some numerical examples.

Let $X/\mathbb{F}_q$ be a smooth affine curve over a finite
field of characteristic $p > 2$. In this talk we discuss the $p$-adic
variation of zeta functions $Z(X_n,s)$ in a pro-covering
$X_\infty:\cdots \to X_1 \to X_0 = X$ with total Galois group
$\mathbb{Z}_p$. For certain ``monodromy stable'' coverings over an
ordinary curve $X$, we prove that the $q$-adic Newton slopes of
$Z(X_n,s)/Z(X,s)$ approach a uniform distribution in the interval
$[0,1]$, confirming a conjecture of Daqing Wan. We also prove a
``Riemann hypothesis'' for a family of Galois representations associated
to $X_\infty/X$, analogous to the Riemann hypothesis for
equicharacteristic $L$-series as posed by David Goss. This is joint
work with Joe Kramer-Miller.

For most microbial communities, the environment and the microbial community structure and function are intimately connected. Most environments outside of the lab are physically and chemically heterogeneous, shaping and complicating the metabolisms of their resident microbial communities: spatial variations introduce physics such as diffusive and advective transport of nutrients and byproducts for example. Conversely, microbial metabolic activity can strongly affect the environment in which the community must function. Hence it is important to link metabolism at the cellular level to physics and chemistry at the community level.
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To introduce metabolism to community-scale population dynamics, many modeling methods rely on large numbers of reaction kinetics parameters that are unmeasured, also making detailed metabolic information mostly unusable. The bioengineering community has addressed this difficulty by moving to kinetics-free formulations at the cellular level, termed flux balance analysis. To combine and connect the two scales, we propose to replace classical kinetics functions in community scale models with cell-level metabolic models, and predict metabolism and how it is influenced by and influences the environment. Further, our methodology permits assimilation of many types of measurement data. We will discuss the background and motivation, model development, and some numerical simulation results.

A foliation on an algebraic variety is a partition of the variety into ``parallel'' disjoint immersed complex submanifolds.
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This turns out to be a very useful notion and holomorphic foliations
have played a central role in several recent developments in the study
of the geometry of projective varieties.
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This is the first part of a two talks series (with Roberto Svaldi)
in which we will explain some recent work building towards the
birational classification of holomorphic foliations on projective
varieties in the spirit of the Minimal Model program. We will explain some applications of these ideas
to the study of the dynamics and geometry of foliations and foliation
singularities.
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Features joint work with P. Cascini and R. Svaldi

Random matrices are an incredibly useful tool for modelling noise and understanding the behavior of large, chaotic systems. They can help us understand the ``average'' complexity of a linear algebra algorithm, model ``on the cheap'' the behavior of large data sets, serve as benchmarks for clustering algorithms, and so much more. And they can do all of this for a very good reason: random matrices are, well, not so random after all. Their spectral asymptotics are highly structured, and highly concentrated, and that is the source of their usefulness.
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This talk will be accessible to an audience familiar with basic linear algebra and probability.