arXiv subjects

Evgeny Goncharov

Publications and source records attributed to Evgeny Goncharov.

6 recordsLinked to original sources

arXiv2026

Mirrors to toric degenerations via intrinsic mirror symmetry

We explore the connection between two mirror constructions in Gross-Siebert mirror symmetry: toric degeneration mirror symmetry (arXiv:1212.4220, arXiv:math/0309070, arXiv:0709.2290, arXiv:math/0703822) and intrinsic mirror symmetry (arXiv:1909.07649, arXiv:2105.02502). After briefly exploring the case of degenerations of elliptic curves, we show that the Gross-Siebert mirror construction for minimal relative log Calabi-Yau degenerations generalizes that for divisorial toric degenerations $\bar{\mathfrak{X}} \to \mathcal{S}$ of K3-s that have a smooth generic fibre. We achieve this by constructing a resolution of $\bar{\mathfrak{X}} \to \mathcal{S}$ to a relative minimal log Calabi-Yau degeneration $\mathfrak{X} \to \mathcal{S}$ and comparing the algorithmic scattering diagram $\bar{\mathfrak{D}}$ giving rise to the toric degeneration mirror $\check{\bar{\mathfrak{X}}}$ and the canonical scattering diagram $\mathfrak{D}$ giving rise to the intrinsic mirror $\check{\mathfrak{X}}$. Moreover, we vastly expand the construction and obtain a correspondence between the restriction of the intrinsic mirror to the (numerical) minimal relative Gross-Siebert locus and the universal toric degeneration mirror. We also discuss generalizing the results to higher dimensions. In particular, we construct log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds.

math.AG↗

arXiv2023

Automatic Adjoint Differentiation for special functions involving expectations

We explain how to compute gradients of functions of the form $G = \frac{1}{2} \sum_{i=1}^{m} (E y_i - C_i)^2$, which often appear in the calibration of stochastic models, using Automatic Adjoint Differentiation and parallelization. We expand on the work of arXiv:1901.04200 and give faster and easier to implement approaches. We also provide an implementation of our methods and apply the technique to calibrate European options.

q-fin.CP↗

arXiv2022

Modifications to a classic BFGS library for use with SIMD-equipped hardware and an AAD library

We introduce certain modifications of the BFGS method for functions that are not parallelizable by nature (having consecutive operations only) taking advantage of SIMD. We also provide a modified LBFGS\texttt{++} library that takes advantage of these modifications, and the use of AAD, and give an interface for AAD users that takes advantage of the modified library automatically. We give two examples to illustrate the performance. The modified library is up to 3.8 times faster for European Swaption curve calibration in ORE (not parallelizable) and 1.4 times faster for calibrating the LMM model by a set of European options.

q-fin.CP↗

arXiv2020

Coulomb branch of a multiloop quiver gauge theory

We compute the Coulomb branch of a multiloop quiver gauge theory for the quiver with a single vertex, $r$ loops, one-dimensional framing, and $\dim V=2$. We identify it with a Slodowy slice in the nilpotent cone of the symplectic Lie algebra of rank $r$. Hence it possesses a symplectic resolution with $2r$ fixed points with respect to a Hamiltonian torus action. We also idenfity its flavor deformation with a base change of the full Slodowy slice.

math.AG↗

arXiv2019

Weil Conjectures I (translation of La Conjecture de Weil I by Pierre Deligne)

I attempted to write the full translation of this article to make the remarkable proof of Pierre Deligne available to a greater number of people. Overviews of the proofs can be found elsewhere. I especially recommend the notes of James Milne on Etale Cohomology that also contain a justification for the theory underlying this article and proofs of the results used by Deligne. The footnotes are mostly claims that some details appear in Milne, clarifications of some of the terminology or my personal struggles. I have also made a thorough overview of the proof together with more detailed explanations - arXiv:1807.10812. Enjoy!

math.AG↗

arXiv2019

Weil Conjectures Exposition

In this paper we provide a full account of the Weil conjectures including Deligne's proof of the conjecture about the eigenvalues of the Frobenius endomorphism. Section 1 is an introduction into the subject. Our exposition heavily relies on the Etale Cohomology theory of Grothendieck so I included an overview in Section 2. Once one verifies (or takes for granted) the results therein, proofs of most of the Weil conjectures are straightforward as we show in Section 3. Sections 4-8 constitute the proof of the remaining conjecture. The exposition is mostly similar to that of Deligne in [7] though I tried to provide more details whenever necessary. Following Deligne, I included an overview of Lefschetz theory (that is crucial for the proof) in Section 6. Section 9 contains a (somewhat random and far from complete) account of the consequences. Numerous references are mentioned throughout the paper as well as briefly discussed in Subsection 1.4.

math.AG↗