Inductive limits are a central construction in C*-theory because they allow one to use well-understood building blocks to naturally construct intricate C*-algebras whose properties remain tractable. One task for the structural theory of operator algebras is to determine which classes of operator algebras arise as inductive limits of nice operator algebras.

The model result in this direction is Connes' 1970's theorem showing that many classes of von Neumann algebras, including semi-discrete von Neumann algebras, arise as inductive limits of finite dimensional von Neumann algebras. For C*-algebras, an analogous result fails outright: many nuclear C*-algebras are not inductive limits of finite dimensional C*-algebras. However, by generalizing our notion of an inductive system, we can in fact describe any nuclear C*-algebra as the limit of a system of finite dimensional C*-algebras. Though seemingly abstract, these generalized inductive systems arise naturally from completely positive approximations of nuclear C*-algebras.

This is based in part on joint work with Wilhelm Winter. 

The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce non-trivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this talk, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology.

 Information theory provides a mathematical framework for quantifying information processing tasks, such as storage, computation, and communication. Connecting the abstract theory to concrete physical systems often gives insight into a system's physics; conversely, physics can often inspire new ideas in information theory itself. This perspective has been particularly fruitful in quantum gravity, for which the essential question is to understand how information is stored and processed by gravitating systems, such as black holes or even the Universe itself. In this talk we will see how quantum gravitational considerations lead to an extended Nyquist-Shannon sampling theorem for fields on Lorentzian manifolds. Applying the results to the physics of the early Universe leads to predictions for cosmological signatures of quantum gravity that can be tested with present-day observations of the cosmos.

We discuss Milnor's link invariants through a geometric lens using intersections of Seifert surfaces. Our work is thus of a similar flavor as that of Cochran from 1990, who based his work on particular choices of Seifert surfaces. But like Mellor and Melvin in 2003, who considered only the first invariant (after linking number), we allow for more arbitrary choices. We conjecture that Milnor’s invariants can be recovered geometrically using the work of Monroe and Sinha on linking of letters and Sinha and Walters on Hopf invariants. We expect our approach to recover Cochran’s work and to extend work of Polyak, Kravchenko, Goussarov, and Viro on Gauss diagrams.

In this work, we present a difference of convex algorithm for solving bilevel programs in which the upper level objective functions are difference of convex functions, and the lower level programs are fully convex. This nontrivial class of bilevel programs provides a powerful modelling framework for dealing with applications arising from hyperparameter selection in machine learning. Thanks to the full convexity of the lower level program, the value function of the lower level program turns out to be convex and hence the bilevel program can be reformulated as a difference of convex bilevel program. We propose an algorithm for solving the reformulated difference of convex program and show its convergence to stationary points under very mild assumptions.

I'm gonna introduce p-adic Analysis and I'm gonna talk about the Skolem-Mahler-Lech theorem and I'm gonna blow your mind.

In this talk, I will explore the rich interplay between differential equations and machine learning. I will highlight the use of collective dynamics and partial differential equations as powerful tools for improving machine learning algorithms and models. (i) In the first half of the talk, I will introduce a novel dynamical system that draws inspiration from collective intelligence observed in biology. This system offers a compelling alternative to gradient-based optimization. It enables gradient-free optimization to efficiently find global minimum in non-convex optimization problems. (ii) In the second half of the talk, I will build the connection between Hamilton-Jacobi-Bellman equations and the multi-armed bandit (MAB) problems. MAB is a widely used paradigm for studying the exploration-exploitation trade-off in sequential decision-making under uncertainty. This is the first work that establishes this connection in a general setting. I will present an efficient algorithm for solving MAB problems based on this connection and demonstrate its practical applications.

We discuss about smooth actions on manifold by higher rank lattices. We mainly focus on lattices in $\mathrm{SL}_n(\mathbb{R})$ ($n$ is at least $3$). Recently, Brown-Fisher-Hurtado and Brown-Rodriguez Hertz-Wang showed that if the manifold has dimension at most $(n-1)$, the action is either isometric or projective. Both cases, we don't have chaotic dynamics from the action (zero entropy). We focus on the case when one element of the action acts with positive topological entropy. These dynamical properties (positive entropy element) significantly constrains the action. Especially, we deduce that if there is a smooth action with positive entropy element on a closed $n$-manifold by a lattice in $\mathrm{SL}_n(\mathbb{R})$ ($n$ is at least $3$) then the lattice should be commensurable with $\mathrm{SL}_n(\mathbb{Z})$. This is the work in progress with Aaron Brown.

 The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed Galois group, and bounded discriminant? In this talk, I will address this question for n = 3, 4, 5. The answers, appropriately interpreted, turn out to be piecewise linear functions on certain convex bodies. If time permits, I will also discuss function field analogues of this problem.

[pre-talk at 1:20PM]

Expander graphs are perhaps one of the most widely useful classes of graphs ever considered. In this talk, we will focus on a fairly weak notion of expanders called sublinear expanders, first introduced by Komlós and Szemerédi around 25 years ago. They have found many remarkable applications ever since. In particular, we will focus on certain robustness conditions one may impose on sublinear expanders and some applications of this very recent idea, which include: 

- recent progress on one of the most classical decomposition conjectures in combinatorics, the Erdős-Gallai Conjecture,

- Rainbow Turan problem for cycles, raised by Keevash, Mubayi, Sudakov and Verstraete, including an application of this result to additive number theory and 

- essentially tight answers to the classical Erdős unit distance and distinct distances problems in "almost all" real normed spaces of any fixed dimension.

Moduli of cubic surfaces can be compactified from the point of view of geometric invariant theory (GIT), and from the point of view of the ball quotient. The Kirwan desingularization resolves the GIT singularities to yield a smooth Kirwan compactification, while the toroidal compactification of the ball quotient is also smooth. We show that these two smooth compactifications are, however, not isomorphic. Based on joint work with S. Casalaina-Martin, K. Hulek, R. Laza