Blocking sets arising from plane curves over finite fields

Speaker: 

Shamil Asgarli

Institution: 

Santa Clara University

Time: 

Thursday, October 6, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

Let F_q be a finite field, and consider the set P^2(F_q) of all F_q-points in the projective plane. A subset B of P^2(F_q) is called a blocking set if B meets every line defined over F_q. Given an algebraic plane curve C in P^2, when does the set of F_q-rational points on C form a blocking set? We will see that curves of low degree do not give rise to blocking sets. As an example of this principle, we will show that cubic plane curves defined over F_q do not give rise to blocking sets whenever q is at least 5. On the other hand, we will describe explicit constructions of smooth plane curves (of large degree) that do give rise to blocking sets. Finding blocking curves of optimal degree over a given finite field remains open. This is joint work with Dragos Ghioca and Chi Hoi Yip.

Quantum money from quaternion algebras

Speaker: 

Shahed Sharif

Institution: 

Cal State University, San Marcos

Time: 

Thursday, October 13, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

Public key quantum money is a replacement for paper money which has cryptographic guarantees against counterfeiting. We propose a new idea for public key quantum money. In the abstract sense, our bills are encoded as a joint eigenstate of a fixed system of commuting unitary operators. We show that the proposal is secure against black box attacks. In order to instantiate this protocol, one needs to find a cryptographically complicated system of computable, commuting, unitary operators. To fill this need, we propose using Brandt operators acting on the Brandt modules associated to certain quaternion algebras. This is joint work with Daniel Kane and Alice Silverberg.

Radical Geometric Monogenicity

Speaker: 

Hanson Smith

Institution: 

Cal State University, San Marcos

Time: 

Thursday, November 3, 2022 - 3:00pm to 4:00pm

Location: 

RH 306
A number field is $\textit{monogenic}$ (over $\mathbb{Q}$) if the ring of integers admits a power integral basis, i.e., a $\mathbb{Z}$-basis of the form $\{1, \alpha, \alpha^2,\dots, \alpha^{n-1}\}$. In this case we call $\alpha$ a $\textit{monogenerator}$. The first portion of the talk will be spent revisiting some classical examples of monogenicity and non-monogenicity. We will touch on some recent work on radical extensions while paying particular attention to obstructions to monogenicity and relations to other arithmetic questions. The latter part of the talk will be devoted to recent work ([1] and [2]) constructing a general moduli space of monogenerators. Specifically, given an extension of algebras $B/A$, we construct a $\textit{scheme}$ $\mathcal{M}_{B/A}$ parameterizing the possible choices of a monogenerator for $B$ over $A$. $$ \ $$ $$\textbf{References} $$ [1] Arpin, S., Bozlee, S., Herr, L., and Smith, H. (2021). The Scheme of Monogenic Generators I: Representability. arXiv: https://arxiv.org/abs/2108.07185. (Accepted to Research in Number Theory.) $$ \ $$ [2] Arpin, S., Bozlee, S., Herr, L., and Smith, H. (2022). The Scheme of Monogenic Generators II: Local Monogenicity and Twists. arXiv: https://arxiv.org/abs/2205.04620.

Matrix enumeration over finite fields (Note the special day!)

Speaker: 

Yifeng Huang

Institution: 

UBC

Time: 

Tuesday, September 27, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

I will investigate certain matrix enumeration problems over a finite field, guided by the phenomenon that many such problems tend to have a generating function with a nice factorization. I then give a uniform and geometric explanation of the phenomenon that works in many cases, using the statistics of finite-length modules (or coherent sheaves) studied by Cohen and Lenstra. However, my recent work on counting pairs of matrices of the form AB=BA=0 (arXiv: 2110.15566) and AB=uBA for a root of unity u (arXiv: 2110.15570), through purely combinatorial methods, gives examples where the phenomenon still holds true in the absence of the above explanation. Time permitting, I will talk about a partial progress on the system of equations AB=BA, A^2=B^3 in a joint work with Ruofan Jiang. In particular, it verifies a pattern that I previously conjectured in an attempt to explain the phenomenon in the AB=BA=0 case geometrically.

Southern California Number Theory Day

Speaker: 

Aaron Landesman, Michelle Manes, Holly Swisher, Stanley Xiao

Institution: 

Harvard University, University of Hawaii, Oregon State University, University of Northern British Columbia

Time: 

Saturday, September 24, 2022 - 9:30am to 5:30pm

Location: 

NS II 1201

Schedule: There will be four one hour invited lectures starting at 10AM and ending around 5:30PM.  A more detailed schedule will be posted soon.

SpeakersAaron Landesman (Harvard University), Michelle Manes (University of Hawaii), Holly Swisher (Oregon State University), Stanley Xiao (University of Northern British Columbia)

Lightning Talks: We are planning a session where number theory graduate students and postdocs are invited to present their research. These talks will be approximately 5-10 minutes.  If you would like to give a lightning talk, please contact Nathan Kaplan by September 9. Please include your name, affiliation, advisor's name, talk title, and a brief abstract.

Registration: There is no registration fee for the conference, but to help our planning please register.

Location: Natural Sciences II, room 1201 (building 402, located at G6 on this map).

Travel support: Some travel funding is available for participants, with preference given to graduate students and postdocs, especially those giving lightning talks. We also encourage applications from members of under-represented groups. If you would like to apply for funding, please contact Nathan Kaplan with an itemized estimate of expenses, preferably by September 16. Please include your name, affiliation, and advisor's name (if applicable). We strongly encourage carpooling.

Dinner: There will be a conference dinner.  Details TBD.

Degree d points on plane curves

Speaker: 

Lea Beneish

Institution: 

UC Berkeley

Time: 

Thursday, October 27, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

Given a plane curve C defined over Q, when the genus of the curve is greater than one, Faltings’ theorem tells us that the set of rational points on the curve is finite. It is then natural to consider higher degree points, that is, points on this curve defined over fields of degree d over Q. We ask for which natural numbers d are there points on the curve in a field of degree d. There is a lot of structure in the set of values d, some of which I will explain in this talk. This talk is based on joint work with Andrew Granville.

A p-adic analogue of an algebraization theorem of Borel

Speaker: 

Abhishek Oswal

Institution: 

Caltech

Time: 

Thursday, November 17, 2022 - 3:00pm to 4:00pm

Location: 

RH 306
Let S be a Shimura variety such that the connected components of the set of complex points $S(\mathbb{C})$ are of the form $D/\Gamma$, where $\Gamma$ is a torsion-free arithmetic group acting on the Hermitian symmetric domain $D$. Borel proved that any holomorphic map from any complex algebraic variety into $S(\mathbb{C})$ is an algebraic map. In this talk I shall describe ongoing joint work with Ananth Shankar and Xinwen Zhu, where we prove a $p$-adic analogue of this result of Borel for compact Shimura varieties of abelian type.

Counting polynomials with a prescribed Galois group

Speaker: 

Vlad Matei

Institution: 

Tel-Aviv

Time: 

Thursday, April 7, 2022 - 10:00am to 11:00am

Location: 

https://uci.zoom.us/j/95268809663

An old problem, dating back to Van der Waerden, asks about counting irreducible polynomials degree $n$ polynomials with coefficients in the box [-H,H] and prescribed Galois group. Van der Waerden was the first to show that H^n+O(H^{n-\delta}) have Galois group S_n and he conjectured that the error term can be improved to o(H^{n-1}). 

Recently, Bhargava almost proved van der Waerden conjecture showing that there are O(H^{n-1+\varepsilon}) non S_n extensions, while Chow and Dietmann showed that there are O(H^{n-1.017}) non S_n, non A_n extensions for n>=3 and n\neq 7,8,10. 

In joint work with Lior Bary-Soroker, and Or Ben-Porath we use a result of Hilbert to prove a lower bound for the case of G=A_n, and upper and lower bounds for C_2 wreath S_{n/2} . The proof  for A_n can be viewed, on the geometric side,  as constructing a morphism \varphi from A^{n/2} into the variety z^2=\Delta(f) where each varphi_i is a quadratic form.  For the upper bound for C_2 wreath S_{n/2} we prove a monic version of Widmer's result four counting polynomials with imprimitive Galois group.

Gaussian distribution of squarefree and B-free numbers in short intervals

Speaker: 

Alexander Mangerel

Institution: 

Durham University

Time: 

Thursday, March 31, 2022 - 10:00am to 11:00am

Location: 

https://uci.zoom.us/j/95268809663

(Joint with O. Gorodetsky and B. Rodgers) It is a classical quest in analytic number theory to understand the fine-scale distribution of arithmetic sequences such as the primes. For a given length scale h, the number of elements of a ``nice'' sequence in a uniformly randomly selected interval $(x,x+h], 1 \leq x \leq X$, might be expected to follow the statistics of a normally distributed random variable (in suitable ranges of $1 \leq h \leq X$).  Following the work of Montgomery and Soundararajan, this is known to be true for the primes, but only if we assume several deep and long-standing conjectures such as the Riemann Hypothesis. In fact, previously such distributional results had not been proven for any (non-trivial) sequence of number-theoretic interest, unconditionally.

 
As a model for the primes, in this talk I will address such statistical questions for the sequence of squarefree numbers, i.e., numbers not divisible by the square of any prime, among other related ``sifted'' sequences called B-free numbers. I hope to further motivate and explain our main result that shows, unconditionally, that short interval counts of squarefree numbers do satisfy Gaussian statistics, answering several old questions of R.R. Hall.

 

The negative Pell equation and applications

Speaker: 

Peter Koymans

Institution: 

Univ. of Michigan

Time: 

Thursday, April 21, 2022 - 10:00am to 11:00am

Location: 

https://uci.zoom.us/j/95268809663

In this talk we will study the negative Pell equation, which is the conic $C_D : x^2  - D y^2 = -1$ to be solved in integers $x, y \in \mathbb{Z}$. We shall be concerned with the following question: as we vary over squarefree integers $D$, how often is $C_D$ soluble? Stevenhagen conjectured an asymptotic formula for such $D$. Fouvry and Kluners gave upper and lower bounds of the correct order of magnitude. We will discuss a proof of Stevenhagen's conjecture, and potential applications of the new proof techniques. This is joint work with Carlo Pagano.

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