Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
To post a seminar which takes place at the Mathematics department, please email with date, time, room, title and possibly with an abstract.
Gakushuin University
Double cover family of K3 surfaces and mirror symmetry
on Monday, February 25, 2019, at 1:30-2:30 PM in CMSA, 20 Garden Street, G10
I will study a family K3 surfaces which are given as double covers of ${\mathbb P}^2$ branched along six lines in general position. Period integrals of this family satisfy the hypergeometric system E(3,6), i.e., Aomoto-Gel'fand hypergeometric system on Grassmannian G(3,6), which were studied in detail by Matsumoto, Sasaki and Yoshida in the '90s. In this talk, I will focus on the parameter space of the E(3,6) system described naturally by GIT or Baily-Borel-Satake compactification. I will find that the E(3,6) system is "locally trivialized" by corresponding GKZ systems. Based on this result, and making suitable resolutions of the compactified parameter space, I will obtain the desired LCSLs (large complex structure limits) where we can read off mirror symmetry by applying the generalized Frobenius method formulated in the '90s. This talk is based on a recent collaboration with B. Lian, H. Takagi and S.-T. Yau (arXiv:1810.00606)

Harvard University
Poncelet's theorem and the birth of modern algebraic geometry
on Monday, February 25, 2019, at 4:30 PM in Science Center 507
In the early 19th century, Poncelet asked a simple question: given two ellipses in the plane, one inside the other, does there exist a polygon inscribed in the outer one and circumscribed about the inner? In this talk, I'll describe the (somewhat surprising) answer, and also describe how consideration of this simple problem led to two major changes in how we do algebraic geometry: the introduction of projective space, and the use of complex numbers. (This talk will be accessible to members of the department at all levels.)

LOGIC SEMINAR: Nathanael Ackerman
Harvard University
Entropy of Invariant Measures
on Monday, February 25, 2019, at 5:40 PM in Science Center 507
The entropy of a probability measure on a finite set quantifies one notion of the complexity of the measure. For a finite relational language $L$, and a measure $\mu$ on $L$-structures with underlying set the natural numbers, if $\mu$ is $S_\infty$-invariant then it is determined by its restriction to $L$-structures of the form $\{0, \ldots , n-1\}$. We can therefore canonically assign to $\mu$ its \emph{entropy function}, $En_\mu$, which sends each natural number $n$ to the entropy of the restriction of $\mu$ to $L$-structures with underlying set $\{0, \ldots , n-1\}$. In this talk I will discuss the relationship between the growth rate of $En_\mu$ and the language $L$. If the largest arity of a relation in $L$ is $k$, then $En_\mu$ grows as $O(n^k)$. I will give a precise expression for the coefficient of $n^k$ based on the hypergraphon representation of $\mu$, and show how these notions generalize what is often called the \emph{entropy of a graphon}. When $\mu$ comes from sampling a Borel structure we will show that the growth rate of the entropy function is $o(n^k)$. We will also discuss lower bounds on the possible growth rates of the entropy function in this case. This generalizes work of Hatami-Norine from graphons to hypergraphons. Joint work with Cameron Freer and Rehana Patel.

Modular vector fields
on Monday, February 25, 2019, at 12:00 - 1:00 PM in CMSA, 20 Garden St, G10
Using the notion of infinitesimal variation of Hodge structures I will define an R-variety which generalizes Calabi-Yau and abelian varieties, cubic four, seven and ten folds, etc. Then I will prove a theorem concerning the existence of certain vector fields in the moduli of enhanced R-varieties. These are algebraic incarnation of differential equations of the generating functions of GW invariants (Lian-Yau 1995), Ramanujan's differential equation between Eisenstein series (Darboux 1887, Halphen 1886, Ramanujan 1911), differential equations of Siegel modular forms (Resnikoff 1970, Bertrand-Zudilin 2005).

COLLOQUIUM: Melanie Wood
University of Wisconsin
Random groups from generators and relations, and unramified extensions of global fields
on Tuesday, February 26, 2019, at 3:00 pm in Science Center 507
We consider a model of random groups that starts with a free group on n generators and takes the quotient by n random relations, and in particular we study the limiting behavior of this model as n goes to infinity. The abelianization of this model is related to the Cohen-Lenstra heuristics, which predict the distribution of unramified abelian extensions of number fields. We use this model as a jumping off point to develop conjectures on the distribution of the Galois groups of the maximal unramified extensions of varying number fields or function fields. We give theorems in the function field case that support these new conjectures. This talk includes joint work with Yuan Liu.

Harvard University
Entropy of arithmetic functions and arithmetic compactifications
on Tuesday, February 26, 2019, at 3:00 PM in Jefferson 356
A new notion of entropy of arithmetic functions is introduced. It is shown that all zero entropy arithmetic functions form a C*-algebra. This entropy also behaves nicely with respect to limits similar to Kolomogorov’s dynamical entropy and has many properties similar to Shannon’s entropy.

Universidade Federal Fluminense
Real inflection points of real linear series on real (hyper)elliptic curves (joint with I. Biswas and C. Garay López)
on Tuesday, February 26, 2019, at 3:00 pm in Science Center B10 *note different location
According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.

Sentiment and Speculation in a Market with Heterogeneous Beliefs
on Wednesday, February 27, 2019, at 2:30-3:30 PM in CMSA, 20 Garden Street, G10
We present a dynamic model featuring risk-averse investors with heterogeneous beliefs. Individual investors have stable beliefs and risk aversion, but agents who were correct in hindsight become relatively wealthy; their beliefs are overrepresented in market sentiment, so “the market” is bullish following good news and bearish following bad news. Extreme states are far more important than in a homogeneous economy. Investors understand that sentiment drives volatility up, and demand high risk premia in compensation. Moderate investors supply liquidity: they trade against market sentiment in the hope of capturing a variance risk premium created by the presence of extremists. [with Dimitris Papadimitriou]

Princeton University
Affine Deligne--Lusztig varieties at infinite level for GLn
on Wednesday, February 27, 2019, at 3:00 PM in Science Center 507
Affine Deligne--Lusztig varieties have been of interest for some time because of their relation to Shimura varieties and the Langlands program. In this talk, we will construct a tower of affine Deligne--Lusztig varieties for GLn and its inner forms. We prove that its limit at infinite level is isomorphic to the semi-infinite Deligne--Lusztig variety of Lusztig and that its cohomology realizes certain cases of automorphic induction and Jacquet--Langlands. This is joint work with A. Ivanov.

Stony Brook
How irrational is an irrational variety?
on Thursday, February 28, 2019, at Tea at 4:00 pm; Talk at 4:30 pm in Science Center Hall A
Recall that an algebraic variety is said to be rational if it contains a Zariski-open subset isomorphic to a Zariski-open subset of projective space. Subtle questions of rationality have seen a great deal of recent interest and progress, but most varieties aren’t rational. I will survey a developing body of work in a complementary direction, centered around measures of irrationality for varieties whose non-rationality is known.

Northwestern University
Introduction and Yau's Theorem
on Friday, March 01, 2019, at 3:00 -4:00 PM in CMSA, 20 Garden St, G10
I will give an introduction to the study of limits of Ricci-flat Kahler metrics on a compact Calabi-Yau manifold when the Kahler class degenerates to the boundary of the Kahler cone. Analytically, the problem is to prove suitable uniform a priori estimates for solutions of a degenerating family complex Monge-Ampère equations, away from some singular set. Geometrically, this can be used to understand the Gromov-Hausdorff limit of these metrics. In the projective case, the limits possess an algebraic structure and are obtained from the initial manifold via contraction morphisms from Mori theory. I will discuss some results on this problem, in both the noncollapsing case and in the much harder collapsing case. I will also discuss a different source of examples, coming from work of McMullen and Cantat in complex dynamics on K3 surfaces.

Northwestern University
D-modules in birational and complex geometry
on Monday, March 04, 2019, at 3:00 pm in Science Center 507
I will give an overview of techniques based on the theory of mixed Hodge modules, which lead to a number of applications of a rather elementary nature in birational and complex geometry, as well as in the study of singularities. One of the main points I will emphasize is the existence (and usefulness) of a package of vanishing and positivity theorems in the context of filtered D-modules of Hodge theoretic origin.

Michigan State University
Super-rigidity and Castelnuovo’s bound
on Friday, March 08, 2019, at 3:30 PM in Science Center 507
Castelnuovo’s bound is a very classical result in algebraic geometry. It asserts a sharp bound on the genus of a curve of degree d in n-dimensional projective space. It is an interesting question to ask whether analogues of Castelnuovo’s bound hold in almost complex geometry. There is a direct analogue in dimension four. In dimension at least eight genus bounds can be established for generic almost complex structures. These results leave open the case of dimension six. Bryan and Panharipande introduced the notion of super-rigidity of an almost complex structure. They also speculated that this condition might hold for a generic almost complex structure (compatible with a fixed symplectic structure). It had been believed for a long time that super-rigidity will play an important role in the proof of the Gopakumar–Vafa conjecture. However, it turned that Ionel and Parker’s recent proof of this conjecture did not make use of it. Nevertheless, super-rigidity has important consequences. I will present one of these consequences, namely, a genus bound for index zero pseudo-holomorphic curves. This is joint work with Aleksander Doan and, heavily, relies on work by De Lellis, Spadaro, and Spolaor and ideas of Taubes'. There has been a lot of progress towards establishing Bryan and Pandharipande’s super-rigidity conjecture in the work of Wendl. In fact, based on his ideas, Aleksander Doan and I have developed an abstract framework for equivariant transversality/Brill–Noether type questions. Wendl’s work shows that the super-rigidity conjecture holds provided generic real Cauchy-Riemann operators satisfy an easy to state analytic condition. I will explain what this condition means and discuss a few cases in which this condition (or versions of it hold). Future schedule is found here:

COLLOQUIUM: Bhargav Bhatt
University of Michigan
Title: Interpolating p-adic cohomology theories
on Monday, March 11, 2019, at 3:00 PM in Science Center 507
Integration of differential forms against cycles on a complex manifold helps relate de Rham cohomology to singular cohomology, which forms the beginning of Hodge theory. The analogous story for p-adic manifolds, which is the subject of p-adic Hodge theory, is richer due to a wider variety of available cohomology theories (de Rham, etale, crystalline, and more) and torsion phenomena. In this talk, I will give a bird's eye view of this picture, guided by the recently discovered notion of prismatic cohomology that provides some cohesion to the story.

COLLOQUIUM: Christopher Hacon
University of Utah
On the geometry of Algebraic varieties
on Wednesday, April 03, 2019, at 3:00 pm in Science Center 507
Algebraic varieties are geometric objects defined by polynomial equations. The minimal model program (MMP) is an ambitious program that aims to classify algebraic varieties. According to the MMP, there are 3 building blocks: Fano varieties, Calabi-Yau varieties and varieties of general type which are higher dimensional analogs of Riemann Surfaces of genus 0,1 or at least 2 respectively. In this talk I will recall the general features of the MMP and discuss recent advances in our understanding of Fano varieties and varieties of general type.

Lecture 4: Cotangent complexes
on Saturday, December 14, 2019, at 3:00 - 4:30 pm in CMSA Building, 20 Garden St, G10
Derived algebraic geometry is a branch of mathematics whose aim is to extend algebraic geometry to study of spaces which are locally defined over topological ring spectra (rather than commutative rings of polynomials) and hence they can come equipped with higher homotopical structure. An object in derived geometry comes with an infinity groupoid of internal symmetries as well as infinity algebra of functions (which we will define in detail during the course). One main application of study of derived algebraic spaces (derived schemes or stacks) has been to provide an algebraic description of configuration spaces in gauge theory. Take for instance the so-called BV-BRST complex which encodes the function algebra on the homological resolution of a locus of solutions to differential equation which encodes movement of particles in certain gauge theoretic mechanical system. The derived (or higher) differential geometry is also similar to differential geometry in higher homotopical setting. In derived differential geometry smooth manifolds are replaced with smooth infinity-stacks or (infinity, 1)-sheaves which live over their site. The course discusses both derived algebraic and derived differential geometries and aims to eventually study application of these two constructions in enumerative geometry of moduli stack of algebraic objects on Calabi-Yau or Fano manifolds (for instance Donaldson-Thomas theory of sheaves on Calabi-Yau 4 folds which have a priori ill-behaved non-perfect deformation-obstruction theories). Some of the main topics covered in the course are: Derived Artin Stacks and their cotangent complexes, De Rham complexes and S1 equivariant schemes (loop spaces), Cyclic homology, Shifted symplectic structures, Lagrangians and Lagrangian fibrations, Derived intersection theory and finally a derived analog of Uhlenbeck-Yau correspondences. The detail syllabus of the course and schedule is provided at: