High-dimensional data is often assumed to be distributed near a smooth manifold. But should we really believe that? In this talk I will introduce HADES, an algorithm that quickly detects singularities where the data distribution fails to be a manifold.

By using hypothesis testing, rather than persistent homology, HADES achieves great speed and a strong statistical foundation. We also have a precise mathematical theorem for correctness, proven using optimal transport theory and differential geometry. In computational experiments, HADES recovers singularities in synthetic data, road networks, molecular conformation space, and images.

Paper link: https://arxiv.org/abs/2311.04171
Github link: https://github.com/uzulim/hades
 

Suppose 𝐺_𝕜 is a connected reductive algebraic 𝕜-group where 𝕜 is an algebraically closed field. If 𝑉_𝕜 is a 𝐺_𝕜-module then, using geometric invariant theory, Kempf has defined the nullcone 𝒩(𝑉_𝕜) of 𝑉_𝕜. For the Lie algebra 𝔤_𝕜 = Lie(𝐺_𝕜), viewed as a 𝐺_𝕜-module via the adjoint action, we have 𝒩(𝔤_𝕜) is precisely the set of nilpotent elements.

We may assume that our group 𝐺_𝕜 = 𝐺 ×_ℤ 𝕜 is obtained by base-change from a suitable ℤ-form 𝐺. Suppose 𝑉 is 𝔤 = Lie(G) or its dual 𝔤* = Hom(𝔤, ℤ) which are both modules for 𝐺, that are free of finite rank as ℤ-modules. Then 𝑉 ⨂_ℤ 𝕜, as a module for 𝐺_𝕜, is 𝔤_𝕜 or 𝔤_𝕜* respectively.

It is known that each 𝐺_ℂ -orbit 𝒪 ⊆ 𝒩(𝑉_ℂ) contains a representative ξ ∈ 𝑉 in the ℤ-form. Reducing ξ one gets an element ξ_𝕜 ∈ 𝑉_𝕜 for any algebraically closed 𝕜. In this talk, we will explain two ways in which we might want ξ to have “good reduction” and how one can find elements with these properties. We will also discuss the relationship to Lusztig’s special orbits.

This is on-going joint work with Adam Thomas (Warwick).

For any smooth complex variety Y with an action of a finite group W, Etingof defines the global Cherednik algebra H_c and its spherical subalgebra B_c as certain sheaves of algebras over Y/W. When Y is an n-dimensional abelian variety, the algebra of global sections of B_c is a polynomial algebra on n generators, as shown by Etingof, Felder, Ma, and Veselov. This defines an integrable system on Y. In the case of Y being a product of n copies of an elliptic curve E and W=S_n, this reproduces the usual elliptic Calogero­­--Moser system. Recently, together with P. Argyres and Y. Lu, we proposed that many of these integrable systems at the classical level can be interpreted as Seiberg­­--Witten integrable systems of certain super­symmetric quantum field theories. I will describe our progress in understanding this connection for groups W=G(m, 1, n), corresponding to the case Y=E^n where E is an elliptic curves with Z_m symmetry, m=2,3,4,6. 

Porta and Yue Yu's model of derived analytic geometry takes as its category of basic, or affine, objects the category opposite to simplicial algebras over the entire functional calculus Lawvere theory. This is analogous to Lurie's approach to derived algebraic geometry where the Lawvere theory is the one governing simplicial commutative rings, and Spivak's derived smooth geometry, using the Lawvere theory of C-infinity-rings. Although there have been numerous important applications including GAGA, base-change, and Riemann-Hilbert theorems, these methods are still missing some crucial ingredients. For example, they do not naturally beget a good definition of quasi-coherent sheaves satisfying descent. On the other hand, the Toen-Vezzosi-Deligne approach of geometry relative to a symmetric monoidal category naturally provides a definition of a category of quasi-coherent sheaves, and in two such approaches to analytic geometry using the categories of bornological and condensed abelian groups respectively, these categories do satisfy descent.  In this talk I will explain how to compare the Porta and Yue Yu model of derived analytic geometry with the bornological one. More generally we give conditions on a Lawvere theory such that its simplicial algebras embed fully faithfully into commutative bornological algebras. Time permitting I will show how the Grothendieck topologies on both sides match up, allowing us to extend the embedding to stacks.

This is based on joint work with Oren Ben-Bassat and Kobi Kremnitzer, and follows work of Kremnitzer and Dennis Borisov.

In this talk, we investigate distributionally robust optimization (DRO) in a dynamic context. We consider a general penalized DRO problem with a causal transport-type penalization. Such a penalization naturally captures the information flow generated by the models. We derive a tractable dynamic duality formula under a measure theoretic framework. Furthermore, we apply the duality to distributionally robust average value-at-risk and stochastic control problems.