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Bridgeland stability conditions were originally motivated by the concept of Pi stability in theoretical physics, as introduced in work of M. Douglas. Pi stability is an attempt to describe BPS states in string theory compactifications. Alternatively, BPS states in string theory can often be described by solutions of certain nonlinear partial differential equations. In this talk I will explain how, starting from nonlinear PDEs, ideas in GIT lead to a version of algebraic stability which is similar to Bridgeland stability. In particular, I will explain how in several examples in dimension 2, GIT stability for line bundles implies Bridgeland stability, but not conversely. In particular, this yields effective tests for Bridgeland stability in many examples.

We are now quite used to the idea that deep neural networks may be trained in a variety of ways to tackle cutting-edge problems in physics and mathematics, sometimes leading to rigorous results. In this talk, however, I will argue that breakthroughs in deep learning theory are also useful for making progress, focusing on applications to field theory and metric flows. Specifically, I will introduce a neural network approach to field theory with a different statistical origin, that exhibits generalized free field behavior at infinite width and interactions at finite width, and that allows for the study of symmetries via the study of correlation functions in a different duality frame. Then, I will review recent progress in approximating Calabi-Yau metrics as neural networks and cast that story into the language of neural tangent kernel theory, yielding a theoretical understanding of neural network metric flows induced by gradient descent and recovering famous metric flows, such as Perelman’s formulation of Ricci flow, in particular limits.

The Ichino–Ikeda conjecture, and its generalization to unitary groups by N. Harris, gives explicit formulas for central critical values of a large class of Rankin–Selberg tensor products. The version for unitary groups is now a theorem, and expresses the central critical value of $L$-functions of the form L(s, Π × Π′) in terms of squares of automorphic periods on unitary groups. Here Π×Π′ is an automorphic representation of GL(n, F) × GL(n − 1, F) that descends to an automorphic representation of U(V) × U(V′), where $V$ and V′ are hermitian spaces over $F$, with respect to a Galois involution c of F, of dimension $n$ and n − 1, respectively.

I will report on the construction of a $p$-adic interpolation of the automorphic period — in other words, of the square root of the central values of the $L$-functions — when Π′ varies in a Hida family. The construction is based on the theory of $p$-adic differential operators due to Eischen, Fintzen, Mantovan, and Varma. Most aspects of the construction should generalize to higher Hida theory. I will explain the archimedean theory of the expected generalization, which is the subject of work in progress with Speh and Kobayashi.

**location in Room G-10 is tentative. Posting will be updated with new location if necessary**

Torus equivariant rank r vector bundles on a toric variety (toric vector bundles) were famously classified by Klyachko (1989) using certain combinatorial data of compatible filtrations in an r-dimensional vector space E. This data can be thought of as a higher rank generalization of an (integer-valued) piecewise linear function. In this talk, we give interpretations of Klyachko data in terms of valuations with values in a certain meet-join lattice as well as points on a tropical linear space. Since tropical linear spaces correspond to linear matroids, this point of view leads us to introduce the notion of a “matroidal vector bundle”, a generalization of toric vector bundles to general matroids (possibly non-representable). The talk focuses on the combinatorial side of the story and I will give a brief review of toric varieties at the beginning. This is a work in progress with Chris Manon (Kentucky).
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For information about the Combinatorics Seminar, please visit…

http://math.mit.edu/seminars/combin/

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This seminar will be broadcast over Zoom: https://harvard.zoom.us/j/98794872462

The sizes of many subcellular structures are coordinated with cell size to ensure that these structures meet the functional demands of the cell. In eukaryotic cells, these subcellular structures are often membrane-bound organelles, whose volume is the physiologically important aspect of their size. Scaling organelle volume with cell volume can be explained by limiting pool mechanisms, wherein a constant concentration of molecular building blocks enables subcellular structures to increase in size proportionally with cell volume. However, limiting pool mechanisms cannot explain how the size of linear subcellular structures, such as cytoskeletal filaments, scale with the linear dimensions of the cell. Recently, we discovered that the length of actin cables in budding yeast (used for intracellular transport) precisely matches the length of the cell in which they are assembled. Using mathematical modeling and quantitative
imaging of actin cable growth dynamics, we found that as the actin cables grow longer, their extension rates slow (or decelerate), enabling cable length to match cell length. Importantly, this deceleration behavior is cell-length dependent, allowing cables in longer cells to grow faster, and therefore reach a longer length before growth stops at the back of the cell. In addition, we have unexpectedly found that cable length is specified by cable shape. Our imaging analysis reveals that cables progressively taper as they extend from the bud neck into the mother cell, and further, this tapering scales with cell length. Integrating observations made for tapering actin networks in other systems, we have developed a novel mathematical model for cable length control that recapitulates our quantitative experimental observations. Unlike other models of size control, this model does not require length-dependent rates of assembly
or disassembly. Instead, feedback control over the length of the cable is an emergent property due to the cross-linked and bundled architecture of the actin filaments within the cable. This work reveals a new strategy that cells use to coordinate the size of their internal parts with their linear dimensions. Similar design principles may control the size and scaling of other subcellular structures whose physiologically important dimension is their length.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/active-matter-seminar/

No additional detail for this event.

We calculate the monopole Floer spectra of almost-rational plumbings, including all Seifert-fibered rational homology spheres, following ideas from lattice homology.  We’ll also talk about a key ingredient, an exact triangle for monopole Floer spectra.  This is joint work with Irving Dai and Hirofumi Sasahira.