Columbia University

Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of $\mathbb{C}^3$, when the Donaldson-Thomas (DT) invariant of sheaves of zero dimensional support and length d is $p(d)$, the number of plane partitions of $d$. The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies a graded vector space with Euler characteristic equal to the DT invariant. I will talk about two other refinements (categorical and K-theoretic) of DT invariants, focusing on the explicit case of $\mathbb{C}^3$. In particular, we show that the K-theoretic DT invariant for $d$ points on $\mathbb{C}^3$ also equals $p(d)$. This is joint work with Yukinobu Toda.

An open market is a subset of a larger equity market, composed of a certain fixed number of top‐capitalization stocks. Though the number of stocks in the open market is fixed, their composition changes over time, as each company's rank by market capitalization fluctuates. When one is allowed to invest also in a money market, an open market resembles the entire “closed” equity market in the sense that the market viability (lack of arbitrage) is equivalent to the existence of a numéraire portfolio (which cannot be outperformed). When access to the money market is prohibited, the class of portfolios shrinks significantly in open markets; in such a setting, we discuss how to construct functionally generated stock portfolios and the concept of the universal portfolio.

This talk is based on joint work with Ioannis Karatzas.

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The Ricci flow on a surface is an intrinsic evolution of the metric converging to a constant curvature metric within the conformal class. It can be seen as an infinite-dimensional gradient flow. We introduce a natural 'Langevin' version of that flow, thus constructing an SPDE with invariant measure expressed in terms of Liouville Conformal Field Theory.
Joint work with Hao Shen (Wisconsin).

We revisit the variational characterization of conservative diffusion as entropic gradient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin–Smoluchowski type, the Fokker–Planck probability density flow maximizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in the backward direction of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the maximal rate of entropy dissipation along the Fokker–Planck flow and measure exactly the deviation from this maximum that corresponds to any given perturbation. As a bonus of our trajectorial approach we derive the HWI inequality relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I).

We "review" how one can expand a filtration by continuously adding a stochastic process. The new results (obtained with Léo Neufcourt) relate to the seimartingale decompositions after the expansion. We give some possible applications. 

We provide a detailed, probabilistic interpretation, based on stochastic calculus, for the variational characterization of conservative diffusion as entropic gradient flux. Jordan, Kinderlehrer, and Otto showed in 1998 that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric in the ambient space of configurations. Using a very direct perturbation analysis we obtain novel, stochastic-process versions of such features. These are valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time. The original results follow then simply by taking expectations. As a bonus, we obtain the HWI inequality of Otto and Villani relating relative entropy, Fisher information and Wasserstein distance; and from it the celebrated log-Sobolev, Talagrand and Poincare inequalities of functional analysis. (Joint work with W. Schachermayer and B. Tschiderer, from the University of Vienna.)

 

The problem of decomposing a given dataset as a superposition of basic motifs arises in a wide range of application areas, including neural spike sorting and the analysis of astrophysical and microscopy data. Motivated by these problems, we study a "short-and-sparse" deconvolution problem, in which the goal is to recover a short motif a from its convolution with a random spike train $x$. We formulate this problem as optimization over the sphere. We analyze the geometry of this (nonconvex) optimization problem, and argue that when the target spike train is sufficiently sparse, on a region of the sphere, every local minimum is equivalent to the ground truth, up to symmetry (here a signed shift). This characterization obtains, e.g., for generic kernels of length $k$, when the sparsity rate of the spike train is proportional to $k^{-2/3}$ (i.e., roughly $k^{1/3}$ spikes in each length-$k$ window). This geometric characterization implies that efficient methods obtain the ground truth under the same conditions. 

Our analysis highlights the key roles of symmetry and negative curvature in the behavior of efficient methods -- in particular, the role of a "dispersive" structure in promoting efficient convergence to global optimizers without the need to explicitly leverage second-order information. We sketch connections to broader families of benign nonconvex problems in machine learning and signal processing, in which efficient methods obtain global optima independent of initialization. These problems include variants of sparse dictionary learning, tensor decomposition, and phase recovery.

Joint work with Yuqian Zhang, Yenson Lau, Han-Wen Kuo, Dar Gilboa, Sky Cheung, Abhay Pasupathy