# Mini-conference on singularities and motivic integration

## November 7-10, 2016

Motivic integration is a relatively new subject which was actively developed in the last couple of decades, starting from the works of Kontsevich, Denef and Loeser. It belongs to the world of algebraic geometry, but has close connections to topology, number theory (in particular, p-adic integration and Langlands program), representation theory and even string theory.

Organizers: Eugene Gorsky (UC Davis) and Michel Raibaut (Universite de Savoie, France)

The conference is supported by UC Davis, NSF grant DMS-1559338 and the Hellman Fellows Fund.

### Speakers

Jozsef Bodnar (Stony Brook University), Manuel Gonzalez Villa (University of Wisconsin-Madison), Julia Gordon (UBC), Eugene Gorsky (UC Davis), Michel Raibaut (Universite de Savoie, France).

### Preliminary schedule

 Time Monday Nov 7 Tuesday Nov 8 Wednesday Nov 9 Thursday Nov 10 10-11 Julia 1 2112 Manuel 2 2112 11-1 Lunch Lunch Lunch Lunch 1-2 Michel 1 3106 Julia 2 3106 Manuel 1 2112 Jozsef 2 3106 2-3 Break Break Break Break 3-4 Discussions Eugene 2112 Discussions Discussions 4-5 Michel 2 1147 Jozsef 1 2112

### Contacts and registration

Please register here. For all inquiries please contact Eugene Gorsky (egorskiy AT math.ucdavis.edu).
Limited travel support may be available for graduate students, please write to Eugene.

### Abstracts

#### Jozsef Bodnar:

Cuspidal curves of higher genus
The theory of complex projective plane curves has a long and rich history. Nevertheless, curves of higher genus were not studied in such a depth as rational ones. Recently, Heegaard--Floer theory of 3- and 4-manifolds turned out to be a useful tool in the classification of possible local topological cusp types on curves of any genus. In a joint work with D. Celoria and M. Golla, using the $d$-invariants of Ozsvath and Szabo, some theory of numerical semigroups, Pell equations and birational transformations, we managed to provide an almost complete classification of possible torus knot types which can arise as singularity links of the cusp on a unicuspidal projective curve of certain genera $g$. Independently, similar results were obtained by M. Borodzik, M. Hedden and C. Livingston.

#### Manuel Gonzalez Villa:

Motivic zeta functions of quasi-ordinary hypersurfaces
We will discuss how to compute motivic invariants associated to irreducible quasi-ordinary hypersurface case. Quasi-ordinary hypersurfaces are a natural generalization of plane curves to arbitrary dimension. We will compare the computation for quasi-ordinary with previous results for the case of curves due to Guibert. Some of the results are based in joint work with P. Gonzalez Perez, with Budur and Gonzalez Perez and with Kennedy and McEwan.

Generalization of the Motivic invariants and the monodromy conjecture
We will present a joint work with A. Libgober and L. Maxim generalizing motivic zeta functions and motivic Milnor functions. We will explain the relations with the work of Denef and Loeser. Finally we will explain some possible applications to the monodromy conjecture.

#### Julia Gordon:

Transfer principles for the Fundamental Lemma
The Fundamental Lemma (proved by B.C. Ngo in positive characteristic in 2009) asserts equality of certain related orbital integrals on a reductive group $G$ and its endoscopic group $H$. I will try to outline the statement of the Fundamental Lemma, and explain how the transfer principle, proved by R. Cluckers, T. Hales and F. Loeser, which is based on motivic integration, can be used to transfer various versions of the Fundamental Lemma to characteristic zero fields. This talk will be based mostly on the work of Cluckers, Hales and Loeser. In particular, this talk will give an introduction to the model-theoretic version of motivic integration by Cluckers and Loeser shortly after the "geometric" and "arithmetic" motivic integration was developed by Denef and Loeser.

Uniform estimates for motivic functions
In 2008, R. Cluckers and F. Loeser defined the class of the so-called "constructible motivic exponential functions". These functions specialize to rather general functions on $p$-adic fields; the Cluckers-Loeser theory of motivic integration for such functions specializes to the classical $p$-adic integration when $p$ is sufficiently large. I will talk about the long-term joint project with R. Cluckers and I. Halupczok, where we explore the delicate properties of constructible functions, such as integrability and boundedness. This leads to the "transfer of integrability principle". Then I will describe the application of these ideas to proving uniform in $p$ bounds for $p$-adic integrals, and in particular, orbital integrals on reductive $p$-adic groups.

#### Eugene Gorsky:

Motivic Poincare series and Heegaard Floer homology
Campillo, Delgado and Gusein-Zade related the coefficients of the Alexander polynomial of an algebraic link to the Euler characteristics of some strata in the space of functions on the corresponding plane curve singularity. I will explain the relation between the homology of these strata, coefficients of the motivic Poincare series, semigroup of the singularity and link Floer homology. This is a joint work with Andras Nemethi.

#### Michel Raibaut:

Introduction to motivic integration
In 95', Kontsevich introduced the theory of motivic integration in order to prove that two birationally equivalent Calabi-Yau varieties have same Hodge numbers. This theory developed by Denef-Loeser and Batyrev is the analog of the $p$-adic integration over $\mathbb{C}((t))$. The measurable sets are parts of arc spaces of algebraic varieties and values are "motives" meaning elements of the Grothendieck ring of varieties $K_0(Var)$. This ring is generated as a group by isomorphism classes of algebraic varieties with scissors relations. In this context, the motive of a variety $X$ is its isomorphism class $[X]$ as element of $K_0(Var)$. In particular, this motive contains all the additive and multiplicative invariants of $X$, and any varieties $X$ and $Y$ have same such invariants as soon as the elements $[X]$ and $[Y]$ are equal in $K_0(Var)$.

Motivic Milnor fibers
Let $f$ be a polynomial with complex coefficients and $x$ be a special point of the special fiber. We will recall the notion of Milnor fiber of $f$ at $x$. Using arcs with origin at $x$, Denef and Loeser introduced a motive $S_{f,x}$ in a Grothendieck ring of varieties with action of roots of unity. This motive encodes all the additive and multiplicative invariants of the Milnor fibration of $f$ at $x$ and its monodromy. In particular, we can recover the monodromy zeta function or the spectrum of $f$ at $x$. For that reason, it is called motivic Milnor fiber. There exists other motives which encode invariants of global Milnor fibrations or more generally of nearby cycles of $f$. If we have time we will present them at the end of the talk.