Math and physics have developed through interactions with each other.
For example, classical mechanics and calculous were born together.
Einstein's theory of gravitation is written in the language of pseudo-Riemann geometry.
Since the late 20th century, physicists centering on Edward Witten have revolutionized modern geometry.
Seiberg-Witten theory is one of such breakthroughs, for both mathematicians and physicists.
In physics it is regarded as a theory describing strong coupling (i.e. low energy) behavior of some supersymmetric gauge theories. It showes confinement (by a mechanism similar to superconductivity) and electric magnetic duality.
Even though this story has not been mathematically justified yet, it is regarded as an important trigger of developments in understanding non perturbative aspects of quantum field theory and string theory, and stimulates broad fields of physics and math.
In math, Seiberg-Witten theory is regarded as a fundamental tool to study 3 and 4-dimensional geometry.
This is based on a PDE called Seiberg-Witten equation, which originates from the "electric magnetic dual description" of monopoles, but people can use it as a tool to study geometry without knowing such a physical origin.
In this talk, developments of Seiberg-Witten theory from both viewpoints will be reviewed and if the time permits, works in math by the speaker and collaborators will be discussed.
The speaker thinks it is unusual for a mathematician to talk about something that has not been mathematically justified yet, but hopes this talk will lead to new interactions between math and physics.

Functional renormalization group (FRG) is a powerful theoretical tool to investigate physical systems described by field theory. Its application covers diverse areas of physics spanning from condensed matter and statistical physics to particle physics and gravity. The purpose of this two-day workshop is to overview the recent applications and progresses of FRG in various fields of physics, to discuss future directions, and to seek for possible new collaborations bridging different fields of physics.

The movement and deformation of the Earth’s crust and upper mantle provide critical insights into the evolution of earthquake processes and future earthquake potentials. Crustal deformation can be modeled by dislocation models that represent earthquake faults in the crust as defects in a continuum medium. In this study, we propose a physics-informed deep learning approach to model crustal deformation due to earthquakes. Neural networks can represent continuous displacement fields in arbitrary geometrical structures and mechanical properties of rocks by incorporating governing equations and boundary conditions into a loss function. The polar coordinate system is introduced to accurately model the displacement discontinuity on a fault as a boundary condition. We illustrate the validity and usefulness of this approach through example problems with strike-slip faults. This approach has a potential advantage over conventional approaches in that it could be straightforwardly extended to high dimensional, anelastic, nonlinear, and inverse problems.

We give an introduction to the formulation towards the quantum theory of gravity using the functional (or exact) renormalization group, the so-called asymptotic safety. First we briefly explain the necessity of quantization of gravity and why the Einstein gravity is not sufficient for this purpose. Second, we introduce the functional renormalization group equation and explain what is the asymptotic safety program to achieve the quantum theory of gravity. This includes the notion of relevant, irrelevant and marginal operators, and it is important that there are finite number of relevant operators to make any prediction of quantum effects. This gives a nonperturbatively renormalizable theory of gravity. We then discuss various examples how the program may be applied to various theories, and summarize the current status of this approach.

(Tentative schedule)

[Day 1: Jan. 24, 2023]
Free discussion: 9:30 - 10:30
Lecture 1: 10:30 - 12:00
Lunch: 12:00 - 13:30
Lecture 2: 13:30 - 15:00
Break: 15:00 - 15:30
Lecture 3: 15:30 - 17:00

[Day 2: Jan. 25, 2023]
Free discussion: 9:30 - 10:30
Lecture 4: 10:30 - 12:00
Lunch: 12:00 - 13:30
Lecture 5: 13:30 - 15:00
Break: 15:00 - 15:30
Lecture 6: 15:30 - 17:00

[Day 3: Jan. 26, 2023]
Q&A + discussion: 9:30 - 15:00

iTHEMS-AIP Joint Colloquium

Optimal transport (OT) has recently gained a lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this talk, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book "Computational Optimal Transport" (see related link below).

Ultra high-energy (UHE) cosmic rays (CRs) from distant sources interact with intergalactic radiation fields, leading to their spallation and attenuation through photo-hadronic processes. Their deflection and diffusion in large scale intergalactic magnetic fields (IGMFs), in particular those associated with Mpc-scale structures, alter the cumulative cooling and interactions of a CR ensemble to modify their spectral shape and composition observed on Earth. In this talk, I will demonstrate the extent to which IGMFs can affect observed UHE CRs, and show that source population models are degenerate with IGMF properties. Interpretation of observations, including the endorsement or rejection of any particular UHE CR source classes, needs careful consideration of the structural properties and evolution of IGMFs. Future observations providing tighter constraints on IGMF properties will significantly improve confidence in assessing UHE CR sources and their intrinsic CR production properties.