(Continued) Even if we know many things about hyperbolic manifolds, there are many open extremal problems on them. To name a few: - How does the maximal systole among closed hyperbolic n-manifolds of volume at most V grow as a function of V? - How does the minimal diameter among closed hyperbolic n-manifolds of volume at least V grow as a function of V? - Are there closed hyperbolic n-manifolds of arbitrarily large volume whose spectral gap is larger than that of hyperbolic n-space? Even for surfaces (i.e n=2), many of these extremal problems are open. In this case, answers to these questions also provide insight into the shape of deformation spaces of hyperbolic surfaces. In these lectures, I will discuss some of these problems. I will talk about what is known about them and how random constructions of hyperbolic manifolds sometimes provide answers.

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Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université mini-course on August 19, 21, 24, 25 (W F M Tu) Online Class only

Arthur packets are certain generalizations of L-packets. Arthur and several others constructed Arthur packets for classical groups. Following ideas of the work of Adams-Barbasch-Vogan on Archimedean groups, Cunningham et al. proposed a purely local way to construct Arthur packets for any algebraic reductive group over p-adic fields. In this talk, I will introduce Cunningham’s proposal using one example for the exceptional group G2. This is a joint work with Cunningham and Fiori.

(Continued) Even if we know many things about hyperbolic manifolds, there are many open extremal problems on them. To name a few: - How does the maximal systole among closed hyperbolic n-manifolds of volume at most V grow as a function of V? - How does the minimal diameter among closed hyperbolic n-manifolds of volume at least V grow as a function of V? - Are there closed hyperbolic n-manifolds of arbitrarily large volume whose spectral gap is larger than that of hyperbolic n-space? Even for surfaces (i.e n=2), many of these extremal problems are open. In this case, answers to these questions also provide insight into the shape of deformation spaces of hyperbolic surfaces. In these lectures, I will discuss some of these problems. I will talk about what is known about them and how random constructions of hyperbolic manifolds sometimes provide answers.

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Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université mini-course on August 19, 21, 24, 25 (W F M Tu) Online Class only

(Continued) Even if we know many things about hyperbolic manifolds, there are many open extremal problems on them. To name a few: - How does the maximal systole among closed hyperbolic n-manifolds of volume at most V grow as a function of V? - How does the minimal diameter among closed hyperbolic n-manifolds of volume at least V grow as a function of V? - Are there closed hyperbolic n-manifolds of arbitrarily large volume whose spectral gap is larger than that of hyperbolic n-space? Even for surfaces (i.e n=2), many of these extremal problems are open. In this case, answers to these questions also provide insight into the shape of deformation spaces of hyperbolic surfaces. In these lectures, I will discuss some of these problems. I will talk about what is known about them and how random constructions of hyperbolic manifolds sometimes provide answers.

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Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université mini-course on August 19, 21, 24, 25 (W F M Tu) Online Class only

Ulrich complexity for a given projective variety X, originally introduced to measure the complexity of polynomials by Bläser-Eisenbud-Schreyer, is defined as the smallest possible rank for the Ulrich sheaves on X. The existence of an Ulrich sheaf on any hypersurface is well-known, however, Ulrich complexity is not very well understood even for cubic hypersurfaces. In this talk, I will review some recent studies on Ulrich complexity for small cubics, in particular, for smooth cubic fourfolds. This is a joint work in progress with D. Faenzi.

Riemann zeta functions and Dirichlet L-functions are first several examples of L-functions. Automorphic L-functions are vast generalizations of these L-functions. In this talk, I will give a quick survey of these L-functions and some related topics, including our recent work on holomorphy of adjoint L-function for GL(3) joint with Joseph Hundley.

Even if we know many things about hyperbolic manifolds, there are many open extremal problems on them. To name a few: - How does the maximal systole among closed hyperbolic n-manifolds of volume at most V grow as a function of V? - How does the minimal diameter among closed hyperbolic n-manifolds of volume at least V grow as a function of V? - Are there closed hyperbolic n-manifolds of arbitrarily large volume whose spectral gap is larger than that of hyperbolic n-space? Even for surfaces (i.e n=2), many of these extremal problems are open. In this case, answers to these questions also provide insight into the shape of deformation spaces of hyperbolic surfaces. In these lectures, I will discuss some of these problems. I will talk about what is known about them and how random constructions of hyperbolic manifolds sometimes provide answers.

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mini-course on August 19, 21, 24, 25 (W F M Tu) Online Class only

In this talk, I will show that secant varieties of a smooth projective curve embedded by a sufficiently large degree has normal Cohen-Macaulay Du Bois singularities. I will also prove that the curve is rational if and only if any secant variety has log terminal singularities, and the curve is elliptic if and only if any secant variety has log canonical singularities that are not log terminal (not even rational). This talk is based on joint work with Lawrence Ein and Wenbo Niu.

In this talk, I will give a brief survey on singularities and log pairs in birational geometry. Log terminal singularities, log canonical singularities, rational singularities, and Du Bois singularities naturally appear in many areas in algebraic geometry such as birational geometry, moduli theory, and Hodge theory.

Grothendieck posed a question of whether the natural map from the Brauer group of a scheme to its cohomological one is an isomorphism of abelian groups. It’s not true in general, but we have some positive results from Grothendieck and Gabber (and de Jong), among many others. After a brief review of Brauer groups in algebraic geometry, I’ll talk about some recent progress in the setting of derived and spectral algebraic geometry, where we can provide an affirmative answer for quasi-compact and quasi-separated (derived/spectral) schemes, and my work which extends the previous results to spectral algebraic stacks.

Derived/spectral algebraic geometry is a relatively new area which features homotopy theory in algebraic geometry. I’ll take deformation theory and intersection theory to provide some flavor of these new fields. There are no prerequisites required other than ordinary algebraic geometry, so everyone is welcome to attend. (There are two lectures; I, II. This is the second of them.)

Derived/spectral algebraic geometry is a relatively new area which features homotopy theory in algebraic geometry. I’ll take deformation theory and intersection theory to provide some flavor of these new fields. There are no prerequisites required other than ordinary algebraic geometry, so everyone is welcome to attend. (There are two lectures; I and II. This is the first of them.)

Modern deep learning (DL) algorithms rely extensively on large amounts of annotated data. Even when a large dataset is available, DL algorithms often fail miserably when deployed to settings with data characteristics significantly differing from those used for training. Domain adaptation (DA) and domain generalization (DG) algorithms aim to mitigate the gap between source (train) and target (test) distributions by learning domain-agnostic features or minimizing the discrepancy in the model’s predictions between the source and target distributions. This issue is prevalent in practical medical imaging settings, as the cost of obtaining both images and annotations is extremely expensive, limiting data accessibility to only a bulk of images collected from a few hospitals or detector devices, but a model must be suitable for multi-center, multi-device settings. In this seminar, we will cover existing literature on DA and DG, discussing their capabilities, assumptions, methodologies, along with their limitations. The session will conclude with research directions relevant to pragmatic industrial settings.

With the goal of reducing the number of annotated data necessary for current deep learning (DL) algorithms, semi-supervised learning (SSL) algorithms use unlabeled data which is vastly more accessible than their labeled counterpart to enhance the performance of deep neural networks (DNNs) when trained on a small number of labeled data. As an example, state-of-the-art SSL algorithms can achieve up to ~84% accuracy on the CIFAR10 dataset using 1 image per class, as long as the single image is of “prototypical” quality. This session will introduce common SSL settings considered in recent works and cover DL-based SSL algorithms in a chronological fashion. While existing SSL algorithms are mainly heuristics (they lack theoretical justifications), the intuition underlying such algorithms will also be discussed in relation to the merging consensus in DL-based generalization theory/studies.

A classical result in Monge-Ampere equation states the paraboloids are the only convex entire solutions to $\det D^2 u = 1$. In this talk, we discuss a recent progress on the generalization of this classification in 2-dimension when the right-hand side is $(1+|Dx|^2)^{\beta}$. This corresponds to the classification of translating solitons to the flow by power of the Gauss curvature. Our proof combines spectral analysis from the linear theory and the theory of Monge-Ampere equation. This is a joint work with Kyeongsu Choi and Soojung Kim.

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Meeting ID: 914 3828 0517 Password: 633013