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I will introduce various heights on number fields and outline how solving polynomial equations with heights conditions is related to Arakelov geometry and a continuous logic theory called GVF.

Tensor decomposition is an unsupervised learning methodology that has applications in a wide variety of domains, including chemometrics, criminology, and neuroscience. We focus on low-rank tensor decomposition using  canonical polyadic or CANDECOMP/PARAFAC format. A low-rank tensor decomposition is the minimizer according to some nonlinear program. The usual objective function is the sum of squares error (SSE) comparing the data tensor and the low-rank model tensor. This leads to a nicely-structured problem with subproblems that are linear least squares problems which can be solved efficiently in closed form. However, the SSE metric is not always ideal. Thus, we consider using other objective functions. For instance, KL divergence is an alternative metric is useful for count data and results in a nonnegative factorization. In the context of nonnegative matrix factorization, for instance, KL divergence was popularized by Lee and Seung (1999). We can also consider various objectives such as logistic odds for binary data, beta-divergence for nonnegative data, and so on. We show the benefits of alternative objective functions on real-world data sets. We consider the computational of generalized tensor decomposition based on other objective functions, summarize the work that has been done thus far, and illuminate open problems and challenges. This talk includes joint work with David Hong and Jed Duersch.

Title: Elliptic curves and modularity

Abstract: The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.

The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.

 I will explore subtle aspects of rank-one 4d N=2 supersymmetric QFTs through their low-energy Coulomb-branch physics. This low-energy Lagrangian is famously encoded in the Seiberg-Witten (SW) curve, which is a one-parameter family of elliptic curves. Less widely appreciated is the fact that various properties of the QFTs, including properties that cannot be read off from the Lagrangian, are nonetheless encoded into the SW curve, in particular in its Mordell-Weil group. This includes the global form of the flavour group, the one-form symmetries under which defect lines are charged, and the "global form" of the theory. In particular, I will discuss in detail the difference between the pure SU(2) and the pure SO(3) N=2 SYM theories from this perspective. I will also comment on 5d SCFTs compactified on a circle in this context.

I will discuss a KLT relation of closed string amplitudes into open string amplitudes for closed string states carrying winding and momentum in toroidal compactifications. The goal is to introduce an interesting D-brane set up in the target space in order to accommodate both quantum numbers of the closed string. I will then discuss KLT factorization of amplitudes for winding closed strings in the presence of a critical Kalb-Ramond field and the relevance of this work for nonrelativistic string theory when taking the zero Regge limit. 

I will review some aspects of gravity in asymptotically flat spacetime and mention important challenges to obtain a holographic description in this framework. I will then present two important approaches towards flat space holography, namely Carrollian and celestial holography, and explain how they are related to each other. Similarities and differences between flat and anti-de Sitter spacetimes will be emphasized throughout the talk.