Search Talks

Mathlib is a vast and constantly growing library of formalized mathematics. As its size increases, it becomes increasingly easy to spend a significant amount of time formalizing a theorem, only to later discover that it was already present in the library. This can be both frustrating and discouraging. In this talk, we will introduce and demonstrate a variety of tools and techniques that can help users efficiently navigate Mathlib, search for existing results, and better understand the structure of the library.

Proteins are biopolymers, composed of repeating sequence of amino acids (AA). In a typical sequence, the constituting AAs have different charges, hydrophobicity, and capacities to form directional and non-directional interactions. Such heterogeneity can results in sequences lacking a stable three dimensional structure. This class of proteins are intrinsically disordered protein (IDPs). A deeper understanding of IDPs require appropriate characterization of the conformations. To this end, scattering and single molecular spectroscopic measurements often assign a single Flory exponent (equivalently fractal dimension) to the IDPs. In this talk, I highlight limitation of this method by enhanced sampling of atomistic resolution conformations of disordered \beta-casein. I will show that the underlying energy landscape of the IDP contains a global minimum along with two shallow funnels. Employing static polymeric scaling laws separately for individual funnels, we find that they cannot be described by the same polymeric scaling exponent. Around the global minimum, the conformations are globular, whereas in the vicinity of local minima, we recover coil-like scaling. To elucidate the implications of structural diversity on equilibrium dynamics, we initiated standard MD simulations in the NVT ensemble with representative conformations from each funnel. Global and internal motions for different classes of trajectories show heterogeneous dynamics with globule to coil-like signatures. Thus, IDPs can behave as entirely different polymers in different regions of the conformational space.

Run-and-tumble (RT) motion is commonly observed in flagellated microswimmers, arising from synchronous and asynchronous flagellar beating. In addition to hydrodynamic interactions, mechanical coupling has recently been recognized to play a key role in flagellar synchronization. To explore this, we design a macroscopic model system that comprises dry, self-propelled robots linked by a rigid rod to model a biflagellated microorganism. To mimic a low Reynolds number environment, we program each robot to undergo overdamped active Brownian (AB) motion. We find that such a system exhibits RT-like behavior, characterized by sharp tumbles and exponentially distributed run times, consistent with real microswimmers. We quantify tumbling frequency and demonstrate its tunability across experimental parameters. Additionally, we provide a theoretical model that reproduces our results, elucidating physical mechanisms governing RT dynamics.

Mathematicians have trained themselves to see objects from many points of view. When considering the real numbers, we can as easily see them as equivalence classes of Cauchy sequences, or Dedekind cuts, or the unique uniformly complete Archimedean field. The computer is not as forgiving and forces us to pick a particular definition that we must stick with. Using examples from the Lean mathematical library, Mathlib, I investigate why it is so important to choose the right definition when formalizing, what makes formal definitions look different from pen and paper definitions, and how we can design our definitions to make proofs flow smoothly.

We study some of the properties of non-equilibrium phase transitions of an interacting system that is in a state with large deviation from equilibrium. We consider a field theoretical model of scalar particles interacting with vector gauge fields with local U (1) gauge symmetry. We show that the evolution of this system towards an equilbrium state can be described using the principle of emergence of global gauge invariance. As a toy model we consider a scalar-vector model with local U (1) gauge invariance. Invoking the assumption of local U (1) gauge invariance breaking we evaluate time evolution of some of the observables of this system. To make our calculations explicit we calculate the time evolution of order parameter of this sytem and evaluate its scaling behaviour near transition region. In the mean-field approximation we show that, for the unperturbed case, that correspond to no external driving the order parameter has a universal algebriac decay m(t) ∼ t −1/2 . However for time-dependent diffusion coefficient, it is found that order parametr has a universal algabriac decay m(t) ∼ t −1/3 . The results are in total agreement with the recent findings using stochastic models of non-equilibrium system

In this work, we studied the three-dimensional random field Potts model (RFPM), focusing on its phase transition, which is governed by a random fixed point located at zero temperature. As finding ground states in RFPM is NP-hard, we employed our recently developed quasi-exact scheme based on graph cuts to determine approximate ground states and analyze critical behavior. We evaluated various key observables, such as magnetization, Binder and energy cumulants, specific heat, and susceptibilities, which we extrapolated to the quasi-exact ground state limit. Their finite-size scaling analyses revealed strong evidence for a continuous transition induced by disorder, in contrast to the first-order transition seen in the pure case. Our results suggest a new universality class for q-state RFPM, distinct from the RFIM.