Scientific

The recipe for moments of $L$-functions

Speaker: 
Siegfried Baluyot
Date: 
Mon, Jul 25, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith formulated a `recipe' that leads to detailed conjectures for the asymptotic behavior of moments of various families of $L$-functions. In this talk, we will survey recent progress towards their conjectures and explore connections with different subjects.

Class: 
Scientific
Subject: 
Mathematics
Number Theory

One-level density of zeros of Dirichlet L-functions over function fields

Speaker: 
Hua Lin
Date: 
Mon, Jul 25, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

We compute the one-level density of zeros of order-$\ell$ Dirichlet $L$-functions over function fields $\mathbb{F}_q[t]$ for $\ell=3,4$ in the Kummer setting ($q\equiv1\pmod{\ell}$) and for $\ell=3,4,6$ in the non-Kummer setting ($q\not\equiv1\pmod{\ell}$). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and a lower order term not predicted by RMT. We also confirm the symmetry type of the family is unitary, supporting the Katz and Sarnak philosophy.

Class: 
Scientific
Subject: 
Mathematics
Number Theory

Distributions of sums of the divisor function over function fields

Speaker: 
Matilde Lalín
Date: 
Mon, Jul 25, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 
The goal of this talk is to discuss the variance of sums of the divisor function leading to certain random matrix distributions. While the knowledge of these problems is quite limited over the natural numbers, much more is known over function fields. We will start by introducing the basics of zeta functions and $L$-functions over function fields. We will then discuss the work of Keating, Rodgers, Roditty-Gershon, and Rudnick on the sums over arithmetic progressions, leading to distributions over unitary matrices by the Katz and Sarnak philosophy and a general conjecture over the natural numbers. Finally, we will present some recent work (in collaboration with Kuperberg) on sums over squares modulo a prime leading to symplectic distributions.
Class: 
Scientific
Subject: 
Mathematics

Selberg's central limit theorem for quadratic Dirichlet $L$-functions over function fields

Speaker: 
Allysa Lumley
Date: 
Mon, Jul 25, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 
In this talk, we will discuss the logarithm of the central value $L\left(\frac{1}{2}, \chi_D\right)$ in the symplectic family of Dirichlet $L$-functions associated with the hyperelliptic curve of genus $g$ over a fixed finite field $\mathbb{F}_q$ in the limit as $g\to \infty$. Unconditionally, we show that the distribution of $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is asymptotically bounded above by the full Gaussian distribution of mean $\frac{1}{2}\log \deg(D)$ and variance $\log \deg(D)$, and also $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is at least $94.27 \%$ Gaussian distributed. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.
Class: 
Scientific
Subject: 
Mathematics
Number Theory

Floer Homotopy 4

Speaker: 
Mohammed Abouzaid
Date: 
Wed, Jul 20, 2022
Location: 
PIMS, University of British Columbia, Zoom, Online
Conference: 
Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory
Abstract: 

A lecture titled "Floer Homotopy" by Mohammed Abouzaid, Columbia University. This is the 4th in a series of 4.

General Description:
The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

Class: 
Scientific
Subject: 
Mathematics

Floer Homology Applications 1

Speaker: 
Jeff Hicks
Date: 
Wed, Jul 20, 2022
Location: 
PIMS, University of British Columbia, Zoom, Online
Conference: 
Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory
Abstract: 

A lecture titled "Floer Homology Applications" by Jeff Hicks, University of Edinburgh. This is the 1st in a series of 3.

General Description:
The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

Class: 
Scientific
Subject: 
Mathematics

Spectra and Smash Products 2

Speaker: 
Cary Malkiewich
Date: 
Wed, Jul 20, 2022
Location: 
PIMS, University of British Columbia, Zoom, Online
Conference: 
Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory
Abstract: 

A lecture titled "Spectra and Smash Products" by Cary Malkiewich, Binghamton University. This is the 2nd in a series of 4.

General Description:
The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

Class: 
Scientific
Subject: 
Mathematics

Floer Homotopy 3

Speaker: 
Mohammed Abouzaid
Date: 
Tue, Jul 19, 2022
Location: 
PIMS, University of British Columbia, Zoom, Online
Conference: 
Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory
Abstract: 

A lecture titled "Floer Homotopy" by Mohammed Abouzaid, Columbia University. This is the 3rd in a series of 4.

General Description:
The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

Class: 
Scientific
Subject: 
Mathematics

Floer Homotopy 2

Speaker: 
Mohammed Abouzaid
Date: 
Tue, Jul 19, 2022
Location: 
PIMS, University of British Columbia, Zoom, Online
Conference: 
Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory
Abstract: 

A lecture titled "Floer Homotopy" by Mohammed Abouzaid, Columbia University. This is the 2nd in a series of 4.

General Description:
The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

Class: 
Scientific
Subject: 
Mathematics

A knot Floer stable homotopy type

Speaker: 
Ciprian Manolescu
Date: 
Mon, Jul 18, 2022
Location: 
PIMS, University of British Columbia, Zoom, Online
Conference: 
Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory
Abstract: 

Given a grid diagram for a knot or link K in the three-sphere, we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)

Class: 
Scientific
Subject: 
Mathematics

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