About
This online conference is hosted by the Department of Mathematics
and the Shenzhen International Center for Mathematics at
SUSTech.
We intend to bring experts in both Number Theory and Topology to share recent advances in these related fields, and to foster
interdisciplinary communication and collaboration.
Speakers
Weiyan Chen, Tsinghua University
Xing Gu, Westlake University
Yiqin He, Peking University
Yongquan Hu, Chinese Academy of Sciences
Zhen Huan, Huazhong University of Science and Technology
Zicheng Qian, Chinese Academy of Sciences
Guozhen Wang, Fudan University
Haoran Wang, Tsinghua University
Seokbeom Yoon, Southern University of Science and Technology
Ningchuan Zhang, University of Pennsylvania
Bin Zhao, Capital Normal University
Schedule (zoom: 863 4229 4780 / passcode: aritopo)
All talks will be China Standard Time (GMT+8)
Friday, December 16
8:50–9:00 am
Opening remarks
9:00–9:50 am
Guozhen Wang Computations of topological cyclic homology
Algebraic $K$-theory for a field is the classification of vector spaces including higher homotopical information.
The cyclotomic trace map from algebraic $K$-theory to topological cyclic homology is a generalization of the Chern
character which approximates $K$-theory by ordinary homology. We will give an introduction to some new developments
in the methods for computing topological cyclic homology and its applications.
Slides
10:00–10:50 am
Weiyan Chen Choosing points on cubic plane curves
It is a classical topic to study structures of certain special points on smooth complex cubic plane curves,
for example, the 9 flex points and the 27 sextactic points. We study the following topological question asked by Farb:
Is it true that the known algebraic structures give all the possible ways to continuously choose $n$ distinct points on every smooth
cubic plane curve, for each given integer $n$? This work is joint with Ishan Banerjee.
Notes
11:00–11:50 am
Zhen Huan Twisted Real quasi-elliptic cohomology
Quasi-elliptic cohomology is closely related to Tate $K$-theory. It is constructed as an object both
reflecting the geometric nature of elliptic curves and more practicable to study than most elliptic cohomology theories.
It can be interpreted by orbifold loop spaces and expressed in terms of equivariant $K$-theories.
We formulate the complete power operation of this theory. Applying that we prove the finite subgroups of Tate curve
can be classified by the Tate $K$-theory of symmetric groups modulo a certain transfer ideal.
In this talk we construct twisted Real quasi-elliptic cohomology as the twisted $KR$-theory of loop groupoids.
The theory systematically incorporates loop rotation and reflection. After establishing basic properties of the theory,
we construct Real analogues of the string power operation of quasi-elliptic cohomology. We also explore the relation of
the theory to the Tate curve. This is joint work with Matthew Young.
Slides
2:00–2:50 pm
Yongquan Hu Multivariable ($\varphi$,
${\mathcal O}_K^\times$)-modules and mod $p$
representations of ${\rm GL}_2$
Let $p$ be a prime number, $K$ a finite unramified extension of ${\mathbb Q}_p$, and $\pi$ a smooth representation of
${\rm GL}_2(K)$ on some Hecke eigenspace in mod $p$ cohomology of Shimura curves. One can
associate to $\pi$ a multivariable $(\varphi, {\mathcal O}_K^\times)$-module. The aim of this talk is to explain the construction
and some recent results around it. This is joint work with C. Breuil, F. Herzig, S. Morra and B.
Schraen.
Notes
3:00–3:50 pm
Haoran Wang On some mod $p$ representations of quaternion algebra over ${\mathbb Q}_p$
Scholze proposed a mod $p$ Jacquet–Langlands correspondence for ${\rm GL}(n, K)$, where $K$ is a
finite extension of ${\mathbb Q}_p$. I will discuss some results about Scholze's functors in the case of ${\rm GL}(2, {\mathbb Q}_p)$.
This is a joint work with Yongquan Hu.
Notes
4:00–4:50 pm
Bin Zhao Slopes of modular form and ghost conjecture
In 2016, Bergdall and Pollack raised a conjecture towards the computation of the $p$-adic slopes of Hecke cuspidal eigenforms whose associated $p$-adic Galois representations satisfy the
assumption that their mod $p$ reductions become reducible when restricted to the $p$-decomposition
group. In this talk, I will report a joint work with Ruochuan Liu, Nha Truong and Liang Xiao to
prove this conjecture under mild assumptions. I will first give the statement of this conjecture and
explain the intuition behind its formulation. I will then explain some key strategies in our proof. If
time permits, I will mention some arithmetic consequences of this conjecture.
Slides
Saturday, December 17
10:00–10:50 am
Ningchuan Zhang A Quillen–Lichtenbaum Conjecture for Dirichlet $L$-functions
The original version of the Quillen–Lichtenbaum Conjecture, proved by Voevodsky and Rost,
connects special values of the Dedekind zeta function of a number field with its algebraic $K$-groups.
In this talk, I will discuss a generalization of this conjecture to Dirichlet $L$-functions.
The key idea is to twist algebraic $K$-theory spectra of number fields with equivariant Moore spectra
associated to Dirichlet characters. Rationally, we obtain a Quillen–Borel type theorem for Artin $L$-functions.
This is joint work in progress with Elden Elmanto.
Slides
11:00–11:50 am
Xing Gu The ordinary and motivic cohomology of $B{\rm PGL}_n({\mathbb C})$
For an algebraic group $G$ over $\mathbb C$, we have the classifying space
$BG$ in the sense of Totaro and Voevodsky, which is an object in the unstable
motivic homotopy category that plays a similar role in algebraic geometry as
the classifying space of a Lie group in topology.
The motivic cohomology (in particular, the Chow ring) of $BG$ is closely
related, via the cycle map, to the singular cohomology of the topological realization of $BG$,
which is the classifying space of $G({\mathbb C})$, the underlying Lie group
of the complex algebraic group $G$.
In this talk we present a work which exploits the above connection between
topological and motivic theory and yields new results on both the ordinary and
the motivic cohomology of $B{\rm PGL}_n({\mathbb C})$, the complex projective linear group.
Slides
2:00–2:50 pm
Seokbeom Yoon Twisted Neumann–Zagier matrices and loop invariants
Let $M$ be a hyperbolic 3-manifold with cusps and $F$ be its trace field.
For a geometric or topological invariant of $M$ that lies in $F$,
it is natural to study its behavior under finite cyclic covers of $M$.
In this talk, we focus on loop invariants, derived from perturbative Chern–Simons theory, and
describe their behaviors under cyclic covers in terms of twisted Neumann–Zagier matrices.
This is joint work with Stavros Garoufalidis.
Notes
3:00–3:50 pm
Zicheng Qian On Breuil–Schraen $\mathscr L$-invariants for ${\rm GL}_n$
The study of Breuil–Schraen $\mathscr L$-invariants is motivated by that of Steinberg case of $p$-adic Langlands correspondence.
Many results are known for ${\rm GL}(2, {\mathbb Q}_p)$ and ${\rm GL}(3, {\mathbb Q}_p)$ due to work of
Breuil, Ding and Schraen. In this talk, we sketch a few recent progress towards general ${\rm GL}_n$.
Notes
4:00–4:50 pm
Yiqin He Parabolic simple $\mathscr L$-invariants and local–global compatibility
Let $L$ be a finite extension of ${\mathbb Q}_p$ and $\rho_L$ be a potentially semistable noncrystalline
$p$-adic representation of ${\rm Gal}_L$ such that the associated $F$-semisimple Weil–Deligne representation is
absolutely indecomposable. Via a study of Breuil's parabolic simple $\mathscr L$-invariants, we attach to $\rho_L$
a locally ${\mathbb Q}_p$-analytic representation $\Pi(\rho_L)$ of ${\rm GL}_n(L)$, which carries the exact
information of the
Fontaine–Mazur parabolic simple $\mathscr L$-invariants of $\rho_L$. When $\rho_L$ comes from a patched automorphic
representation of $G({\mathbb A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$
which is compact at
infinite places and ${\rm GL}_n$ at $p$-adic places), we prove under mild hypothesis that $\Pi(\rho_L)$ is a
subrepresentation of the associated Hecke-isotypic subspace of the Banach spaces of (patched) $p$-adic
automophic forms on $G({\mathbb A}_{F^+})$.
Scientific Committee
Stavros Garoufalidis, Southern University of Science and Technology
Organizers
Hui Gao, Southern University of Science and Technology
Yifei Zhu, Southern University of Science and Technology
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