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# Quantum money from quaternion algebras

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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.

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

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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

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**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.

**Speakers****: **Aaron 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.

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# Counting polynomials with a prescribed Galois group

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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

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(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.