Calendar

Given an arithmetic subgroup Γ in a semi-simple Lie group G, the multiplicity of an irreducible representation of G in L^2(Γ\G) is unknown in general.
We observe the multiplicity of any discrete series representation pi of SL (2, R) in L^2 (Γ(n)\SL (2, R)) is close to the von Neumann dimension of pi over the group algebra of Γ(n).
We extend this result to other Lie groups and bounded families of irreducible representations of them.
By applying the trace formulas, we show the multiplicities are exactly the von Neumann dimensions if we take certain towers of descending lattices in some Lie groups.

Please see website for more details: www.math.harvard.edu/~ctm/sem.

Deligne connects the weight-zero compactly supported cohomology of a moduli space to the combinatorics of its compactifications. This gives a method for using combinatorics to compute a piece of the cohomology of a moduli space. In this talk, we discuss this method for the cases of the moduli space of abelian varieties and the moduli space of n-marked hyperelliptic curves.

===============================

For more info, see https://math.mit.edu/combin/

The theory of “random surfaces” has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, combinatorics, analysis and mathematical physics. Just as “Brownian motion” is a special kind of random path, there is a similarly special kind of random surface.

Random surfaces are often motivated by physics: statistical physics, string theory, quantum field theory, and so forth. They have also been independently studied by mathematicians working in random matrix theory and enumerative graph theory. But even without that motivation, one may be drawn to wonder what a “typical” two-dimensional manifold looks like, or how one can make sense of that question.

I will give an overview of what this theory is about, including many computer simulations and illustrations. In particular, I will discuss the so-called Liouville quantum gravity surfaces, and explain how they are approximated by discrete random surfaces called random planar maps.

===============================

https://people.math.harvard.edu/~gammage/ons/

A central goal of condensed matter physics is to understand which quantum phases of matter can emerge in a quantum material. For this purpose, one should be able to not only describe the quantum phases using some effective field theories, but also capture the important microscopic information of the material via mathematical formulation. In this talk, I will present a framework to classify quantum phases in quantum materials, where the microscopic information of a material is encoded in its quantum anomaly. I will talk about the application of this framework to classify various exotic quantum phases of matter in different lattice systems. Using our framework, we have obtained many results unexpected from the previous literature.

Zoom: https://harvard.zoom.us/j/977347126
Password: cmsa

Translational tiling is a covering of a space (such as Euclidean space) using translated copies of one building block, called a “translational tile”, without any positive measure overlaps.
Can we determine whether a given set is a translational tile? Does any translational tile admit a periodic tiling? A well known argument shows that these two questions are closely related. In the talk, we will discuss this relation and present some new developments, joint with Terence Tao, establishing answers to both questions.

===============================

For more info, see https://math.mit.edu/combin/

In mirror symmetry, the dual object to a Fano variety is a Landau-Ginzburg model. Broadly, a Landau-Ginzburg model is quasi-projective variety Y with a superpotential function w, but not all such pairs correspond to Fano varieties under mirror symmetry, so a very natural question to ask is: Which Landau-Ginzburg models are mirror to Fano varieties? In this talk, I will discuss a cohomological characterization of mirrors of (semi-)Fano varieties, focusing on the case of threefolds. I’ll discuss how this characterization relates to the deformation and Hodge theory of (Y,w), and in particular, how the classification of (semi-)Fano threefolds is related to questions about moduli spaces of lattice polarized K3 surfaces.

Zoom: https://harvard.zoom.us/j/7855806609

Password: cmsa

Due to unforeseen circumstances, his event has been CANCELLED.

I will discuss an elementary notion — the angle rank of a polynomial — and an application to the Tate conjecture for Abelian varieties over finite fields.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09
Password: cmsa

I will discuss recent joint work with Jeremy Rouse and Drew Sutherland on Mazur’s “Program B” — the classification of the possible “images of Galois” associated to an elliptic curve (equivalently, classification of all rational points on certain modular curves $Xʜ$. The main result is a provisional classification of the possible images of $\ell$-adic Galois representations associated to elliptic curves over ℚ and is provably complete barring the existence of unexpected rational points on modular curves associated to the normalizers of non-split Cartan subgroups and two additional genus 9 modular curves of level 49.

I will also discuss the framework and various applications (for example: a very fast algorithm to rigorously compute the $\ell$-adic image of Galois of an elliptic curve over ℚ, and then highlight several new ideas from the joint work, including techniques for computing models of modular curves and novel arguments to determine their rational points, a computational approach that works directly with moduli and bypasses defining equations, and (with John Voight) a generalization of Kolyvagin’s theorem to the modular curves we study.

Given n-3 subsets of {1, …, n} of size 4, the cross-ratio degree counts the number of ways to place n marked points on the Riemann sphere such that the n-3 cross-ratios are prescribed generic complex numbers. If more than k-3 of the sets involve just k of the points, then those k points are overdetermined and the cross-ratio degree vanishes. We show that this is the only reason why a cross-ratio degree can vanish: if no subset of the points is overdetermined, then the cross-ratio degree is positive. This gives a new proof of Laman’s theorem characterizing graphs whose generic embedding in the plane is rigid. Joint with Joshua Brakensiek, Christopher Eur, and Shiyue Li.

===============================

For more info, see https://math.mit.edu/combin/

Nucleus of a living cell houses a cell genome – a polymer called chromatin, which is a functional form of DNA. It is very long, e.g., 2 meters long for every human cell. Nucleus is also an arena of incessant energy-driven activity. Experiments show that chromatin undergoes large scale motions sustained over long times of order seconds. In the talk, after reviewing the phenomenology, I will show how these flows may arise due to a phase transition in which chromatin-driving motors, such as RNA polymerase, form a polar (“ferromagnetic”) order controlled by hydrodynamic interactions. The talk is based on the joint work with I.Eshghi and A.Zidovska.

Lunch served at 12:30.

This seminar will be held in person and on Zoom.

https://harvard.zoom.us/j/96657833341

Password: cmsa

Thursday Seminar given by Natalia Pacheco-Tallaj on “Invertible field theories and stable homotopy theory.”

A hyperbolic 3-manifold M carries a flat PSL(2;C)-connection whose Chern-Simons invariant has been much studied since the early 1980’s. For example, its real part is the volume of M. Explicit formulas in terms of a triangulation involve the dilogarithm. In joint work with Andy Neitzke we use 3-dimensional spectral networks to abelianize the computation of complex Chern-Simons invariants. The locality of the Chern-Simons invariant, expressed in the language of topological field theory, plays an important role. The dilogarithm arises from a novel construction involving Chern-Simons invariants of flat C*-connections over a 2-torus.

A determinantal octic double solid is the double cover X of P^3 branched along the degree 8 determinant of a symmetric matrix of homogeneous forms on P^3. These X are nodal CY threefolds which do not admit a projective small resolution. B-model techniques can be applied to compute GV invariants up to g \le 32. This raises the question: what is the geometric meaning of these invariants?

Evidence suggests that these enumerative invariants are associated with moduli stacks of coherent sheaves of modules over a sheaf B of noncommutative algebras on X constructed by Kuznetsov. One of these moduli stacks is a stacky small resolution X’ of X itself. This leads to another geometric interpretation of the invariants as being associated with moduli of sheaves on X’ twisted by a Brauer class. Geometric computations based on these working definitions always agree with the B-model computations.

This talk is based on joint work with Albrecht Klemm, Thorsten Schimannek, and Eric Sharpe

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith posed a conjecture on the asymptotics of moments of quadratic L-functions. While this conjecture originates as a question about number fields, it has a more geometric version when posed over function fields in positive characteristic. I’ll talk about how one can reinterpret the central object in this conjecture in terms of the action of the Galois group of a finite field on the cohomology of braid groups with certain coefficients coming from the braid group’s interpretation as the hyperelliptic mapping class group. We will see the “arithmetic factor” in this conjecture appear in the part of this cohomology that is accessible through tools of homological stability. This is joint work with Jonas Bergström, Adrian Diaconu, and Dan Petersen.

We will describe appearances of the algebraic phenomena of Koszul duality in the context of boundary conditions and defects in quantum field theory. Primarily motivated by topological string theory, this point of view was pioneered by Costello and Li in their proposal for a twisted version of the AdS/CFT correspondence. Since then, many important examples of (twisted) holographic dualities in string and M-theory have been studied in work of Costello, Gaiotto, Paquette and many others. I will survey some of these examples and some current work with Raghavendran and Saberi which uses this formalism to predict exceptional symmetries present in M-theory.

Given a fibration $f\colon X\to Y$ of smooth complex projective with general fiber $F$, the celebrated Iitaka conjecture predicts the inequality $\kappa(K_X)\geq \kappa(K_F)+\kappa(K_Y)$. Recently Chang showed that, under some natural conditions, the inequality $\kappa(-K_X)\leq \kappa(-K_F)+\kappa(-K_Y)$ holds.

In this talk I will show that, despite the failure in positive characteristic of both the Iitaka conjecture and Chang’s theorem, it is possible to recover the latter for “tame” positive characteristic fibrations. This is based on joint work with M. Benozzo and C.-K. Chang.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

Agenda:
3:00-4:00 PM EST (GMT -4:00): PhD Admissions Panel
4:00-5:00 PM EST (GMT -4:00): Breakout Session Q&As with PhD Programs

We hope you will join us to:
–Learn about physical and mathematical sciences PhD programs at Harvard
–Connect with current faculty, staff, and students
–Learn about how to put together a strong application

At the end of the Admissions Panel, you will receive Zoom links for the individual PhD program breakout sessions

Registration link
https://harvard.zoom.us/webinar/register/WN_CEaHSF-pTA6z-Xix62P5tw

https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09
Password: cmsa

Consider the setting of a smooth variety $S$ over $\mathbb{F}_q$, and an $\ell$-adic local on $S$ which has finite determinant and is geometrically irreducible. Work of Lafforgue proves that such a local system must be pure, and it is conjectured that the action of Frobenius at closed points is semisimple. I will sketch a proof of this conjecture in the setting of mod $p$ Shimura varieties, and will deduce applications to the existence of CM lifts of certain mod p points. If time permits, I will also address the question of integral canonical models of Shimura varieties.
This is joint work with Ben Bakker and Jacob Tsimerman.

Please see website for more details: www.math.harvard.edu/~ctm/sem.

Discrete Morse theory, developed by Forman, is an efficient tool to determine the homotopy type of a regular CW complex. The theory has been reformulated by Chari in purely combinatorial terms of acyclic matchings on the face poset. In this talk, I will discuss explicit constructions of such acyclic matchings on Bruhat intervals using reflection orders. As an application, we show the totally nonnegative Springer fibres are contractible, verifying a conjecture of Lusztig. This is based on work in progress with Xuhua He.

===============================

For more info, see https://math.mit.edu/combin/

===============================

https://people.math.harvard.edu/~gammage/ons/

Suppose that we only see small “k x k snapshots” of a random two-dimensional “n x n picture”: can we piece the original picture back together? Motivated by the one-dimensional problem of shotgun sequencing DNA, Mossel and Ross raised several interesting questions (like the one aforementioned) about reconstructing random structures from “small snapshots” in two (and higher) dimensions. In this talk, I will sketch how we can now answer some of these two-dimensional reconstruction questions: in particular, it turns out that the answer to the problem mentioned above exhibits somewhat surprising “two-point concentration,” and getting to this answer involves a combination of entropic methods and tools from percolation.

===============================

For more info, see https://math.mit.edu/combin/

ng-Mills gauge theory with gauge group SU(2) has played a significant role in the study of the topology of 3- and 4-manifolds. It is natural to ask whether we obtain more topological information by working with other choices of gauge groups such as SU(n) for higher values of n. Mariño and Moore formulated a conjecture essentially stating that there is no new information in Donaldson invariants of smooth 4-manifolds defined using SU(n) Yang-Mills gauge theory. Despite this “negative” prediction, one might still hope that there is still novel information about 3-manifolds in higher rank gauge theory.
In this talk, I will discuss a result about the topology of 3-manifolds obtained using gauge theory with respect to the Lie group SU(3): for any knot K in the 3-dimensionl sphere (or more generally an integer homology sphere) there is a non-abelian representation of the knot group of K into SU(3) such that the homotopy class of the meridian of K is mapped to a matrix with eigenvalues 1, w, w^2 with w being a primitive third root of unity. As a byproduct of the proof, we obtain a structure theorem for SU(3) Donaldson invariants of 4-manifolds, analogous to Kronheimer and Mrowka’s structure theorem for SU(2) Donaldson invariants. This can be regarded as a piece of evidence supporting Mariño and Moore’s conjecture. This talk is based on a recent joint work with Nobuo Iida and Chris Scaduto.

For more information and to register online, please see: https://cmsa.fas.harvard.edu/event/mathscilit2023/

Recent progress in generating synapse-level maps of brains, a field known as connectomics, brings both opportunities and challenges. The upside is that the biophysical instantiation of memories, behaviors, and knowledge will soon be before us. The downside is that no one knows exactly how to make sense of this data. I will show what connectomics data sets are and attempt to explain why it is so difficult to unravel their meaning.

Solomon Lefschetz showed that the Picard group of a general surface in P3 of degree greater than three is ZZ. That is, the vast majority of surfaces in P3 have the smallest possible Picard group. The set of surfaces of degree greater than 3 on which this theorem fails is called the Noether-Lefschetz locus. This locus has infinite components and their dimensions are somehow mysterious.

In this talk, I will calculate the dimension of infinite Noether-Lefschetz components that are simple in a sense, but still give us an idea of the complexity of the entire Noether-Lefschetz locus. This is joint work with Montserrat Vite and Manuel Leal.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

We shall first explain the relation between a family of deformations of genus zero p-adic string worldsheet action and Tate’s thesis. We then propose a genus one p-adic string worldsheet action. The key is the definition of a p-adic Laplacian operator on the Tate curve. We show that the genus one p-adic Green’s function exists, is unique under some obvious constraints, is locally constant off diagonal, and has a reflection symmetry. It can also be numerically computed exactly off the diagonal, thanks to some simplifications due to the p-adic setup. Numerics suggest that at least in some special cases, the asymptotic behavior of the Green’s function near the diagonal is a direct p-adic counterpart of the familiar Archimedean case, although the p-adic Laplacian is not a local operator. Joint work in progress with Rebecca Rohrlich.

A striking correspondence between dynamics of automorphism groups of countable first order structures and Ramsey theory of finitary approximation of the structures was established in 2005 by Kechris, Pestov, and Todocevic. Since then, their work has been generalized and applied in many directions. It also struck a fresh wave of interest in finite Ramsey theory. Many classes of finite structures are shown to have the Ramsey property by encoding their problem in a known Ramsey class and translating a solution back. This is often a case-by-case approach and naturally there is a great need for abstracting the process. There has been much success on this front, however, none of the tools captures every situation. We will discuss one such encoding via a model-theoretic notion of semi-retraction introduced by Lynn Scow in 2012. In a joint work, we showed that a semi-retraction transfers the Ramsey property from one class of structures to another under quite general conditions. We compare semi-retractions to a category-theoretical notion of pre-adjunction revived by Mašulović in 2016. If time permits, I will mention a transfer theorem of the Ramsey property from a class of finite structures to their uncountable ultraproducts, which is an AIMSQuaRE project with Džamonja, Patel, and Scow.

Zoom: https://harvard.zoom.us/j/7855806609

Password: cmsa

We will investigate if some well known properties of log canonical singularities over the complex numbers still hold true over perfect fields of positive characteristic and over excellent rings with perfect residue fields. We will discuss both pathological behavior in characteristic p as well as some positive results for threefolds. We will see that the pathological behavior of these singularities in positive characteristic is closely linked to the failure of certain vanishing theorems in positive characteristic. Additionally, we will explore how these questions are related to the moduli theory of varieties of general type.
This is based on joint work with F. Bernasconi and Zs. Patakfalvi, as well as joint work with Q. Posva.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html

Please see website for more details: www.math.harvard.edu/~ctm/sem.

Discrete Morse theory, developed by Forman, is an efficient tool to determine the homotopy type of a regular CW complex. The theory has been reformulated by Chari in purely combinatorial terms of acyclic matchings on the face poset. In this talk, I will discuss explicit constructions of such acyclic matchings on Bruhat intervals using reflection orders. As an application, we show the totally nonnegative Springer fibres are contractible, verifying a conjecture of Lusztig. This is based on work in progress with Xuhua He.

===============================

For more info, see https://math.mit.edu/combin/