# Queen's University - Department of Mathematics and Statistics

 Tuesday, January 30 Seminar in Free Probability and Random Matrices Time: 3:30 p.m. Place: Jeffery 319 Speaker: Neha Prabhu, Queen’s University Title: Semicircle distribution in number theory, Part II Abstract Attached Wednesday, January 31 Number Theory Seminar Time: 2:15 p.m. Place: Jeffery 319 Speaker: Vaidehee Thatte Title: Defect Extensions - I Abstract Attached Thursday, February 1 Math Club Time: 5:30 p.m. Place: Jeffery 118 Speaker: Peter Taylor Title: The sum of the cubes is the square of the sum Abstract Attached Friday, February 2 Dynamics Seminar Time: 10:30 a.m. Place: Jeffery 422 Speaker: Francesco Cellarosi Title: Limit theorems for rotations Abstract Attached Friday, February 2 Department Colloquium Time: 2:30 p.m. Place: Jeffery 234 Speaker: Jory Griffin, Queen’s University Title: The Lorentz Gas – Macroscopic Transport from Microscopic Dynamics Abstract Attached Monday, February 5 Geometry and Representation Theory Seminar Time: 4:30 p.m. Place: Jeffery 319 Speaker: David Wehlau Title: Khovanski Bases and Derivations Abstract Attached Wednesday, February 7 Curves Seminar Time: 3:30 p.m. Place: Jeffery 319 Speaker: Daniel Erman, Wisconsin – Madison Title: Big polynomial rings and Stillman’s conjecture 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, January 30, 3:30 p.m. Jeffery 319

Seminar in Free Probability and Random Matrices

Speaker: Neha Prabhu

Title: Semicircle distribution in number theory, Part II

Abstract: In free probability theory, the role of the semicircle distribution is analogous to that of the normal distribution in classical probability theory. However, the semicircle distribution also shows up in

number theory: it governs the distribution of eigenvalues of Hecke operators acting on spaces of modular cusp forms. In this talk, I will give a brief introduction to this theory of Hecke operators and sketch the proof of a result which is a central limit type theorem from classical probability theory, that involves the semicircle measure.

Wednesday, January 31, 2:15 p.m. Jeffery 319 - Number Theory Seminar

Speaker: Vaidehee Thatte

Title: Defect Extensions – I

Abstract: Let $K$ be a valued field of characteristic $p > 0$ with henselian valuation ring $A$. Let $L$ be a non-trivial Artin-Schreier extension of $K$ with $B$ as the integral closure of $A$ in $L$. In

the classical theory of complete discrete valuation rings, $B$ is generated as an $A$-algebra by a single element. This in particular, is not true in the defect case. We will discuss a result that allows us to write $B$ as a "filtered union over $A$", when there is defect.

Similar results can be obtained in the mixed characteristic case.

Thursday, February 1, 5:30 p.m. Jeffery 118 - Math Club

Speaker: Peter Taylor

Title: The sum of the cubes is the square of the sum

Abstract: We have all seen the lovely formula: 13+23+33+⋯+n3=(1+2+3+⋯+n)2. Often attributed to the Greek mathematician Nicomachus of Gerasa (circa AD 100). Is this a "one-off" result or are there other simple examples of a sequence for which the sum of the cubes is the square of the sum? We will see that there are some unexpectedly simple examples of such sequences.

Friday, February 2, 10:30 a.m. Jeffery 422 - Dynamics Seminar

Speaker: Francesco Cellarosi

Title: Limit theorems for rotations

Abstract: I will review the rich literature on the various limit theorems one can achieve for ergodic averages of irrational rotation of the circle.

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

Speaker: Jory Griffin

Title: The Lorentz Gas – Macroscopic Transport from Microscopic Dynamics

Abstract: The Lorentz Gas is microscopic model for conductivity in which a point particle representing an electron moves through an infinite array of scatterers representing the background medium. On the macroscopic scale the dynamics can instead be modelled by the linear Boltzmann transport equation, an irreversible equation where motion of particles appears to be stochastic. How can these two pictures be reconciled? Can we 'derive' the macroscopic picture from the microscopic one? I will talk about the solution to this problem as well as its quantum mechanical analogue where much less is currently known.

Monday, February 5, 4:30 p.m. Jeffery 319 - Geometry and Representation Theory Seminar

Speaker: David Wehlau

Title: Khovanski Bases and Derivations

Abstract: Let $R=K[x_1,,\dots,x_m,y_1,\dots,y_m,z_1,\dots,z_m]$ be a polynomial algebera over a field $K$ of characteristic zero, Let $\Delta$ be the locally nilpotent derivation on $R$ determined by $\Delta(z_i) = y_i$, $\Delta(y_i) = x_i$ and $\Delta(x_i)=0$ for $i=1,2,\dots,m$. This is an example of

a Weitzenb\"ock derivation. We exhibit a minimal set of generators $\mathcal G$ for the algebra of

constants $R^\Delta = \ker \Delta$. We also construct a Khovanski (or sagbi) basis for this algebra.

Even though this basis is infinite our proof yields an algorithm to express any element of $R^\Delta$ as
a polynomial in the elements of $\mathcal G$. In particular, this method shows how the classical techniques of polarization and restitution may be used in combination with Khovanski bases to yield a constructive method for expressing elements of a subalgebra as a polynomials in its generators.

Wednesday, February 7, 3:30 p.m. Jeffery 319 - Curves Seminar

Speaker: Daniel Erman

Title: Big polynomial rings and Stillman’s conjecture

Abstract: Ananyan–Hochster's recent proof of Stillman's conjecture is based on a key principle: if f_1,.., f_r are sufficiently general forms in a polynomial ring, then as the number of variables tends to infinity, they will behave increasingly like independent variables. We show that this principle becomes a theorem if ones passes to a limit of polynomial rings, using either the inverse limit or the ultraproduct. This yields the surprising fact that these limiting rings are themselves polynomial rings (in uncountably many variables). It also yields two new proofs of Stillman's conjecture. This is joint work with Steven Sam and Andrew Snowden.

### 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