In this talk, I will explain first what is the representation theory with an easy example from finite groups. Next, I will introduce certain algebraic objects called the quantum affine algebras and the quiver Hecke algebras, and explain a connection between those objects via representation theory of associative algebras.

Hessian operators are, roughly speaking, the ones that depend on the eigenvalues of the Hessian matrix. Classical examples include the Laplacian and the real and complex Monge-Amp\`ere operator. Typically discussion of Hessian equations is restricted to subfamilies of functions, so that the problem becomes (degenerate) elliptic. In my talk I will discuss the basics of general Hessian equations and explain its links to problems arising in geometric analysis. If time permits I will focus on more specific examples admitting a richer theory.

Deep neural networks usually act on fixed dimensional items. However, many real-world problems are formulated as learning mappings from sets of items to outputs. Such problems include multiple-instance learning, visual scene understandings, few-shot classifications, and even generic Bayesian inference procedures. Recently, several methods have been proposed to construct neural networks taking sets as inputs. The key properties required for those neural networks are permutation invariance and equivariance, meaning that intermediate outputs and final values of a network should remain unchanged with respect to the processing order of items in sets. This talk discusses recent advances in permutation invariant and equivariant neural networks, and discuss their theoretical properties, especially their universalities. The later part of the talk will also introduce interesting applications of the permutation invariant/equivariant neural networks.

(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.)