Title: A finiteness criterion for 2-dimensional representations of surface groups.

Abstract: Let C be a a complex algebraic curve of genus \geq 1, and let pi be its fundamental group. Let \rho: pi\rightarrow \GL_2(\C) be a semisimple 2-dimensional representation, such that \rho(\alpha) has finite order for every simple closed loop \alpha. We will prove that \rho has finite image. If time permits, we will mention applications of this result to the Grothendieck-Katz p-curvature conjecture. This is joint work with Anand Patel and Junho Peter Whang.

Title: Connections between the Circle method, trace formula and bounds for the Subconvexity problem.

Abstract: After introducing the sub-convexity problem for L-functions in a general context, we will focus our attention to the particular case of Rankin-Selberg L-functions. We will briefly trace the history of this particular problem starting from Kowalski, Michel, and Vanderkam, with a lot of authors in between upto the seminal work of Michel, Venkatesh. I will then try to explain how the circle method enters this question by sketching an argument of Munshi for what is perhaps the simplest case i.e character twists of GL(2) L-functions. I will end the talk by explaining how we can solve the Subconvexity problem for Rankin-Selberg L-functions in the combined level aspect by a very easy version of the circle method and see how this approach is connected to earlier work on the same problem.

Title: Geometric invariants and geometric consistency of Manin's conjecture.

Abstract: Let X be a Fano variety with an associated height function defined over a number field. Manin's conjecture predicts that, after removing a thin set, the growth of the number of rational points of bounded height on X is controlled by certain geometric invariants (e.g. the Fujita invariant of X). I will talk about how to use birational geometric methods to study the behaviour of these invariants and propose a geometric description of the thin set in Manin's conjecture. Part of this is joint work with Brian Lehmann and Sho Tanimoto.

Title: A concrete approach to virtual classes in genus 0 Gromov--Witten theory.

Abstract: Gromov--Witten theory is concerned with counting curves inside (smooth) projective varieties satisfying some incidence conditions (e.g. how many rational degree d curves pass through 3d-1 generic points in the complex projective plane?). In general (due to complications arising from the fact that spaces of curves may not be smooth), instead of counting curves directly, we need to use intersection theory on the space of curves to define certain "virtual counts". In the first half of the talk, we will provide the necessary background (spaces of curves, "expected dimension", compactness and some examples of curve counting). In the second half of the talk, we will describe a concrete differential geometric approach to these "virtual counts" for genus 0 curves in projective varieties (using ideas coming from the theory of psuedo-holomorphic curves in symplectic manifolds).

Title: A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory.

Abstract: Min-max theory for the area functional was developed by Almgren and Pitts to construct closed minimal hypersurfaces in an arbitrary closed Riemannian manifold. There is an alternate approach via PDE to the construction of minimal hypersurfaces. This approach is based on the study of the limiting behaviour of solutions to the Allen-Cahn equation. In my talk, I will briefly describe the Amgren-Pitts min-max theory and the Allen-Cahn min-max theory and discuss the question to what extent these two theories agree.

Title: Singularities in Positive Characteristics.

Abstract: In the first half, we shall look at the several notions of singularities of a polynomial function at a point from both analytic and algebro-geomteric point of view. We will indicate the surprising similarities of the results coming from these two seemingly different directions. In the second half, these ideas will be put into a more general context detailing more on the positive characteristic side. We shall discuss the notion of F-split, F-regular schemes, how these notions are related to the characteristic zero singularities. We shall end by mentioning some open problems relating singularities in characteristic zero and positive characteristic. This talk is an exposition of the ideas developed in the last fifty years in different contexts.

Click here for a recording of the lecture.

This talk is based mostly on the following two references: arXiv:1309.4814 and arXiv:1409.1169.

Title: *p*-adic Hodge theory and delta geometry.

Abstract: We will talk about a new *p*-adic Galois representation that comes from δ-geometry. Here, we construct a new filtered Isocrystal associated to an abelian scheme that is different from the usual crystalline cohomology. In the case of elliptic curves, depending on a modular parameter, this Isocrystal is also weakly admissible which leads to a new crystalline Galois representation attached to the elliptic curve via the Fontaine functor. This is joint work with Jim Borger.
We will dedicate the first half of the talk on giving an overview of *p*-adic Hodge theory and δ-geometry.

Title: Arithmetic nature of special values of L-functions.

Abstract: The study of L-functions has occupied a center stage in number theory since the work of Riemann and Dirichlet. A standard example of an L-function is the Riemann zeta-function, $\zeta(s)$, given by the series $\sum_{n=1}^{\infty} n^{-s}$ when $\Re(s)>1$. The aim of this talk will be to discuss the question of determining the arithmetic nature (that is, rational/irrational and algebraic/transcendental) of values of L-functions at positive integers. For example, it is expected that the values $\zeta(m)$ are transcendental for all integers $m >1$. However, the only known cases of this conjecture are the even zeta-values, which Euler had explicitly evaluated in the 1730s. Among the odd zeta-values, Apery proved that $\zeta(3)$ is irrational, whereas the irrationality of the remaining odd zeta-values remains a mystery. In this talk, we will discuss various facets of this problem. If time permits, we will prove that for a fixed odd positive integer m, all the values $\zeta_K(m)$ are irrational as K varies over imaginary quadratic fields, with at most one possible exception. This is joint work with Ram Murty. This talk will be accessible to a wide audience.

Title: Squarefree sieves in arithmetic statistics.

Abstract: A classical question in analytic number theory is: given a polynomial with integer coefficients, how often does it take squarefree values? In arithmetic statistics, we are particularly interested in the case of discriminant polynomials. In this talk, I will present several different cases of this question. First, we will consider a classical result of Davenport--Heilbronn which considers the case of discriminants of binary cubic forms. Then, I will discuss joint work with Bhargava in which we consider the case of discriminants of ternary cubic forms. Third, I will describe joint and ongoing work with Bhargava and Wang, in which we consider different families of degree-n polynomials in one variable, and determine the proportion of those having squarefree discriminant. Finally, I will describe various applications of these results.

Title: A non-archimedean definable Chow theorem.

Abstract: O-minimality has had some striking applications to number theory. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the rigidity that it imposes, the theory is sufficiently flexible to allow for a range of analytic constructions. An illustration of this `tame' property is the following surprising generalization of Chow's theorem proved by Peterzil and Starchenko - A closed analytic subset of a complex algebraic variety that is also definable in an o-minimal structure, is in fact algebraic. While the o-minimal machinery aims to capture the archimedean order topology of the real line, it is natural to wonder if such a machinery can be set up over non-archimedean fields. In this talk, we explore a non-archimedean analogue of an o-minimal structure and describe a version of the definable Chow theorem in this context.

Title: Construction of p-adic L-functions.

Abstract: In this talk, I will discuss some of the constructions of p-adic L-functions that interpolates families of classical special values. Finally, I will talk about the construction of the p-adic adjoint L-functions using overconvergent cohomology. I will try to keep this talk accessible to as wide an audience as possible.

Title: Topological invariants in arithmetic geometry.

Abstract: This will be a gentle introduction to two independent themes. The first half of the talk will focus on conductors and discriminants, two invariants that measure degeneration in a family of a hyperelliptic curves. We will show how a combinatorial refinement helps us compare these two invariants. The second half of the talk will be be an introduction to A^1 enumerative geometry, i.e., how we may use quadratic forms to encode additional arithmetic information in enumerative problems in algebraic geometry.

Title: (Inverse)-Hessian type equations and positivity in complex algebraic geometry.

Abstract: In the early 2000's Demailly and Paun proved that a (1,1) cohomology class on a K\"ahler manifold is positive if and only if certain intersection numbers are positive. This is a generalization of the classical Nakai criteria for ampleness of line bundles on projective manifolds. The proof, somewhat surprisingly, relies on Yau's work on the complex Monge-Ampere equation, and his solution to the Calabi conjecture. In 2019, Gao Chen extended the method of Demailly-Paun to prove that another important PDE in Kahler geometry, namely the J-equation, is equivalent to the positivity of certain (twisted) intersection numbers, thereby settling a conjecture of Lejmi and Szekelyhidi. In my talk, I will describe this circle of ideas, concluding with a recent result obtained in collaboration with Vamsi Pingali extending the work of Gao Chen to more general inverse Hessian type equations, thereby settling a conjecture of Szekelyhidi for projective manifolds. In the process we obtain an equivariant version of Gao Chen's result, and in particular recover some results of Collins and Szekelyhidi on the J-equation on toric manifolds.

Title: Graph potentials and Moduli spaces of vector bundles of curves.

Abstract: We construct and study graph potentials, a collection of Laurent polynomials associated with colored graphs of small valency. The potentials we construct are related to the moduli spaces of vector bundles of rank two with fixed determinant on algebraic curves. We will discuss relations between these graph potentials and Gromov-Witten invariants of the moduli spaces. This is joint work with P. Belmans and S. Galkin.

Title: Positivity of direct image sheaves.

Abstract: Positivity properties of direct image sheaves have deep implications in the geometry of families of varieties. For instance, the existence of enough global sections of pushforwards of higher tensors of the relative canonical bundle of a family, puts certain restrictions on the kinds of varieties that can appear on the fibres etc. I will discuss some these positivity properties, especially the ones that come as a generalisation of the Fujita conjecture and its application to the Iitaka conjecture. This is partially a joint work with Takumi Murayama.

Title: Brauer p-dimensions of complete discretely valued fields.

Abstract: (This is joint work with Bastian Haase) To every central simple algebra A over a field F, one can associate two numerical Brauer class invariants called the index(A) and the period(A). It is well known from that index(A) divides high powers of per(A). The Brauer dimension of a field F is defined to be the least number n such that index(A) divides period(A)^n for every central simple algebra A defined over any finite extension of F. Similarly there exist analogous notions of Brauer-p-dimensions of fields. The 'period-index' questions revolve around bounding the Brauer (p) dimensions of arbitrary fields.

In this talk, we will look at the period-index question over complete discretely valued fields in the so-called 'bad characteristic' case (i.e when the residue field has characteristic p). We will give a flavour of the known results for this question and discuss progress for the cases when the residue fields have small 'p-ranks'. Finally, we will propose a (still open!) conjecture which very precisely relates the Brauer p-dimensions of the complete discretely valued fields to the p-ranks of the residue fields, along with some evidence via a family of examples. The key idea involves working with Kato's filtrations and bounding the symbol length of the second Milnor K group modulo p in a concrete manner, which further relies on the machinery of differentials in characteristic p as developed by Cartier.

Title: The Kardar-Parisi-Zhang equation in d ≥ 3 and the Gaussian free field.

Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a singular stochastic partial differential equation (SPDE) and belongs to a large class of models known as the KPZ universality class, which is believed to exhibit very different behavior than Gaussian universality class and describe the long-time of a wide class of systems including some noisy SPDEs, driven lattice gases, randomly growing interfaces and directed polymers in random media. In spatial dimension one, recently this class has been studied extensively based on approximations by exactly solvable models. which no longer exist if perturbations appear in the approximating models, or when higher dimensional models are investigated. When the spatial dimension is at least three, it was conjectured that two disjoint universality classes co-exist when the long-time/large-scale behavior of the solutions are studied. We will report some recent progress along these directions.

Title: The structure of the regular and the singular set of the free boundary in the obstacle problem for fractional heat equation.

Abstract: In this talk, I will discuss the structure of the free boundary in the obstacle problem for fractional powers of the heat operator. Our results are derived from the study of a lower dimensional obstacle problem for a class of local, but degenerate, parabolic equations. The analysis will be based on new Almgren, Weiss and Monneau type monotonicity formulas and the associated blow-up analysis. This is a joint work with D. Danielli, N. Garofalo and A. Petrosyan.

Title: Noncollinear points in the projective plane.

Abstract: We will look at the space of n points in the projective plane such that no three lie on the same line. There is an action of the symmetric group by permuting the points and we can compute the action on cohomology (for n < 7) by counting the numbers of such n tuples over the finite field F_q, with a 'twist'. Unfortunately for n > 6 such a computation is still hard and we expect the answer for large n to be arbitrarily complicated (in the sense of Mnëv's universality). This talk will be based on joint work with Ben O'Connor.

Title: Unstable motivic homotopy theory and few commutative algebra problems.

CMI Online Seminar Series

Last modified: July 10, 2020