Time: 12:00pm EST

Zoom Meeting ID: 967 6311 6726

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An information-theoretic approach towards the Kneser-Poulsen conjecture

Abstract: The Kneser-Poulsen conjecture in discrete geometry asserts that the volume of a union of balls in Euclidean space must decrease if their centres are brought closer. In this talk, we will propose an information-theoretic approach to tackle this problem. Our approach revolves around a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a 1-Lipschitz map. Some techniques that we have used to settle several cases of this question will be presented along with a selection of open questions we encountered. (Based on joint work with Irfan Alam, Dongbin Li, Sergii Myroshnychenko, and Oscar Zatarain-Vera)

Time: 2:00pm EST

Zoom Meeting ID:  676 808 9500

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Slicing all edges of an n-cube requires n2/3 hyperplanes

Abstract: Consider the n-cube graph in Rn, with vertices {0,1}n and edges connecting vertices with Hamming distance 1. How many hyperplanes are required in order to dissect all edges? This problem has been open since the 70s. We will discuss this and related problems.

Puzzle: Show that n hyperplanes are sufficient, while sqrt(n) are not enough.

Time: 3:30pm EDT

Zoom Meeting ID:  931 5882 2271

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The Subspace Flatness Conjecture and Faster Integer Programming

Abstract: In a seminal paper, Kannan and Lovász (1988) considered a quantity \mu_{KL}(\Lambda,K) which denotes the best volume-based lower bound on the covering radius \mu(\Lambda,K) of a convex body K with respect to a lattice \Lambda. Kannan and Lovász proved that \mu(\Lambda,K) \leq n \cdot \mu_{KL}(\Lambda,K) and the Subspace Flatness Conjecture by Dadush (2012) claims a O(\log n) factor suffices, which would match the lower bound from the work of Kannan and Lovász.

We settle this conjecture up to a constant in the exponent by proving that \mu(\Lambda,K) \leq O(log^3(n)) \cdot \mu_{KL} (\Lambda,K). Our proof is based on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz (2017). Following the work of Dadush (2012, 2019), we obtain a (log n)^{4n}-time randomized algorithm to solve integer programs in n variables. Another implication of our main result is a near-optimal flatness constant of O(n log^3(n)). Joint work with Thomas Rothvoss.

Time: 3:00 pm EST

Zoom Meeting ID:  991 5787 8710

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A generic Lipschitz function is extremely far from being differentiable

Abstract: The classical Rademacher Theorem guarantees that every Lipschitz function between finite-dimensional spaces is differentiable almost everywhere. There are, however, null subsets S of R^n (with n>1) with the property that every Lipschitz function on R^n has points of differentiability in S; one says that S is a universal differentiability set (UDS).

It turns out that some sets T which are not UDS still have the property that a typical Lipschitz function (understood in the sense of Baire category) has points of differentiability in T. We characterise such sets completely in the language of Geometric Measure Theory: these are exactly the sets which cannot be covered by countably many closed 1-purely unrectifiable sets. 

Surprisingly though, no matter how good the set T is, we show that a typical 1-Lipschitz function is non-differentiable at a typical point of T in a very strong sense: the derivative ratio approximates every linear operator of norm at most 1. For "coverable" sets as above we show that "typical point" may be strengthened to "every point". One of the steps in the proof leads to an interesting problem in combinatorial geometry.

This is a joint work with Michael Dymond.

Time: 3:00pm EST

Zoom Meeting ID:  991 7010 4891

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Testing Convex Truncation

Abstract: We study the basic statistical problem of testing whether normally distributed n-dimensional data has been truncated, i.e. altered by only retaining points that lie in some unknown truncation set S \subseteq \mathbb{R}^n. As our main algorithmic results,

1. We give a computationally efficient O(n)-sample algorithm that can distinguish the standard normal distribution N(0,I_n) from N(0,I_n) conditioned on an unknown and arbitrary convex set S. 

2. We give a different computationally efficient O(n)-sample algorithm that can distinguish N(0,I_n) from N(0,I_n) conditioned on an unknown and arbitrary mixture of symmetric convex sets.
   
   

These results stand in sharp contrast with known results for learning or testing convex bodies with respect to the normal distribution or learning convex-truncated normal distributions, where state-of-the-art algorithms require essentially n^{O(\sqrt{n})} samples. An easy argument shows that no finite number of samples suffices to distinguish N(0,I_n) from an unknown and arbitrary mixture of general (not necessarily symmetric) convex sets, so no common generalization of results (1) and (2) above is possible. 

We also prove lower bounds on the sample complexity of distinguishing algorithms (computationally efficient or otherwise) for that match our algorithms up to constant factors. 

 Based on joint work with Anindya De and Rocco A. Servedio that appeared in SODA 2023. 

Time: 3:00pm EST

Zoom Meeting ID:  954 5747 4477

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Two weight inequalities for Calderon-Zygmund operators

Abstract: Weighted inequalities for Calderón-Zygmund operators are central in the study of harmonic analysis. Two weight inequalities for Calderón-Zygmund operators, in particular attracted attention as they can be essential in the perturbation theory of unitary matrices, spectral theory of Jacobi matrices and PDE’s. In this talk, I will discuss several results concerning the two weight inequalities for various Calderon-Zygmund operators in both Euclidean setting and in the more generic setting of spaces of homogeneous type in the sense of Coifman and Weiss.

Time: 3:00pm EST

Zoom Meeting ID:  676 808 9500

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The Stochastic Geometry of Randomized Decision Trees and Forests

Abstract: Random forests are a popular class of algorithms used for regression and classification that are ensembles of randomized decision trees built from axis-aligned partitions of the feature space. One particular variant, called Mondrian forests, were proposed to handle the online setting and are the first class of random forests for which minimax optimal rates were obtained in arbitrary dimension. However, the restriction to axis-aligned splits fails to capture dependencies between features, and random forest algorithms using oblique splits have shown improved empirical performance.

By viewing the partitioning process generating Mondrian forests as a special case of the stable under iteration (STIT) process in stochastic geometry, we utilize the theory of stationary random tessellations to show that a large class of random forests with oblique splits can achieve minimax optimal rates of convergence and illustrate how they can obtain improved rates in high dimensional feature space depending on the choice of split directions. This talk is based on joint work with Ngoc Mai Tran.

Time: 4:00pm EST

Zoom Meeting ID:  449 762 7644

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Some problems related to floating bodies

Abstract: Assume that all sections of an origin-symmetric convex body $K$ in $R^n$, $n\ge 3$, have a symmetry of the cube. Does it follow that $K$ is a Euclidean ball? We will discuss this and other  problems of uniqueness related to symmetries of sections and projections of convex bodies and to floating bodies.

Time: 3:00pm EST

Zoom Meeting ID:  [to appear]

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[Title: to appear]

Abstract: [to appear]

Time: 3:00pm EST

Zoom Meeting ID:  991 5533 5217

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Title: Matrix displacement convexity and intrinsic dimensional functional inequalities

Abstract: The discovery by McCann of displacement convexity had a significant impact on probability, analysis, and geometry. I will introduce a new and stronger notion of displacement convexity which operates on the matrix level. I will then show that a broad class of flows satisfy matrix displacement convexity: heat flow, optimal transport, entropic interpolation, mean-field games, and semiclassical limits of non-linear Schrödinger equations. Consequently, the ambient dimensions of functional inequalities describing the behavior of these flows can be replaced by their intrinsic dimensions, capturing the behavior of the  flows along different directions in space. This leads to intrinsic dimensional functional inequalities which provide a systematic improvement on numerous classical functional inequalities.

Time: 3:00pm EST

Zoom Meeting ID:  954 5747 4477

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On sharp bilinear Fourier Hoermander multiplier with Lipschitz singularity in the local L2 region

Abstract: In this talk, I will discuss the Lp boundedness of bilinear Fourier Hoermander multiplier with singularity a Lipschitz curve in the local L2 range. I will show a modified local paraproduct estimate in order to obtain the sharp regularity for the multiplier. In particular, this result also leads to a generalization of the boundedness of bilinear Hilbert transform.

Time: 3:30pm EST

Zoom Meeting ID:  927 1536 9496

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On star-convex bodies with rotationally invariant sections

Abstract: We will outline the proof that an origin-symmetric star-convex body K with sufficiently smooth boundary and such that every hyperplane section of K passing through the origin is a body of affine revolution, is itself a body of affine revolution. This will give a positive answer to the recent question asked by G. Bor, L. Hernández-Lamoneda, V. Jiménez de Santiago, and L. Montejano-Peimbert [Remark 2.9 in the article "On the isometric conjecture of Banach", Geometry & Topology 25 (2021), no. 5, 2621–2642], though with slightly different prerequisites. Our argument is built mainly upon the tools of differential geometry and linear algebra, but occasionally we will need to use some more involved facts from other fields like algebraic topology or commutative algebra. The talk is based on the article [B. Zawalski, "On star-convex bodies with rotationally invariant sections", Beiträge zur Algebra und Geometrie (2023)].