I will survey some mathematics around the application of Goodwillie-Weiss Embedding Calculus to knot theory and some then share some related problems.  These problems (and related comments and questions) include:

- Are the cochains of the n-disks operad formal, say as diagrams of E_infinity algebras?  (This would imply a spectral sequence collapse result, which would in turn imply major results in knot theory and its interface with mathematical physics and Lie theory.  What machinery is there to establish formality of cochains integrally?  Could intersection
theoretic models serve over the integers some of the roles that differential forms play over the reals?)

- Can one explicitly and geometrically define invariants of homotopy classes of maps between say finite complexes which are associated to a bar construction on their cochains over the Koszul dual of the E_infty operad (that is, the derivatives of the identity functor)?  (Rationally, such explicit and geometric "homotopy periods" defined using the bar
construction over the Lie cooperad are a pretty story, useful for applications to geometric questions in which homotopy arises.)

- What are some other consequences of formality of the disks operad, including a strong version in dimension two?  (This is less my direct interest and more something to bring to the attention of people doing things with these operads, including in calculus and factorization homology.  Some like Kontsevich have gotten quite a lot of mileage from this formality over the reals.)

To lead up to these questions, I'll start with simple models for the Goodwillie-Weiss tower, inspired by Gauss's definition of linking number.

Motivic integration was invented by Kontsevich about 20 years ago for proving that Hodge numbers of Calabi-Yau are birational invariants and has developed at a fast pace since. I will start by presenting informally a device for defining and computing motivic integrals due to Cluckers and myself. I will then focus on  some recent applications. In particular I plan to explain how motivic integration provides tools for proving uniformity results for $p$-adic integrals occuring in the Langlands program and  how one may use "motivic harmonic analysis" to study certain generating series counting curves in enumerative algebraic geometry.

9:30 - 10:20 am: Alexandru Buium (U. New Mexico) Curvature on the integers

Abstract: The talk explains how one can introduce an arithmetic analogue of connection and curvature on principal bundles (over flat space). In the arithmetic  framework functions on manifolds are replaced by integer numbers, partial derivatives  are replaced by Fermat quotients with respect to different primes, and linear connections with values in classical Lie algebras are replaced by non-linear objects naturally attached to the corresponding classical groups.

Our main results are vanishing/non-vanishing theorems for the curvatures of these connections. This is joint work in progress, with Malik Barrett.

10:30 - 11:20 am: James Freitag (UC Berkeley) Superstability and central extensions of algebraic groups

Abstract: Altinel and Cherlin proved that any perfect central extension of an algebraic group over an algebraically closed field which happens to be of finite Morley rank as a group is actually a finite central extension and is itself an algebraic group. We will prove a generalization of their result in the infinite rank setting with an additional hypothesis on the center of the group, while giving an example which shows the necessity of this hypothesis. The inspiration for the work comes from differential algebra; namely, this work answers a question of Cassidy and Singer in the (more general) setting of superstable groups.

11:30 am - 12:20 pm: Rahim Moosa (U. Waterloo) Differential-algebraic tangent spaces and internality to the constants

Abstract: Motivated on the one hand by phenomena in bimeromorphic geometry (so compact complex manifolds) and on the other by model-theoretic consideration arising from the study of the canonical base property, I asked some years ago whether the Kolchin tangent bundle of a finite dimensional differential-algebraic variety has the property that the restriction to any C-algebraic subvariety is C-algebraic. Here C is the field of constants in a saturated differentially closed field of characteristic zero, and C-algebraic means being generically in finite-to-finite correspondence with the C-points of an algebraic variety over C. Actually, one is interested not only in the Kolchin tangent bundle but also its higher order incarnations appearing in the work of Pillay and Ziegler. In recent work with Zoé Chatzidakis and Matthew Harrison-Trainor, we show that while the answer to the question as stated is no, an appropriate generic formulation has a positive answer. I will give motivations for the problem and discuss some aspects of its solution.

2:00 - 2:50 pm: Thomas Scanlon (UC Berkeley) Algebraic differential equations from covering maps

Abstract: We show as a consequence of two model theoretic theorems, elimination of imaginaries in differentially closed fields and the Peterzil-Starchenko GAGA theorem, that in wide generality for analytic covering maps π :U → X(C) of algebraic varieties that there are differential constructible functions χ defined on X which "nearly invert'' π in the same way in which the logarithmic derivative nearly inverts the exponential function.   As consequences we deduce the finite dimensionality of the Kolchin closure of the Hecke orbits of arbitrary points in moduli spaces of abelian varieties generalizing a theorem of Buium.

3:00 - 3:20pm : Tea break

3:20 -4:05pm: Michael Singer (U. North Carolina) Direct and Inverse Problems for Parameterized Linear Differential Equations

Abstract: I will give an introduction to a Galois theory for differential equations of the form Y_x = A(x,t)Y where A(x,t) is an mxm matrix with entries that are functions of the principal variable x and the parameter t . The Galois groups in this theory are linear differential algebraic groups, that is, groups of mxm matrices whose entries are functions of t satisfying some fixed set of differential equations. This theory has been successfully used to decide when solutions of such equations satisfy additional differential equations with respect to the parameters and in particular questions of integrability. I will describe recent work of Minchenko, Ovchinnikov and myself concerning the direct problem calculating the Galois group of a given equation) and the inverse problem (which groups can occur as Galois groups).

4:10 - 4:55pm: Lucia di Vizio (Versailles Saint Quentin) Linear Differential Equations with a discrete parameter: Theory

5:00 - 5:45pm: Charlotte Hardouin (Toulouse) Linear Differential Equations with a discrete parameter: Applications

Common abstract: We will discuss an analogous situation to the one described by M. Singer, apart from the fact that we consider a discrete parameter. In this case the theory produce a Galois group, which is a difference scheme and can be non-\sigma-reduced. We will explain how we get around this difficulty and the implications on the theory.

3:45: David Eisenbud: Assymptotic Boij-Soederberg theory and resolutions over complete intersections of quadrics

I'll describe the cone of resolutions of high syzygies over a complete intersection of quadrics, and other related phenomena.

5:00 David Berlekamp: Castelnuovo-Mumford regularity and log-canonical thresholds

The regularity of an ideal is a measure of its computational complexity.  The log-canonical threshold is a measure of singularity.  Not surprisingly, these things are related - the log-canonical threshold of an ideal sheaf on projective space is (sharply) bounded below by the inverse of its regularity.  This and related bounds are worked out with multiplier ideals in recent work of Kuronya and Pintye.  I will attempt to explain what all of these words mean, and how this is done.  

I'll describe the computational tools needed to understand the proof of the results in the main talk.

Rohlin's theorem on the signature of Spin 4-manifolds can be restated in terms of the connection between real and complex K-theory given by homotopy fixed points. This comes from a bordism result about Real manifolds versus unoriented manifolds, which in turn, comes from a C2-equivariant story. I'll describe a surprising analogue of this for
larger cyclic 2 groups, showing that the element eta cubed is never detected! In particular, for any bordism theory orienting these generalizations of Real manifolds, the three torus is always a boundary.

Over a field in characteristic zero commutative dgas are well-behaved; otherwise they are not. For instance there isn't a
right-induced model structure on them in the general case. I'll explain some of these issues and advertise symmetric sequences to repair that defect.

Eilenberg-Mac Lane spectra represent singular cohomology and they allow for rather algebraic considerations in stable homotopy theory. Shipley proved that there is a Quillen equivalence between algebras over the Eilenberg-Mac Lane spectrum of the integers, HZ, and differential graded rings. I'll talk about ongoing work with her where we aim at extending her result to commutative HZ-algebras. This builds on a Dold-Kan type theorem for commutative monoids in symmetric sequences.

This talk--aimed for a general audience of neither topologists nor model theorists--will discuss applications of cobordisms to Kontsevich's mirror symmetry conjecture. We'll begin by stating a rough version of the
conjecture, which builds a bridge between symplectic geometry on one hand, and on the other hand, algebraic geometry over the complex numbers. We then discuss how the theory of cobordisms, which studies when two manifolds can be the boundary of another manifold, sheds light on how to generalize the mirror symmetry conjecture, while giving us information about objects in symplectic geometry. (For example, two Lagrangians related by a compact cobordism are equivalent in the Fukaya category.)

The talk will be built around  examples of how model theory informs our understanding in areas of mathematics such as algebra, number theory, algebraic geometry and analysis. The model theory behind these applications includes concepts such as compactness, definability, stability and o-minimality. However, we assume  no special expertise in model theory, but rather aim  to illustrate the kinds  of  mathematical questions for which a model theoretical perspective has proven to be useful.

3:45pm
Speaker: Zvi Rosen (UC Berkeley)
Title:   Computing Algebraic Matroids

Abstract: Algebraic matroids characterize the combinatorial structure of an algebraic variety. By decorating the matroid with circuit polynomials and base degrees, we capture even more information about the variety and its coordinate projections.  In this talk, we will introduce algebraic matroids, discuss algorithms for their computation, and present some motivating examples.

5:00pm
Speaker: Noah Giansiracusa (UC Berkeley)
Title:  Equations of Tropical Varieties

Abstract: I'll discuss joint work with J.H. Giansiracusa (Swansea) in which we study scheme theory over the tropical semiring T, using the notion of semiring schemes provided by Toen-Vaquie, Durov, or Lorscheid. We define tropical hypersurfaces in this setting and a tropicalization functor that sends closed subschemes of a toric variety over a field with non-archimedean valuation to closed subschemes of the corresponding toric variety over T. Upon passing to the set of \mathbb{T}-valued points this yields Kajiwara-Payne's extended tropicalization functor. We prove that the Hilbert polynomial of any projective subscheme is preserved by our tropicalization functor, so the scheme-theoretic foundations developed here reveal a hidden flatness in the degeneration sending a variety to its tropical skeleton.