# Scientific

## String Topology 1

A lecture titled "String Topology" by Katherine Poirier, New York City College of Technology. 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.

- Read more about String Topology 1
- 804 reads

## Floer Homology Fundamentals 3

A lecture titled "Floer Homology Fundamentals" by Catherine Cannizzo, SCGP. This is the 3rd in a series of 9.

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.

## The equivalence of the Ekeland-Hofer and equivariant symplectic homology capacities

The Ekeland-Hofer capacities are some of the earliest symplectic capacities. They were defined without Floer theory and their calculation for ellipsoids and polydisks laid the foundation for the understanding of symplectic embeddings for a long time. More recently, Gutt and Hutchings defined a sequence of capacities using positive S^1 equivariant symplectic homology, which are harder to define, but much easier to compute. In this talk, I will explain how there is an isomorphism from the Hamiltonian Floer homology of a class of Hamiltonians to its H^{1/2}-Morse homology and how this implies that those two sequences of capacities coincide. This is joint work with J. Gutt.

## Floer Homology Applications 2

A lecture titled "Floer Homology Applications" by Jeff Hicks, University of Edinburgh. This is the 2nd 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.

## Spectra and Smash Products 3

A lecture titled "Spectra and Smash Products" by Cary Malkiewich, Binghamton University. This is the 3rd in a series of 4.

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.

## Moments of the Hurwitz zeta function

The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, for shift parameters $0<\alpha\leqslant 1$. We consider the integral moments of the Hurwitz zeta function on the critical line $\Re(s)=\frac12$. We focus on rational $\alpha$. In this case, the Hurwitz zeta function decomposes as a linear combination of Dirichlet $L$-functions, which leads us into investigating moments of products of $L$-functions. Using heuristics from random matrix theory, we conjecture an asymptotic of the same form as the moments of the Riemann zeta function. If time permits, we will discuss the case of irrational shift parameters $\alpha$, which will include some joint work with Winston Heap and Trevor Wooley and some ongoing work with Heap.

## An extension of Venkatesh's converse theorem to the Selberg class

In his thesis, Venkatesh gave a new proof of the classical converse theorem for modular forms of level~$1$ in the context of Langlands' ``Beyond Endoscopy". We extend his approach to arbitrary levels and characters. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree~$2$ gamma factors of Selberg class type.

This is joint work with Andrew R. Booker and Michael Farmer.

## Averages of long Dirichlet polynomials with modular coefficients

We study the moments of $L$-functions associated with primitive cusp forms, in the weight aspect. In particular, we present recent joint work with Brian Conrey, where we obtain an asymptotic formula for the twisted $r$-th moment of a long Dirichlet polynomial approximation of such $L$-functions. This result, which is conditional on the Generalized Lindel\"of Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.

## Local statistics for zeros of Artin--Schreier $L$-functions

We discuss the local statistics of zeros of $L$-functions attached to Artin--Scheier curves over finite fields, that is, curves defined by equations of the form $y^p-y=f(x)$, where $f$ is a rational function with coefficients in $F_q$ ($q$ a power of~$p$).

We consider three families of Artin--Schreier $L$-functions: the ordinary, polynomial (the $p$-rank $0$ stratum) and odd-polynomial families.

We present recent results on the $1$-level zero-density of the first and third families and the $2$-level density of the second family, for test functions with Fourier transform supported in suitable intervals. In each case we obtain agreement with a unitary or symplectic random matrix model.

## Moments of $L$-functions in the world of number field counting

We discuss some appearances of $L$-function moments in number field counting problems, with a particular focus on counting abelian extensions of number fields with restricted ramification.