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Initial degenerations of flag varieties
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DANIEL COREY, JORGE ALBERTO OLARTE
Journal:

Mathematical Proceedings of the Cambridge Philosophical Society , First View

Published online by Cambridge University Press:

10 November 2025, pp. 1-29
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We prove that the initial degenerations of the flag variety admit closed immersions into finite inverse limits of flag matroid strata, where the diagrams are derived from matroidal subdivisions of a suitable flag matroid polytope. As an application, we prove that the initial degenerations of $\mathrm{F}\ell^{\circ}(n)$–the open subvariety of the complete flag variety $\mathrm{F}\ell(n)$ consisting of flags in general position—are smooth and irreducible when $n\leq 4$. We also study the Chow quotient of $\mathrm{F}\ell(n)$ by the diagonal torus of $\textrm{PGL}(n)$ and show that, for $n=4$, this is a log crepant resolution of its log canonical model.

On The Schinzel–Wójcik Problem Under Hypothesis H
Part of
- Elementary number theory
M. ANWAR
Journal:

Mathematical Proceedings of the Cambridge Philosophical Society , First View

Published online by Cambridge University Press:

03 November 2025, pp. 1-13
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Given r non-zero rational numbers $a_1, \ldots, a_r$ which are not $\pm1$, we complete, under Hypothesis H, a characterisation of the Schinzel–Wójcik r-rational tuples (i.e. r-tuples of rational numbers for which the Schinzel–Wójcik problem has an affirmative answer) which satisfy that the sum of the exponents of the positive elements $a_i$ in the representation of $-1$ in terms of the elements $a_i$ in the multiplicative group $\langle a_1,\dots, a_r\rangle\subset \mathbb{Q}^*$ is even whenever $-1 \in \langle a_1,\dots, a_r\rangle.$

Silting, cosilting and extensions of commutative rings
Part of
SIMION BREAZ, MICHAL HRBEK, GEORGE CIPRIAN MODOI
Journal:

Mathematical Proceedings of the Cambridge Philosophical Society , First View

Published online by Cambridge University Press:

27 October 2025, pp. 1-26
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We study the transfer of (co)silting objects in derived categories of module categories via the extension functors induced by a morphism of commutative rings. It is proved that the extension functors preserve (co)silting objects of (co)finite type. In many cases the bounded silting property descends along faithfully flat ring extensions. In particular, the notion of bounded silting complex is Zariski local.

The structure of Lonely Runner spectra
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VIKRAM GIRI, NOAH KRAVITZ
Journal:

Mathematical Proceedings of the Cambridge Philosophical Society , First View

Published online by Cambridge University Press:

24 October 2025, pp. 1-19
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For each closed subtorus T of $(\mathbb{R}/\mathbb{Z})^n$, let D(T) denote the (infimal) $L^\infty$-distance from T to the point $(1/2,\ldots, 1/2)$. The nth Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values achieved by D(T) as T ranges over the 1-dimensional subtori of $(\mathbb{R}/\mathbb{Z})^n$ that are not contained in the coordinate hyperplanes. The Lonely Runner Conjecture predicts that $\mathcal{S}(n) \subseteq [0,1/2-1/(n+1)]$. Rather than attack this conjecture directly, we study the qualitative structure of the sets $\mathcal{S}(n)$ via their accumulation points. This project brings into the picture the analogues of $\mathcal{S}(n)$ where 1-dimensional subtori are replaced by k-dimensional subtori or k-dimensional subgroups.

20 - A Few Explicit Examples of Causal Variational Principles
from Part IV - Applications and Outlook
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 353-365
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Summary

In this chapter, we introduce a few examples of causal variational principles and analyze them in detail.

6 - Causal Variational Principles
from Part II - Causal Fermion Systems: Fundamental Structures
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 135-147
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Summary

Causal variational principles are introduced as a generalization of the causal action principle. An overview of different cases and settings is given.

Frontmatter
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp i-vi
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11 - Topological and Geometric Structures
from Part II - Causal Fermion Systems: Fundamental Structures
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 199-212
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Summary

This chapter is devoted to the topological and geometric structures of a causal fermion system.

Notation Index: Thematic Order
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 401-401
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Part III - Mathematical Methods and Analytic Constructions
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 213-214
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12 - Measure-Theoretic Methods
from Part III - Mathematical Methods and Analytic Constructions
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 215-242
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Summary

The existence of minimizers is proven using measure-theoretic tools.

5 - A Brief Introduction to Causal Fermion Systems
from Part II - Causal Fermion Systems: Fundamental Structures
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 93-134
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Summary

We define and explain the basic objects and structures of a causal fermion system.

1 - Physical Preliminaries
from Part I - Physical and Mathematical Background
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 3-20
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Summary

We summarize some basics on quantum mechanics and relativity theory as needed in order to understand the physical content and context of the theory of causal fermion systems.

14 - Energy Methods for the Linearized Field Equations
from Part III - Mathematical Methods and Analytic Constructions
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 268-278
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Summary

The initial-value problem for the linearized field equations is developed.

4 - Spinors in Curved Spacetime
from Part I - Physical and Mathematical Background
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 68-90
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Summary

A brief introduction to spinors in curved spacetime is given in a formulation as a U(2,2) gauge theory.

Appendix: - The Spin Coefficients
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 388-390
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The spin coefficients are computed in detail.

Subject Index
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

Published online:

15 November 2025

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23 October 2025, pp 402-407
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22 - Connection to Quantum Field Theory
from Part IV - Applications and Outlook
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

Published online:

15 November 2025

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23 October 2025, pp 379-387
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Summary

An outlook is given on how to get a connection between the causal action principle and the dynamics of quantum fields.

Part I - Physical and Mathematical Background
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

Published online:

15 November 2025

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23 October 2025, pp 1-2
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9 - Surface Layer Integrals and Conservation Laws
from Part II - Causal Fermion Systems: Fundamental Structures
Felix Finster, Universität Regensburg, Germany, Sebastian Kindermann, Comenius Gymnasium Deggendorf, Germany, Jan‐Hendrik Treude, Universität Konstanz, Germany
Book:

Causal Fermion Systems

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15 November 2025

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23 October 2025, pp 171-189
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Summary

Surface layer integrals are introduced as an adaptation of surface integrals to causal fermion systems and causal variational principles. Conservation laws are derived and formulated in terms of surface layer integrals.

Theoretical Physics and Mathematical Physics

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26203 results in Theoretical Physics and Mathematical Physics

Initial degenerations of flag varieties

On The Schinzel–Wójcik Problem Under Hypothesis H

Silting, cosilting and extensions of commutative rings

The structure of Lonely Runner spectra

20 - A Few Explicit Examples of Causal Variational Principles

Summary

6 - Causal Variational Principles

Summary

Frontmatter

11 - Topological and Geometric Structures

Summary

Notation Index: Thematic Order

Part III - Mathematical Methods and Analytic Constructions

12 - Measure-Theoretic Methods

Summary

5 - A Brief Introduction to Causal Fermion Systems

Summary

1 - Physical Preliminaries

Summary

14 - Energy Methods for the Linearized Field Equations

Summary

4 - Spinors in Curved Spacetime

Summary

Appendix: - The Spin Coefficients

Summary

Subject Index

22 - Connection to Quantum Field Theory

Summary

Part I - Physical and Mathematical Background

9 - Surface Layer Integrals and Conservation Laws

Summary

Theoretical Physics and Mathematical Physics

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26203 results in Theoretical Physics and Mathematical Physics

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