Wed, 28 Feb 2024
15:00
Lecture room 5

Mathematics of magic angles for twisted bilayer graphene.

Prof Maciej Zworski
(University of California, Berkeley)
Further Information

This is a joint seminar with Random Matrix Theory and Oxford Centre for Nonlinear Partial Differential Equations.

Abstract

Magic angles refer to a remarkable theoretical (Bistritzer--MacDonald, 2011) and experimental (Jarillo-Herrero et al 2018) discovery, that two sheets of graphene twisted by a certain (magic) angle display unusual electronic properties such as superconductivity.

 

Mathematically, this is related to having flat bands of nontrivial topology for the corresponding periodic Hamiltonian and their existence be shown for the chiral model of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019). A spectral characterization of magic angles (Becker--Embree--Wittsten--Z, 2021, Galkowski--Z, 2023) also produces complex values and the distribution of their reciprocals looks remarkably like a distribution of scattering resonances for a two dimensional problem, with the real magic angles corresponding to anti-bound states. I will review various results on that distribution as well as on the properties of the associated eigenstates.

 

The talk is based on joint works with S Becker, M Embree, J Galkowski, M Hitrik, T Humbert and J Wittsten

Tue, 21 Nov 2023

17:00 - 18:00
L1

Advances in Advancing Interfaces: The Mathematics of Manufacturing of Industrial Foams, Fluidic Devices, and Automobile Painting

James Sethian
(University of California, Berkeley)
Further Information

https://www.maths.ox.ac.uk/events/special-lectures/brooke-benjamin-lect…

Abstract

Complex dynamics underlying industrial manufacturing depend in part on
multiphase multiphysics, in which fluids and materials interact across
orders of magnitude variations in time and space. In this talk, we will
discuss the development and application of a host of numerical methods for
these problems, including Level Set Methods, Voronoi Implicit Interface
Methods, implicit adaptive representations, and multiphase discontinuous
Galerkin Methods.  Applications for industrial problems will include modeling
how foams evolve, how electro-fluid jetting devices work, and
the physics and dynamics of rotary bell spray painting across the automotive
industry.

 

Thu, 02 Feb 2023
17:00
L3

Geometric Stability Theory and the Classification of Unstable Structures

Scott Mutchnik
(University of California, Berkeley)
Abstract

The equivalence of NSOP${}_1$ and NSOP${}_3$, two model-theoretic complexity properties, remains open, and both the classes NSOP${}_1$ and NSOP${}_3$ are more complex than even the simple unstable theories. And yet, it turns out that classical geometric stability theory, in particular the group configuration theorem of Hrushovski (1992), is capable of controlling classification theory on either side of the NSOP${}_1$-SOP${}_3$ dichotomy, via the expansion of stable theories by generic predicates and equivalence relations. This allows us to construct new examples of strictly NSOP${}_1$ theories. We introduce generic expansions corresponding, though universal axioms, to definable relations in the underlying theory, and discuss the existence of model companions for some of these expansions. In the case where the defining relation in the underlying theory $T$ is a ternary relation $R(x, y, z)$ coming from a surface in 3-space, we give a surprising application of the group configuration theorem to classifying the corresponding generic expansion $T^R$. Namely, when $T$ is weakly minimal and eliminates the quantifier $\exists^{\infty}$, $T^R$ is strictly NSOP${}_4$ and TP${}_2$ exactly when $R$ comes from the graph of a type-definable group operation; otherwise, depending on whether the expansion is by a generic predicate or a generic equivalence relation, it is simple or NSOP${}_1$.

Thu, 19 May 2022

11:30 - 12:45
L6

Skew-invariant curves and algebraic independence

Thomas Scanlon
(University of California, Berkeley)
Abstract
A $\sigma$-variety over a difference field $(K, \sigma)$ is a pair $(X, \varphi)$ consisting of an algebraic variety $X$ over $K$ and $\varphi : X \rightarrow X^{\sigma}$ is a regular map from $X$ to its transform $X^{\sigma}$ under $\sigma$. A subvariety $Y \subseteq X$ is skew-invariant if $\varphi(Y) \subseteq Y^{\sigma}$. In earlier work with Alice Medvedev we gave a procedure to describe skew-invariant varieties of $\sigma$-varieties of the form $(\mathbb{A}^n, \varphi)$ where $\varphi(x_1, \dots, x_n) = (P_1(x_1), \dots, P_n(x_n))$. The most important case, from which the others may be deduced, is that of $n=2$. In the present work we give a sharper description of the skew-invariant curves in the case where $P_2 = P_1^{\tau}$ for some other automorphism of $K$ which commutes with $\sigma$. Specifically, if $P \in K[x]$ is a polynomial of degree greater than one which is not eventually skew-conjugate to a monomial or $\pm$ Chebyshev (i.e. $P$ is "nonexceptional") then skew-invariant curves in $(\mathbb{A}^2, (P, P^{\tau}))$ are horizontal, vertical, or skew-twists: described by equations of the form $y = \alpha^{\sigma^n} \circ P^{\sigma^{n-1}} \circ \dots \circ P^{\sigma} \circ P(x)$ or $x = \beta^{\sigma^{-1}} \circ P^{\tau \sigma^{-n-2}} \circ P^{\tau \sigma^{-n-3}} \circ \dots \circ P^{\tau}(y)$ where $P = \alpha \circ \beta$ and $P^{\tau} = \alpha^{\sigma^{n+1}} \circ \beta^{\sigma^n}$ for some integer $n$. 
We use this new characterization to prove that a function $f(t)$ which satisfies $p$-Mahler equation of nonexceptional polynomial type, by which we mean $f(t^p) = P(f(t))$ for $p \in \mathbb{Q}_{+} \setminus \{1\}$ and $P \in \mathbb{C}(t)[x]$ a nonexceptional polynomial, is necessarily algebraically independent from functions satisfying $q$-Mahler equations with $q$ multiplicatively independent from $p$. 
This is a report on joint work with Khoa Dang Nguyen and Alice Medvedev available at arXiv:2203.05083.  
Mon, 24 Oct 2011

17:00 - 18:00
Gibson 1st Floor SR

Partial Regularity Results for A Variational Problem for Nematic Liquid Crystal.

Hung Tran
(University of California, Berkeley)
Abstract

This is a joint work with Craig Evans. We study the partial regularity of minimizers for certain functionals in the calculus of variations, namely the modified Landau-de Gennes energy functional in nematic liquid crystal theory introduced by Ball and Majumdar.

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