This talk will feature an introduction to the Weil height machine, line bundles on abelian varieties, Neron--Tate heights, and a discussion of the Silverman--Tate theorem on heights in families.

TBA

I will show how to calculate 1/N expansion of the vacuum energy of the 2D SU(N) Principal Chiral Model for a certain profile of chemical potentials. Combining this expansion with strong coupling I will identify double-scaling limit which bears striking similarities to the c = 1 non-critical string theory and suggests that the double-scaled PCM is dual to a non-critical string with a (2 + 1)-dimensional target space where an additional dimension emerges dynamically from the SU(N) Dynkin diagram. Developing this idea further, I will show how to solve large-N PCM for an arbitrary set of chemical potentials and any interaction strength, a unique result of such kind for an asymptotically free QFT. The solution matches one-loop perturbative calculation at weak coupling, and in the opposite strong-coupling regime exhibits an emergent spacial dimension from the continuum limit of the SU(N) Dynkin diagram. In the second part of my talk I will show that the calculation of the expectation value of half-BPS circular Wilson loops in N = 2 superconformal A_{n�������1} quiver gauge theories trivialises in the large n limit (similarly to PCM), construct 1/n expansion, identify DS limit and solve it for any finite value of DS parameter and any profile of coupling constants.

Solving discrete optimization problems is useful in many modern applications, but they are well known for their hardness. With advances in technologies such as the Coherent Ising Machine (CIM) and Simulated Bifurcation (SB), there is an emergent interest in using physical dynamics of continuous variables to solve hard combinatorial optimizations. An important issue is whether such continuous dynamics can lead to solutions coincident with those of the discrete problem. In particular, we are interested in bifurcations of the continuous dynamics when external parameters (such as the pump rate in CIMs or SBs) are tuned, and their role in finding the optimal solution of the discrete problem. When all nodal states undergo bifurcation dynamics at the same tuning value of the external parameter, we derive sufficient conditions that the transition contains enough information for exactly solving the discrete optimization problem. When synchronous bifurcations are not possible due to frustration effects, subsequent cascades of bifurcations become necessary to reveal further details of the discrete optimization landscape. When the pump rate increases further, we derive the pump rate above which there is a guaranteed existence of the steady state of the continuous dynamics that can be binarized to map to the ground state of the discrete system. Inspired by the observation that nodes which bifurcate early tend to maintain their signs during the dynamical evolution, we devise a new trapping-and-correction (TAC) approach, which can be applied to various physical solvers, including CIMs and SBs and their variants. The proposed approach takes advantages of fixing the early bifurcated ���trapped nodes��� to enable updates of other nodes, effectively reducing computation time of the Ising dynamics. Using problem instances from the Biq-Mac library benchmark and random Ising models, we validated TAC approach's superior convergence and accuracy.

References
[1] Juntao Wang, Daniel Ebler, K. Y. Michael Wong, David Shui Wing Hui, and Jie Sun, Nature Communications, v. 14, May 2023, article number 2510.

In this talk, we will study invariants of complex isolated hypersurface singularities. In the first half I will review the basics of Floer theory, and I will describe Fukaya-Seidel categories, a powerful and geometric derived invariant of singularities. In the second half, I will describe invariants of a special family of isolated singularities, whose Fukaya-Seidel categories play an important role in bordered Heegaard Floer theory. Motivated by representation theory, I will relate these singularities to abstract objects associated to algebras of type A (named after the quiver of Dynkin type A). I will introduce ���type A symplectic Auslander correspondence���, a purely geometrical construction which realises a notable result in representation theory. Most of the talk will be example-based.

In this work we consider a time-periodic and random version of the ABC flow. We are concerned with two main subjects. On the one hand, we study the mixing problem of a passive tracer in the three-dimensional torus by the action of the random ABC vector field. On the other hand, we investigate the effect of the ABC flow on the growth of a magnetic field described by the kinematic dynamo equations. To deal with these questions we analyse the ABC flow as a random dynamical system and examine the ergodic properties of its associated one-point, two-point, and projective Markov chains, as well as its top Lyapunov exponent. This work settles that the random ABC vector field is an example of a space-time smooth universal exponential mixer in the three dimensions, and in addition, we obtain that it is an ideal kinematic fast dynamo. This is a joint work with Michele Coti Zelati (Imperial College London).

Duffin-Schaeffer meets Littlewood and related topics

Khintchine's Theorem is one of the cornerstones in metric Diophantine approximation. The question of removing the monotonicity condition on the approximation function in Khintchine's Theorem led to the recently proved Duffin-Schaeffer conjecture. Gallagher showed an analogue of Khintchine's Theorem for multiplicative Diophantine approximation, again assuming monotonicity. In this talk, I will discuss my joint work with L. Fr��hwirth about a Duffin-Schaeffer version for Gallagher's Theorem. Furthermore, I will give a broader overview on various questions in metric Diophantine approximation and demonstrate the deep connection to analytic number theory that lies in the heart of the corresponding proofs.

This term we will have a study group on the work of Dimitrov--Gao--Habegger and K��hne on uniformity in the Mordell conjecture. The first half of the schedule is meant to be an introduction to the area for non-specialists. In the second half, we will try to introduce some of the ideas from functional transcendence, dynamics and the moduli of abelian varieties which go into the proof.

The first talk will be an introduction to the history and statement of the results, with a vague hint at the methods of proof. A plan of the rest of the study group can be found here: https://sites.google.com/site/netandogra/seminars/uniform-mordell