Talks and presentations
Seminar Talks
Nov 4 & 11, 2024, SUSTech Graduate Topology Seminar
Perspectives from differential and algebraic geometry on ruled surfaces
- In the first lecture, we will begin by reviewing some basic concepts of classical differential geometry like curvature and torsion within the context of three-dimensional Euclidean space. Then we will briefly recall the structure theorems pertaining to curves and surfaces. Following this foundation, we will concentrate on the properties and classifications of ruled surfaces and developable surfaces. We will then proceed to demonstrate several explicit examples that are naturally derived from discriminant varieties of polynomial equations, highlighting some intriguing generalizations along the way.
- Building upon last week's introduction to ruled surfaces, the second talk focuses on discriminant surfaces arising from characteristic polynomials of 3-order matrices with special symmetries. Our investigation is motivated by condensed matter physics, particularly the study of Hamiltonians exhibiting Parity-Time symmetry and Minkowski-like pseudo-Hermitian symmetry. We will present several explicit examples of such discriminant surfaces and explore their geometric properties. Notably, we will showcase numerous counterexamples of non-ruled discriminant surfaces, leading to an intriguing question: what conditions ensure that a discriminant surface is ruled?
- Slides
Sep 10, 2024, Workshop on Computer-assisted Research in Geometry and Topology, Kunming, China
Periodicity in classical and motivic homotopy theory
- In this lightning talk, I will introduce the concept of exotic periodicity in the motivic stable homotopy category. Additionally, I will discuss the discovery and detection of Krause's families of elements related to the $\beta_{ij}$ periodicity.
- Slides
May 9, 2024, SUSTech Applied and Computational Topology Seminar
The discovery of murmuration phenomenon for elliptic curves
- Topological, statistical, and machine learning methods in computational number theory
- Machine learning meets number theory: The data science of Birch–Swinnerton-Dyer, or arxiv version
- Nina Zubrilina - Murmurations (modular forms)
- Elliptic Curve "Murmurations" Found With AI Take Flight
- Advancing mathematics by guiding human intuition with AI, Nature
Main references:
Apr 16, 2024, SUSTech Graduate Topology Seminar
From Adams's $v_1$-periodicity to Andrews's motivic $w_1$-periodicity in homotopy groups
- The purpose of this talk is to introduce Michael Andrews's work on discovering exotic periodicity in the homotopy groups of the motivic sphere spectrum and the identification of six infinite families of elements. Firstly, we will provide a brief overview of the chromatic phenomenon in classical homotopy theory and Adams's 8-periodic isomorphism, expressed by a Massey product in the Adams spectral sequence. Then we will delve into motivic homotopy theory and explore the philosophy behind Haynes Miller's square, which serves as an analogy and connection between the motivic Adams spectral sequence and the classical Adams–Novikov spectral sequence. This perspective allows us to consider multiplication by the non-nilpotent element $\eta$ as a $w_0$-periodic map, similar to the role played by $2$ for the $v_0$- periodic map. After defining a Thom reduction map to compare the $E_2$-pages of two spectral sequences, we can reduce the existence of motivic $w_1^4$-periodic maps to the existence of classical $v_1^4$-periodic maps, as initially found by Adams in the early 1960s.
- New families in the homotopy of the motivic sphere spectrum - J. Andrews
- Notes
Apr 11, 2024, SUSTech Applied and Computational Topology Seminar
Categorical, statistical, and linear algebra methods in natural language processing
- With the recent success of large language models, exploring the underlying structure of linguistic meaning has become not only a question of linguistics and engineering but also a profound mathematical and philosophical problem. To what extent can established mathematical tools—such as mathematical analysis, linear algebra, probability, and statistics—describe and characterize the semantics of language? Remarkably, these seemingly fundamental theories, refined over many years, have already enabled groundbreaking engineering achievements. But what about more complex and abstract tools like category theory, topology, and others? Can they provide deeper insights into the nature of meaning?
- The purpose of this talk is to introduce the interesting ideas from Juan Luis Gastaldi’s paper in Notices of the American Mathematical Society, titled "The structure of meaning in language: Parallel narratives in linear algebra and category theory," to the research group of Prof. Yifei Zhu at SUSTech. We also extensively reference Gastaldi’s insightful and well-crafted slides, for which we express our sincere gratitude to the author. Main references:
- The structure of meaning in language: Parallel narratives in linear algebra and category theory
- Understanding Language (Models)-AI Language Models From a Structuralist Perspective
- The Language of Mathematics-Epistemological Consequences of Applying AI Methods to Mathematics
- Peeking into the AI Language Modeling Black Box
Mar 12, 2024, SUSTech Graduate Topology Seminar
An introduction to singularity theory, Milnor's fibration theorem and Brieskorn's construction of exotic spheres
- Through this lecture, I will provide an overview of the singularity theory of complex analytic functions and polynomials, appreciating its wide-ranging applications and contributions to various branches of mathematics. I will begin by introducing fundamental concepts and definitions in this field, then proceed to discuss the invariants and classification of simple singular points, highlighting their significance. Furthermore, I will illustrate the intriguing relationship between the ADE classification and other mathematical objects such as finite subgroups of the orthogonal group or the intersection form of a fiber manifold. This connection sheds light on the broader mathematical landscape and enriches our understanding of singularity theory. Moving forward, I will introduce the Milnor fibration of a singular point and describe its topological structure by Morse theory. Finally, I will provide a brief introduction to the groundbreaking work of Milnor and Kervaire on the discovery and classification of exotic diffeomorphism structures on 7-dimensional spheres. Additionally, I will touch upon Brieskorn's explicit construction, which involves a family of polynomial singularities that gives rise to all exotic spheres. This fascinating exploration demonstrates the profound impact of singularity theory in the realm of differential topology.
- Slides
Feb 17, 2024, Winter Holiday Online Seminar
An introduction to topological quantum field theory
- This self-organized winter seminar focused on an introductory exploration of Topological Quantum Field Theory (TQFT), emphasizing its foundational definitions and physical motivations. We began by contextualizing TQFT within statistical mechanics (via partition functions) and quantum mechanics principles, elucidating why TQFT is formally defined as a monoidal functor from the cobordism category to the category of vector spaces—specifically mapping disjoint unions of manifolds to tensor products of vector spaces. Central to our discussion was the application of TQFT in defining topological invariants for closed manifolds, revealing how its mathematical physics framework extracts global geometric and topological features of manifolds through axiomatic constructions. Furthermore, we established the equivalence between 2D TQFTs and Frobenius algebras, demonstrating how axiomatic properties of TQFTs naturally align with the algebraic framework of Frobenius pairs.
- Slides
Dec 19, 2023, SUSTech Doctoral Dissertation Proposal, M1001
Chromatic structures in motivic and equivariant stable homotopy categories
Nov 28, 2023, SUSTech Graduate Topology Seminar
On "The special fiber of the motivic deformation of the stable homotopy category is algebraic"
- In this talk, I will introduce the main content of the paper ''The special fiber of the motivic deformation of the stable homotopy category is algebraic'' by Bogdan Gheorghe, Guozhen Wang, and Zhouli Xu. They define a Chow-Novikov t-structure on the category of module spectra over cofiber $S^{0,0}/\tau$, and establish that this category is equivalent to the derived category of $BP_*BP$-comodule category as stable $\infty$-categories equipped with t-structures.
An exciting application is the isomorphism between the motivic Adams spectral sequence for $S^{0,0}/\tau$ and the algebraic Novikov spectral sequence. This isomorphism enables Isaksen, Wang, and Xu to compute the stable homotopy groups of spheres up to the 90-stem. - Slides
May 30, 2023, SUSTech Graduate Topology Seminar
Functor calculus and chromatic homotopy theory
May 11, 2023, SUSTech Undergraduate Geometry and Topology Seminar
Intoduction to Floer homology and knot Floer homology
- Through this lecture
- Slides
Oct 10, 2022, SUSTech Applied and Computational Topology Seminar
Periodicity detection of biomedical video and human speech signal
- A progress report investigates periodicity properties in video and audio data through two approaches: (1) detection of mouse scratching behavior through analysis of one-dimensional time series data generated by video frame summation, using sliding window embedding and geometric statistics; and (2) introduction of the moduli space theoretical framework for vowels and consonants in linguistics, with exploration of topological data analysis applications to phonetic signals.
May 26, 2022, BNU Master's Thesis Defense
Some topics on smooth quandles
- Quandles are algebraic structures derived from knots theory, smooth quandles have both algebraic structures and compatible smooth manifold structures, so they are rich in content. In this thesis, we study the construction and classification of smooth quandles on homogeneous spaces in detail. Then given a fixed Lie group, we define a series of mapping type knot invariants and compute this invariant for all torus knots and Borromean links. At last, we study the deformation of smooth quandle structures, and define a new quandle cohomology theory and corresponding Kodaira-Spencer mapping by analogizing the deformation theories of complex geometry and associative algebra.
- Slides