Supersymmetric gauge theories encode deep structures in algebraic geometry, and geometric engineering gives a powerful way to understand the underlying structures by string/M theory. In this talk we will see how the dynamics of M5 branes tell us about the motivic and semiorthogonal decompositions of moduli of bundles on curves.

Please note that there will be a pre-talk by Kai Xu from 10:00 am before the main talk at 10:30

Join us for the second annual Jameel Al-Aidroos Mathematical Pedagogy Lecture Series

When: October 2, 2023 | 3 p.m. – 4:30 p.m.

Where: Science Center Room 507, 1 Oxford Street, Cambridge, MA, 02138

Speaker: Gregory R. Goldsmith

• What does it mean to mentor?
• Every mentor is different. Every mentee is different.

Greg Goldsmith is the Associate Dean for Research and Development and an Associate Professor of Biological Sciences in the Schmid College of Science and Technology at Chapman University. He served for six years as the first director of the Grand Challenges Initiative, building it into one of the nation’s largest and most comprehensive postdoctoral training programs.

REGISTER

This speaker series is a small way to remember Jameel Al-Aidroos, his extraordinary warmth of character and pedagogical skills, and his contributions and dedication to teaching and learning at Harvard. He motivated and inspired his students and colleagues; through this series, we hope to celebrate and keep alive that legacy by bringing speakers who share new perspectives on mathematics and pedagogy, and motivate us to reflect on our professional roles.

Organizers

• Robin Gottlieb | Harvard Professor of the Practice in Teaching of Mathematics
• Brendan Kelly | Harvard Department of Mathematics Senior Preceptor, Director of Introductory Mathematics

Gravitational instantons were introduced by Hawking as building blocks of his Euclidean quantum gravity theory back in the 1970s. These are non-compact Calabi-Yau surfaces with L2 curvature and thus can be viewed as the non-compact analogue of K3 surfaces. K3 surfaces are 2-dimensional Calabi-Yau manifolds and are usually the testing stone before conquering the general Calabi-Yau problems. The moduli space of K3 surfaces and its compactification on their own form important problems in various branches in geometry. In this talk, we will discuss the Torelli theorem of gravitational instantons, how the cohomological invariants of a gravitational instanton determine them. As a consequence, this leads to a description of the moduli space of gravitational instantons.

Classification of singularities is an interesting problem in many areas of algebraic geometry, like the minimal model program. One classical approach is to assign to a variety a rational number, its log canonical threshold. For complex hypersurface singularities, this invariant has been refined by M. Saito to the minimal exponent. This invariant is related to Bernstein-Sato polynomials, Hodge ideals and higher du Bois and higher rational singularities.

In joint work with Qianyu Chen, Mircea Mustață and Sebastián Olano, we defined the minimal exponent for LCI subvarieties of smooth complex varieties. We relate it to local cohomology, higher du Bois and higher rational singularities. I will describe what was done in the hypersurface case, give our definition in the LCI case and explain the relation to local cohomology modules and the classification of singularities.

Weissauer proved using the theory of endoscopy that the Galois representations associated to classical modular forms of weight two appear in the middle cohomology of both a modular curve and a Siegel modular threefold. Correspondingly, there are large families of Tate classes on the product of these two Shimura varieties, and it is natural to ask whether one can construct algebraic cycles giving rise to these Tate classes. It turns out that a natural algebraic cycle generates some, but not all, of the Tate classes: to be precise, it generates exactly the Tate classes which are associated to generic members of the endoscopic L-packets on GSp₄. In the non-generic case, one can at least show that all the Tate classes arise from Hodge cycles. For this talk, I’ll focus on the behavior of the algebraic cycle class. NB: This talk is independent of the one in last week’s number theorists’ seminar.

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

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

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

Password: cmsa

We will discuss a result which states that every projective family over ℂℙ¹ with at most two singular fibres is uniruled by rational (multi)sections. We obtain these rational curves by using techniques from symplectic geometry. In this talk, we will focus on (1) discussing the motivation for this work from Hodge theory and (2) presenting the geometric constructions and ideas involved in our proofs. No knowledge of symplectic geometry is required.

Please note MIT location

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

This seminar will be held in person and on Zoom: https://harvard.zoom.us/j/97514733653?pwd=Q05XN3oxSnYvaXlnS0dsRnVyMXZMUT09

Password: 353114

This seminar will be held in person and on Zoom: https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09

The Neron–Ogg–Shafarevich criterion asserts that an abelian variety over ℚp has good reduction if and only if the Galois action on its ℤℓ-linear Tate module is unramified (for ℓ different from p). In 1995, Oda formulated and proved an analogue of the Neron–Ogg–Shafarevich criterion for smooth projective curves X of genus at least two: X has good reduction if and only if the outer Galois action on its pro-ℓ geometric fundamental group is unramified. In this talk, I will explain a relative version of Oda’s criterion, due to myself and Netan Dogra, in which we answer the question of when the Galois action on the pro-ℓ torsor of paths between two points x and y is unramified in terms of the relative position of x and y on the reduction of X. On the way, we will touch on topics from mapping class groups and the theory of electrical circuits, and, time permitting, will outline some consequences for the Chabauty–Kim method.

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

Abstract: TBA

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

Various processes in living cells depend on contractile forces that are often generated by myosin motors in concert with polar actin filaments. A textbook example of this is the actomyosin contractile ring that forms during cell division. Recent evidence, however, has begun to suggest alternate or redundant mechanisms that do not depend on myosin. Experiments on simplified, reconstituted systems also point to contractility and structure formation in disordered, apolar arrays of filaments. We propose a motor-free mechanism that can generate contraction in biopolymer networks without the need for motors such as myosin or polar filaments such as actin. This mechanism is based on active binding and unbinding of cross-linkers that breaks the principle of detailed balance, together with the asymmetric force-extension response of semiflexible biopolymers. We discuss the resulting force-velocity relation and other implications of this, as well as possible evidence for non-motor force generation.

This seminar will be held in person and on Zoom.

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

Password: cmsa

Thursday Seminar given by Dan Freed on “Fun with defects (in topological quantum field theories).”

Heilbronn’s triangle problem asks, how small is the smallest triangle formed by a set of points? Suppose n points are placed in the unit square and Delta is the smallest area triangle formed by three of these points. It is not too hard to see that Delta < C n^{-1}. Komlos, Pintz, and Szemeredi proved in 1981 that Delta << C n^{-8/7} by building on an ingenious method of Roth. We improve the bound to Delta < C n^{-8/7-ep}. The improvement comes from establishing new connections between Heilbronn’s problem and projection theory.

All joint with Cosmin Pohoata.

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

Freedman recently posed a new question in quantitative topology about link packings. Given a link L, define the $\epsilon$-diagonal packing number $n_{L(\epsilon)}$ to be the number of copies of L that can be simultaneously embedded in $[0,1]^3$ so that (1) Each copy of $L$ is contained in a ball which is disjoint from the other copies. (2) Within each copy, the components are separated by a distance of at least $\epsilon$. We’ll discuss a new construction for obtaining a lower bound on $n_{L(\epsilon)}$ and expand on Freedman’s ideas to obtain an upper bound on $n_{L(\epsilon)}$ when $L$ has a non-trivial Milnor Invariant. At the end we’ll mention several related open problems about link packings. This is joint work with Fedya Manin.

The theories of KSBA stability and K-stability furnish compact moduli spaces of general type pairs and Fano pairs respectively. However, much less is known about the moduli theory of Calabi-Yau pairs. In this talk I will present an approach to constructing a moduli space of Calabi-Yau pairs which should interpolate between KSBA and K-stable moduli via wall-crossing. I will explain how this approach can be used to construct projective moduli spaces of plane curve pairs. This is based on joint work with K. Ascher, H. Blum, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.

Please note that there will be a pretalk by Rosie Shen (Harvard Math) from 10:00 am before the main talk at 10:30. The title of this pretalk is “Introduction to the singularities of the MMP.”

Abstract TBA

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

Course decision-making is incredibly important for the unfolding of undergraduate pathways, yet consistently understudied. Using data from Harvard’s Math Department and “Western University” another highly selective college, I will highlight some of the mechanisms at play and discuss which factors of students’ backgrounds (gender, class, relationship to math) and university constraints impact these decisions. Building on developing work at Harvard in the fall 2023 semester, I will have some preliminary findings to present and discuss.

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

We prove results in the direction of showing that for some affine curves, Baker’s method applied to finite étale covers is insufficient to determine the integral points.

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

The amplituhedron is an object introduced by physicists in 2013 arising from their study of scattering amplitudes which has garnered much recent attention from physicists and mathematicians alike. Mathematically, it is a linear projection of a nonnegative Grassmannian to a smaller Grassmannian, via a map induced by a totally positive matrix. A Grassmann polytope, or Grasstope, is a generalization of the amplituhedron, defined to be such a projection by any matrix, removing one of the positivity conditions. In this talk, I will discuss joint work with Dmitrii Pavlov and Lizzie Pratt in which we study these objects, with the hope that we may gain new insights by broadening our horizons and studying all Grasstopes.

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

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

Thursday Seminar given by Sanath Devalapurkar on “Topological quantum field theories from finite homotopy types.”

Let f be a rational map on P^1 defined over the rationals Q whose second iterate f^2 is not a polynomial. For x in Q, Silverman has shown that there are only finitely many integers in the forward orbit {x,f(x),…,f^n(x),…}. In this talk, we present results about the more general setting of S-integers in orbits, where S is a finite set of primes, and provide new upper bounds on the number of S-integers in orbits in terms of the size |S|.

Please see http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for more information

The joints problem asks to determine the maximum number of joints N lines can form, where a joint in a d-dimensional space is a point on d lines in linearly independent directions. Recently, we determined the maximum exactly for k choose d-1 lines in d-dimensional space, namely k choose d. What is more important is that we are able to prove a structural result determining all optimal configurations, and this is the first success of the polynomial method in this direction. It turns out that our result implies a conjecture of Bollobás and Eccles as an immediate corollary regarding a generalization of the Kruskal–Katona theorem. In this talk, we will talk about the connection to that conjecture and also give a high-level overview of the key ideas.

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

We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil which are different from the analogous groups in other Floer theories such as Heegaard Floer and monopole Floer. Finally, we sketch the proof of the triangle.

This talk is dedicated to the memory of my friend and collaborator Bumsig Kim and based on joint work with Ciocan-Fontanine–Guere–Kim–Shoemaker. Gauged Linear Sigma Models (GLSMs) serve as a means of interpolating between Kahler geometry and singularity theory. In enumerative geometry, they should specialize to both Gromov-Witten and Fan-Jarvis-Ruan-Witten theory. In joint work with Bumsig Kim (see arXiv:2006.12182), we constructed such enumerative invariants for GLSMs. Furthermore, we proved that these invariants form a Cohomological Field Theory. In this lecture, I will describe GLSMs and Cohomological Field Theories, review the history of their development in enumerative geometry, and discuss the construction of these general invariants. Briefly, the invariants are obtained by forming the analogue of a virtual fundamental class which lives in the twisted Hodge complex over a certain “moduli space of maps to the GLSM”. This virtual fundamental class roughly comes as the Atiyah class of a “virtual matrix factorization” associated to the GLSM data.

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

Password: cmsa

I will talk about work with Hacking, Keel and Siebert on using mirror constructions to provide partial compactifications of the moduli of K3 surfaces. Starting with a one-parameter maximally unipotent degeneration of Picard rank 19 K3 surfaces, we construct, using methods of myself and Siebert, a mirror family which is defined in a formal neighbourhood of a union of strata of a toric variety whose fan is defined, to first approximation, as the Mori fan of the original degeneration. This formal family may then be glued in to the moduli space of polarized K3 surfaces to obtain a partial compactification. Perhaps the most significant by-product of this construction is the existence of theta functions in this formal neighbourhood, certain canonical bases for sections of powers of the polarizing line bundle.

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

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

p-adic heights have been a rich source of explicit functions vanishing on rational points on a curve. In this talk, we will outline a new construction of canonical p-adic heights on abelian varieties from p-adic adelic metrics, using p-adic Arakelov theory developed by Besser. This construction closely mirrors Zhang’s construction of canonical real valued heights from real-valued adelic metrics. We will use this new construction to give direct explanations (avoiding p-adic Hodge theory) of the key properties of height pairings needed for the quadratic Chabauty method for rational points. This is joint work with Amnon Besser and Steffen Mueller.

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

We organize a number of analytic and combinatorial properties of Boolean functions into a hierarchy of equivalence classes in a similar style as quasi-random graphs, but depending on ‘local’ parameters. We construct quasi-random Boolean functions that separate different levels of the quasi-random hierarchy. In addition, we will briefly survey various well known notions of pseudo-randomness for Boolean functions and explore their relations to the quasi-random hierarchy.

This is a joint work with N. Sieger.

Please note: **special location** MIT 2-449

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

The cell nucleus is enveloped by a complex membrane, whose wrinkling has been implicated in disease and cellular aging. The biophysical dynamics and spectral evolution of nuclear wrinkling during multicellular development remain poorly understood due to a lack of direct quantitative measurements. We characterize the onset and dynamics of nuclear wrinkling during egg development in the fruit fly when nurse cell nuclei increase in size and display stereotypical wrinkling behaviour. A spectral analysis of three-dimensional high-resolution live-imaging data from several hundred nuclei reveals a robust asymptotic power-law scaling of angular fluctuations consistent with renormalization and scaling predictions from a nonlinear elastic shell model. We further demonstrate that nuclear wrinkling can be reversed through osmotic shock and suppressed by microtubule disruption, providing tunable physical and biological control parameters for probing the mechanical properties of the nuclear envelope, highlighting in passing the importance of nonlinear response to biological robustness.

This seminar will be held in person and on Zoom.

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

Password: cmsa

Thursday Seminar given by Cameron Krulewski on “The Ising model as a boundary field theory.”

Mathematics in Science: Perspectives and Prospects

A showcase of mathematics in interaction with physics, computer science, biology, and beyond.

October 27–28, 2023

Location: Harvard University Science Center Hall D & via Zoom.

For more information, please see: https://cmsa.fas.harvard.edu/event/mathematics-in-scien

If a set of integers is syndetic (finitely many translates cover the integers), must it contain two integers whose ratio is a square? No one knows. In the broader context of the disjointness between additive and multiplicative configurations and actions in ergodic Ramsey theory, it makes sense to ask similar questions about dynamically syndetic sets, those sets that contain the visit times of a point to an open set in a minimal topological dynamical system. The main result of the talk is that almost every dynamically syndetic set is multiplicatively very rich: it is “thick” in some coset of a multiplicative subsemigroup. We will discuss some applications: a “thick-starters” van der Waerden theorem; the existence of multiplicative structure in sets of the form A – A + t; and the topological disjointness of minimal niltranslations and minimal, aperiodic multiplicative actions. Time permitting, we will discuss three tools that proved useful in the topic: the prolongation relation (the closure of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the theory of rational points and polynomials on nilmanifolds developed by Leibman, Green, Tao; and the machinery of topological characteristic factors developed recently by Glasner, Huang, Shao, Weiss, and Ye. This talk is based on work in https://arxiv.org/abs/2207.00098.

Please note MIT location.

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

Quantum UV-IR map (a.k.a. q-nonabelianization map), introduced by Neitzke and Yan, is a map from UV line defects in a 4d N=2 theory of class S to those of the IR. Mathematically, it can be described as a map between skein modules and is a close cousin of quantum trace map of Bonahon and Wong.
In this talk, I will discuss how quantum UV-IR map can be generalized to a map between HOMFLYPT skein modules, using skein-valued curve counts of Ekholm and Shend

In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbert correspondence. After that, I will explain how similar ideas can be applied in the study of mixed characteristic singularities. This is based on a joint work (in progress) with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron.

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