Yujie Xu

I'm a Joseph F. Ritt Assistant Professor and NSF Research Fellow at Columbia University. My research is partially funded by an NSF (U.S. National Science Foundation) award.

I received my PhD in May 2022 from Harvard University, and spent the academic year 2022-2023 at MIT. Previously, I was an undergraduate student at Caltech (B.S. 2017, class rank no.1).

I can be reached at xu.yujie[at]columbia[dot]edu or yujiexu[at]mit[dot]edu or yujiexu[at]alumni[dot]harvard[dot]edu (note that my math.harvard.edu email address no longer works and incoming mails will not be forwarded).

CV available upon request.

Research

I'm interested in Number Theory (NT), Representation Theory (RT) and Arithmetic Geometry (AG).

Some of my papers (chronological order; with math subject classification):

Uniformizing the moduli stacks of global G-Shtukas, with U. Hartl (2023); [AG/NT: p-adic Hodge theory]

We show that the moduli spaces of bounded global G-Shtukas with colliding legs admit p-adic uniformization isomorphisms by Rapoport-Zink spaces. Moreover, we deduce the Langlands-Rapoport Conjecture over function fields in the case of colliding legs using our uniformization theorem.

The Explicit Local Langlands Correspondence for GSp4, Sp4 and stability (with an application to Modularity Lifting), with K. Suzuki, (2023); [RT/NT: geometric representation theory, with applications to number theory] (Here is a recording of a conference talk related to this work.)

We give a purely local proof of the explicit Local Langlands Correspondence for GSp(4) and Sp(4). Moreover, we give a unique characterization in terms of stability of L-packets and other properties. Finally, in the appendix, we give an application of our explicit local Langlands correspondence to modularity lifting.

The Explicit Local Langlands Correspondence for G2 II: character formulas and stability, with K. Suzuki, (2023) [RT/NT: geometric representation theory, with applications to number theory] (Here is a recording of a conference talk related to this work.)

We write down character formulas for representations of G_2 considered in [LLC-G2], and show that stability for L-packets uniquely pins down the Local Langlands Correspondence constructed in [LLC-G2], thus proving unique characterization of the LLC in loc.cit.

Geometrization of the Satake transform for mod p Hecke algebras, with R. Cass (2022) [RT/NT: geometric representation theory, with applications to number theory]

We geometrize the mod p Satake isomorphism of Herzig and Henniart--Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirkovic--Vilonen cycles, for the Satake transform of an arbitrary pararhoric mod p Hecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric mod p Hecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic.

The Explicit Local Langlands Correspondence for G2, with A.-M. Aubert (2022); [RT/NT: geometric representation theory, with applications to number theory] (Here is a recording of a conference talk related to this work.)

We develop a general strategy for constructing explicit Local Langlands Correspondences for p-adic reductive groups via reduction to LLC for supercuspidal representations of proper Levi subgroups, using Hecke algebra techniques. As an example of our general strategy, we construct explicit Local Langlands Correspondence for the exceptional group G_2 over a nonarchimedean local field, with explicit L-packets and explicit matching between the group and Galois sides. We also give a list of characterizing properties for our LLC. For intermediate series, we build on our previous results on Hecke algebras. Moreover, we show the existence of non-unipotent singular supercuspidal representations of G_2, and exhibit them in mixed L-packets mixing supercuspidal representations with non-supercuspidal ones. Furthermore, our LLC satisfies a list of expected properties, including the compatibility with cuspidal support.

The connected components of affine Deligne--Lusztig varieties, with I. Gleason and D.G. Lim (2022); [AG/NT/RT: p-adic Hodge theory, p-adic geometry, with applications to representation theory]

We compute the connected components of arbitrary parahoric level affine Deligne--Lusztig varieties and local Shimura varieties, thus resolving the conjecture raised in [He] in full generality (even for non-quasisplit groups). We achieve this by relating them to the connected components of infinite level moduli spaces of p-adic shtukas, where we use v-sheaf-theoretic techniques such as the specialization map of kimberlites. Along the way, we give a p-adic Hodge-theoretic characterization of HN-irreducibility. As applications, we obtain many results on the geometry of integral models of Shimura varieties at arbitrary parahoric levels. In particular, we deduce new CM lifting results on integral models of Shimura varieties for quasi-split groups at arbitrary connected parahoric levels (which were used to strengthen some results in my integral model paper).

Hecke algebras for p-adic reductive groups and Local Langlands Correspondence for Bernstein blocks, with A.-M. Aubert (2022);

We construct local Langlands correspondences for Bernstein blocks for arbitrary reductive groups using Hecke algebra techniques. Some results from §4 are used by the same authors to construct a full local Langlands correspondence in [AX-LLC-G2]. Moreover, we also prove a reduction to depth zero result for the Bernstein components attached to regular supercuspidal representations of Levi subgroups.

A comparison of the Kisin-Pappas model with Rapoport-Smithling-Zhang model (2022); [AG/NT: p-adic Hodge theory]

We prove that Kisin-Pappas integral models for Shimura varieties agree with Rapoport-Smithling-Zhang's exotic construction, using methods inspired from my PEL paper and my normalization paper. This is included as an appendix to C. Qiu's "Modularity of arithmetic special divisors for unitary Shimura varieties" and serves as a technical ingredient for the proof of the main results.

Normalization in integral models of Shimura varieties of abelian type (2020); [AG/NT: p-adic Hodge theory, algebraic geometry]

(Here is a recording of a talk related to an earlier version of this work. The final version of the main theorem has been strengthened using my joint paper on ADLV.)

We show that the normalization step in the construction of integral models for abelian type Shimura varieties is redundant, and that the flat closure model is already smooth at hyperspecial level (resp. normal at parahoric level). As a consequence, we show that there exist closed embeddings of toroidal compactifications of integral models of Hodge type into toroidal compactifications of Siegel integral models, for suitable choices of cone decompositions.

Finiteness of fppf cohomology (2021); [AG/NT: p-adic Hodge theory, algebraic geometry]

We show that H^1_fppf (Spec R, G) is finite, for R a Henselian DVR with finite residue field and G a finite type, flat R-group scheme (not necessarily commutative) with smooth generic fiber. We then give an application of the global analogue of this finiteness result to PEL type integral models of Shimura varieties.

On the Hodge embedding for PEL type integral models of Shimura varieties (2021); [AG/NT: p-adic Hodge theory, algebraic geometry]

We give a simple proof that Kottwitz’s PEL type integral models of Shimura varieties admit closed embeddings into Siegel integral models. We also show that Rapoport’s and Kottwitz’s integral models agree with Kisin’s integral models for relevant Shimura data.

Bounding Selmer Groups for the Rankin-Selberg Convolution of Coleman Families, with A. Graham and D. Gulotta (2020); [AG/NT: number theory]

For two cuspidal modular forms f and g, consider a Coleman family passing through f. For an open affinoid subdomain V of weight space W, we construct a coherent sheaf over V x W which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e the range where the p-adic L-function L_p interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of L_p.

Surjectivity of Galois representations in rational families of abelian varieties, with A. Landesman, A. Swaminathan and J. Tao (2019); [AG/NT: arithmetic geometry]

We show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-one subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension g greater than or equal to three, there are infinitely many abelian varieties over Q with adelic Galois representation having image equal to all of GSp_2g(\widehat{\mathbb Z}).

Some old REU/SURF (undergraduate) papers:

(REU="Research Experience for Undergraduates" in the U.S., and SURF="Summer Undergraduate Research Fellowships" at Caltech)

Lifting subgroups of symplectic groups over \mathbb{Z}/\ell\mathbb{Z}, with A. Landesman, A. Swaminathan and J. Tao (2017); [AG/NT: arithmetic geometry]

Hyperelliptic curves with maximal Galois action on the torsion points of their Jacobians, with A. Landesman, A. Swaminathan and J. Tao (2017); [AG/NT: arithmetic geometry]

Quantum statistical mechanics in Arithmetic Topology, with M. Marcolli (2017); [Algebraic topology, Mathematical Physics]


Other Writings

Here is (the expository part of) my Harvard minor thesis on the Geometric Langlands Conjectures.


Teaching

Instructor/Lecturer for Math 21b Linear Algebra and Differential Equations, Harvard University (Awarded a Bok Center Certificate of Distinction and Excellence in Teaching)

Here you can find excerpts of some extremely positive student comments from the freshman linear algebra class I taught at Harvard.

Teaching Fellow for Math 221 Commutative Algebra, Fall 2019, Harvard University (Awarded a Bok Center Certificate of Distinction and Excellence in Teaching)

Teaching Fellow for Math 137 Algebraic Geometry, Spring 2020, Harvard University (due to the pandemic, this course switched to Zoom instruction halfway, and there was no quantitive student evaluation nor Bok Center Teaching Award given out)

Teaching Fellow, Math 122 Algebra I: Theory of Groups and Vector Spaces, Fall 2020 (Zoom instruction), Harvard University (Awarded a Bok Center Certificate of Distinction and Excellence in Teaching)

Elliptic Curves and Beyond, Summer Tutorial 2019, Harvard University

Seminars

The STAGE seminar in Spring 2023 is on Shimura Varieties. Here are my English translations of Deligne's Trauvaux de Shimura and Varieties de Shimura. (Let me know if you spot typos, errors etc.)

I co-organized the STAGE seminar in Fall 2021 on Uniform Mordell. You can find notes for some of the talks here.

MIT-WIM

Accessibility


Last revised: October 2023

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