# Queen's University - Department of Mathematics and Statistics

 Tuesday, February 28 Seminar in Free Probability and Random Matrices Time: 2:30 p.m. Place: Jeffery 222 Speaker: Mario Diaz, Queen’s University Title: The linearization technique. Part II: non-commutative rational functions and their linearizations Abstract Attached Tuesday, February 28 Dynamics Seminar Time: 3:30 p.m. Place: Jeffery 422 Speaker: Francesco Cellarosi, Queen’s University Title: Introduction to SUSY quantum mechanics Abstract Attached Thursday, March 2 Math Club Time: 5:30 p.m. Place: Jeffery 118 Speaker: Ivan Dimitrov Title: The one-sentence proof Abstract Attached Friday, March 3 Number Theory Seminar Time: 10:30 a.m. Place: Jeffery 422 Speaker: Sacha Mangerel, University of Toronto Title: Some Applications of Pretentiousness in the Theory of Dirichlet Characters Abstract Attached Friday, March 3 Department Colloquium Time: 2:30 p.m. Place: Jeffery 234 Speaker: Enrico Arbarello, La Sapienza – Università di Roma Title: Hyperplane sections of K3 surfaces Abstract Attached

Items for the Info Sheet should reach Anne ( burnsa@queensu.ca) by noon on Monday. The Info Sheet is published every Tuesday.

Tuesday, February 28, 2:30 p.m. Jeffery 222

Seminar in Free Probability and Random Matrices

Speaker: Mario Diaz

Title: The linearization technique. Part II: non-commutative rational functions and their linearizations

Abstract: Last time we showed that every complex polynomial in non-commutative variables can be linearized into a linear polynomial with matricial coefficients. In this talk we will show that this is also true for a non-commutative rational function.

Seminar website: http://www.mast.queensu.ca/~mingo/seminar/

Tuesday, February 28, 3:30 p.m. Jeffery 422 - Dynamics Seminar

Speaker: Francesco Cellarosi

Title: Introduction to SUSY quantum mechanics

Abstract: I will give an elementary introduction to supersymmetric quantum mechanics, motivated by Dirac's derivation of the spectrum of the harmonic oscillator in 1935.

Thursday, March 2, 5:30 p.m. Jeffery 118 - Math Club

Speaker: Ivan Dimitrov

Title: The one-sentence proof

Abstract: This talk will discuss the one-sentence proof that every prime congruent to 1 modulo 4 can be written as a sum of squares.

Friday, March 3, 10:30 a.m. Jeffery 422 - Number Theory Seminar

Speaker: Sacha Mangerel

Title: Some Applications of Pretentiousness in the Theory of Dirichlet Characters

Abstract: ''Pretentious'' methods in analytic number theory, as introduced by Granville and Soundararajan, are powerful tools in the analysis of mean values and correlations of multiplicative functions. In this talk, we will give two applications of these tools to problems involving Dirichlet characters.

i) For a non-principal Dirichlet character $\chi$ modulo $q$, define $M(\chi)=\max_{t } |\sum_{n \leq t} \chi(n)|$. The classical P\'{o}lya-Vinogradov inequality asserts that $M(\chi) \ll \sqrt{q} \log q$ unconditionally, and on the Generalized Riemann Hypothesis (GRH), Montgomery and Vaughan showed that $M(\chi) \ll \sqrt{q} \log \log q$. We discuss a recent improvement, in joint work with Y. Lamzouri, to both of these results in the case that $\chi$ has odd order. This improvement on GRH is best possible up to a factor of $\log \log \log \log q$.

One of the key ingredients in the proof of the upper bounds is a new Hal\'asz-type inequality for logarithmic mean values of completely multiplicative functions.

ii) Time permitting, we will also discuss the following rigidity theorem related to binary correlations of Dirichlet characters.

If $f : \mathbb{N} \rightarrow \mathbb{C}$ is a 1-bounded multiplicative function for which there is a primitive Dirichlet character $\chi$ of conductor $q$ such that $$\sum_{n \leq x} f(n)\bar{f(n+h)} = (1+o(1))\sum_{n \leq x} \chi(n)\bar{\chi(n+h)}$$ for all shifts $|h| \leq H$, then if $H \rightarrow \infty$ with $x$ then $f$ ''pretends'' to be $\chi$ in a precise sense. This is joint work with O. Klurman.

Friday, March 3, 2:30 p.m. Jeffery 234 - Department Colloquium

Speaker: Enrico Arbarello

Title: Hyperplane sections of K3 surfaces

Abstract: K3 surfaces and their hyperplane sections play a central role in algebraic geometry. This is a survey of the work done during the past five years to characterize which smooth curves lie on a K3 surface. Related topics will be discussed. These are joint works with a combination of the following authors: Andrea Bruno, Gavril Farkas, Edoardo Sernesi and Giulia Saccà.

### Contact Info

 Department of Math & Stats Jeffery Hall, 48 University Ave. Kingston, ON Canada, K7L 3N6 Phone: (613) 533-2390 Fax: (613) 533-2964 mathstat@mast.queensu.ca Office Hours: 8:30am-12:00pm & 1:00pm-4:30pm