NTS ABSTRACTFall2024

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Sep 12

Non-reductive special cycles and arithmetic fundamental lemmas

Zhiyu Zhang

We care about arithmetic invariants of polynomial equations e.g. L-functions, which (conjecturally) are often automorphic and related to special cycles on Shimura varieties (or Shimura sets) based on the relative Langlands program. Arithmetic fundamental lemmas reveal such relations in the p-adic local world. In this talk, I will study certain ``universal'’ non-reductive special cycles on local GL_n Shimura varieties, and give applications e.g. the proof of twisted arithmetic fundamental lemma for the tuple (U_n, GL_n, U_n). Time permitting, I will explain some global analogs where at least the (Betti) cohomology class of special cycles could be defined. It turns out that algebraic special cycles are often pullbacks of ``universal’’ non-algebraic cycles (e.g. from Kudla-Millson theory on non-Hermitian symmetric spaces).

Sep 19

Random walks on group extensions

Lindenstrauss and Varju asked the following questions: for every prime p, let S_p be a symmetric generating set of G_p:=SL(2,F_p)xSL(2,F_p). Suppose the family of Cayley graphs {Cay(SL(2,F_p), pr_i(S_p))} is a family of expanders, where pr_i is the projection to the i-th component. Is it true that the family of Cayley graphs {Cay(G_p, S_p)} is a family of expanders? We answer this question and go beyond that by describing random-walks in various group extensions. (This is a joint work with Srivatsa Srinivas.)

Sep 26

Rational points on/near homogeneous hyper-surfaces

How many rational points are on/near a compact hyper-surface? This question is related to Serre's Dimension Growth Conjecture. We survey the state of the art, and explain a standard random model. Furthermore, we report on recent joint work with Rajula Srivastava (Uni/MPIM Bonn). Our arguments are rooted in Fourier analysis and, in particular, clarify the role of curvature in the random model.

Oct 31

Faster isomorphism testing of p-groups of Frattini class-2

The finite group isomorphism problem asks to decide whether two finite groups of order N are isomorphic. Improving the classical $N^{O(\log N)}$-time algorithm for group isomorphism is a long-standing open problem. It is generally regarded that p-groups of class 2 and exponent p form a bottleneck case for group isomorphism in general. The recent breakthrough by Sun (STOC '23) presents an $N^{O((\log N)^{5/6})}$-time algorithm for this group class. Our work sharpens the key technical ingredients in Sun's algorithm and further improves Sun's result by presenting an $N^{\tilde O((\log N)^{1/2})}$-time algorithm for this group class. Besides, we also extend the result to the more general p-groups of Frattini class-2, which includes non-abelian 2-groups. In this talk, I will present the problem background and our main algorithm in detail, and introduce some connections with other research topics. For example, one intriguing connection is with the maximal and non-commutative ranks of matrix spaces, which have recently received considerable attention in algebraic complexity and computational invariant theory. Results from the theory of Tensor Isomorphism complexity class (Grochow--Qiao, SIAM J. Comput. '23) are utilized to simplify the algorithm and achieve the extension to p-groups of Frattini class-2.