Seminars

Poincaré Duality in p-adic Analytic Geometry Rigid-analytic spaces are geometric objects described by convergent  power series over the field of p-adic numbers Q_p. Just as complex-analytic spaces provide a robust framework for analytic geometry over C, rigid-analytic spaces offer a natural setting for analytic geometry over Q_p. In this talk, I will give a gentle introduction to the theory of rigid-analytic spaces and then discuss a version of Poincaré Duality for these spaces, as conjectured by Peter Scholze in 2012.

Tuesday, January 7, 2025, 4 - 5pm
Widtsoe Building - John A. (JWB) - 335
Seminar Type: Departmental Colloquium

Gromov-Witten theory of complete intersections The Gromov-Witten invariants of a space X record the number of curves in X of a specified genus and degree that intersect a given collection of cycles in X, called "insertions''. These numbers play a central role in modern enumerative algebraic geometry and have been extensively studied over the last 30 years. However, calculating them in concrete terms is still technically challenging, and explicit results are known in a limited number of cases. I will explain how to calculate all Gromov-Witten invariants of complete intersections in projective spaces. While the focus of previous work in this case has been primarily on invariants with insertions pulled back from the ambient projective space, we will study invariants with arbitrary insertions. I will describe several techniques we developed to achieve this, utilizing monodromy, degeneration, and nodal relative geometry. This is joint work with Pierrick Bousseau, Rahul Pandharipande, and Dimitri Zvonkine.

Thursday, January 9, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Local symmetries of geometric Eisenstein series I'll begin with the story of an sl_2-triple, discovered by  Finkelberg and Kuznetsov in the late 90s, which emerges a little  unexpectedly from a very classical geometric object: the space of  endomorphisms of the Riemann sphere. Then I'll discuss some recent joint  work with Andreas Hayash in which we reinterpret and generalize this  construction in the context of geometric Eisenstein series, giving a  local-to-global construction of certain symmetries predicted by the  geometric Langlands conjecture (now a theorem).

Tuesday, January 14, 2025, 4 - 5pm
Widtsoe Building - John A. (JWB) - 335
Seminar Type: Departmental Colloquium

Why do cells migrate collectively in different modes? Collective cell migration underpins key physiological processes, ranging from embryonic development to wound healing and cancer metastasis. While notable progress has been made in elucidating the required mechanisms, such as contact inhibition of locomotion, contact following of locomotion, and supracellular organization, the classification of collective motility modes remains incomplete. In this study, we focus on the migration patterns of small cell groups, specifically cohesive pairs of cells. Experimental observations reveal two distinct motility modes in Dictyostelium discoideum tandem pairs: the individual contributor (IC) mode, where each cell generates its own traction force dipole, and the supracellular (S) mode, characterized by a single traction force dipole. Intriguingly, IC mode predominates in Dictyostelium pairs, but S mode prevails in MDCK doublets, highlighting an apparent discrepancy in emergent modes between cell types. To uncover the mechanisms driving these diverse motility modes, we developed a new two-dimensional biophysical model incorporating mechanochemical details such as intercellular interactions, membrane contractility, and cell-surface adhesions, along with a new quantification method. Our model was capable of recapitulating experimental observations; IC mode emerged naturally in amoeboid doublets when both cells exerted similar traction forces, while S mode dominated with “stronger” leaders that essentially pull on trailers. In contrast, simulations of MDCK-like pairs show roughly equal distribution of motility modes, but tunable with variations in cell-cell interactions, underscoring cell-type-specific adaptations in migration strategies. Our findings reveal how cell mechanics, particularly cell-surface interactions and cell membrane properties, drive collective migration of small cell groups. Extending our model to longer cell trains, we demonstrate its applicability across scales, providing a foundation for exploring collective migratory behavior in other physiological and pathological contexts.

Thursday, January 16, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Polynomial functors and stable cohomology The theory of polynomial representations of the general linear group goes back to the thesis of Issai Schur at the turn of the 20th century. Such representations include the tensor, symmetric, and exterior powers of a vector space, and have been completely classified in the work of Schur when the underlying field is the complex numbers. While there has been significant progress since the work of Schur, the story over a field of positive characteristic remains largely unknown. In my talk I will describe some novel stabilization results for sheaf cohomology, and explain their connection to the study of polynomial representations / functors. This is based on joint work with Keller VandeBogert.

Tuesday, January 21, 2025, 4 - 5pm
Widtsoe Building - John A. (JWB) - 335
Seminar Type: Departmental Colloquium

A versatile model for excitable activator-inhibitor systems driven by heterogeneities Ever since Turing’s famous work on pattern formation, biology has continued to provide new examples of how activator-inhibitor dynamics drive self-organization. One example is the coupling between the GTPase Rho and actin filaments in the cytoskeleton, where Rho drives filament assembly, and actin filaments recruit an inhibitor of Rho. In this talk, I will demonstrate that models that account for actin heterogeneities can reproduce experimental data that cannot be explained with classical models. To do so, I will introduce a hybrid discrete-continuum forward model, then use a Bayesian inverse framework to infer the distribution of actin dynamics parameters associated with experimental data in C. elegans and starfish embryos. The inferred parameter values demonstrate how varying actin kinetics can explain changing patterns of RhoA excitability observed across multiple experimental systems.

Wednesday, January 22, 2025, 4 - 5pm
Widtsoe Building - John A. (JWB) - 335
Seminar Type: Departmental Colloquium

An introduction to the relative Langlands program The Langlands program is a web of far-reaching and influential conjectures about connections between number theory, representation theory and geometry proposed. Within this program, the relative Langlands program has emerged as one of its most important and productive branches. In this talk, I will give an overview of key problems in the relative Langlands program, with a focus on the elegant theory of relative Langlands duality, recently developed by Ben-Zvi, Sakellaridis, and Venkatesh. I will also present some of my works in this area.

Thursday, January 23, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Modular forms from Betti numbers Modular forms are complex analytic functions with striking symmetries, which play fundamental role in number theory. In the last few decades there have been a series of astonishing predictions from theoretical physics that various basic mathematical numbers when put in a generating series, end up being modular forms, when there is no known mathematical reason for such hidden structure. In this talk, we will first provide a gentle introduction to modular forms, suitable for a broad mathematical audience. We will then focus on spaces parametrizing complex plane algebraic curves with line bundles, and prove that generating series of their Betti numbers are modular forms. This verifies physical predictions, using various tools of modern enumerative algebraic geometry. Part of this is joint work with Honglu Fan, Shuai Guo, and Longting Wu.

Tuesday, January 28, 2025, 4 - 5pm
Widtsoe Building - John A. (JWB) - 335
Seminar Type: Departmental Colloquium

Learning the spatial stochastic dynamics of gene expression from static images The molecules inside cells operate with behavior dominated by randomness and disorder. Yet, from these ingredients, robust life emerges. Understanding this paradox demands new mathematics that bridges mechanistic models that capture molecular-scale stochasticity with statistical approaches for extracting patterns from large-scale, noisy, heterogeneous data. This talk presents a case study in bridging these gaps to infer gene expression dynamics from static spatial patterns of mRNA molecules in cells. The approach links spatial point processes for individual molecule locations with tractable solutions to stochastic partial differential equations. This framework combines the strengths of mechanistic insight with the power of modern data science, enabling discoveries from challenging biological datasets while raising mathematical questions about inference in stochastic systems. I will discuss advances and future directions in connecting spatial stochastic dynamics with biological data.

Wednesday, January 29, 2025, 4 - 5pm
Widtsoe Building - John A. (JWB) - 335
Seminar Type: Departmental Colloquium

Representations of p-adic groups: "elementary particles" and the "periodic table"

The representation theory of p-adic groups lies at the interface of algebra, geometry, and combinatorics, and it sheds light, through the Langlands program, on fundamental problems in arithmetic geometry. As with any branch of representation theory, two key problems in the subject are to construct the "elementary particles" that make up all other representations and to organize these building blocks into a "periodic table". In this talk, I'll begin with a gentle introduction to representation theory, starting from scratch. Then I'll turn to p-adic groups and what is known about the "elementary particles" and "periodic table" of their representations, highlighting several contributions of mine in this area.

Thursday, January 30, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Hybrid imaging modalities and the range of the spherical means transform The last two decades saw proliferation of novel coupled-physics imaging modalities. A variety of sensitive but safe and inexpensive medical imaging methods has been developed, that hold great promise for the breast imaging for cancer, detection of ischemia, hemorrhaging, blood clots, etc. These imaging techniques work by combining high-resolution ultrasound with electromagnetic fields that are highly sensitive to the features of interest. In the first part of my talk I will overview the most interesting of these modalities, with emphasis on the underlying mathematics. Inevitably, the introduction of the new techniques has posed a variety of new and exciting inverse problems. In particular, a prominent role in this field is played by the spherical means operator, that maps a function into a collection of integrals over spheres with centers lying on a given surface. The problem I will discuss is the description of the range of this operator. As it happens, all of the classical results are suboptimal, in that they use twice the amount of needed data. I will present a novel range description, that overcomes this issue. (This is a joint work with Peter Kuchment, Texas A&M University)

Monday, February 10, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 222
Seminar Type: Applied Mathematics Seminar

Cell Growth, Homeostasis, and Antibiotic Response In this talk, I will discuss two problems in single-cell dynamics. The first half will cover cell-size homeostasis and models of single-cell biomass growth. Exponentially proliferating cells must compensate for fluctuations in growth and cell-cycle progression to maintain stable size distributions at the population level. Prior work has examined whether these compensations occur through negative correlations between single-cell growth rates and cell size or through negative correlations between single-cell generation times and cell size. I will introduce novel ways to approach this question and present insights gained from high-resolution mass measurements of lymphocytic leukemia cells. Next, I will discuss new work on gene expression under antibiotic stress. Experiments reveal that abrupt antibiotic exposure leads to divergent cell fates and expression profiles in E. coli, which cannot be explained by conventional models of gene expression noise. Instead, these dynamics require a stochastic framework for resource allocation. In both projects, I will highlight open questions at the interface of biology and mathematics.  

Tuesday, February 11, 2025, 2 - 3pm
Cowles Building - LeRoy E. (LCB) - 215
Seminar Type: Math Biology Seminar

Subshifts with very low word complexity The word complexity function p(n) of a subshift X measures the number of n-letter words appearing in sequences in X, and X is said to have linear complexity if p(n)/n is bounded. It’s been known since work of Ferenczi that linear word complexity highly constrains the dynamical behavior of a subshift. In recent work with Darren Creutz, we show that if X is a transitive subshift with limsup p(n)/n < 3/2, then X is measure-theoretically isomorphic to a compact abelian group rotation. On the other hand, limsup p(n)/n = 3/2 can occur even for X measurably weak mixing. Our proofs rely on a substitutive/S-adic decomposition for such subshifts. I’ll give some background/history on linear complexity, discuss our results, and will describe several ways in which 3/2 turns out to be a key threshold (for limsup p(n)/n) for several different types of dynamical behavior.

Wednesday, February 12, 2025, 3:15 - 4:15pm
Cowles Building - LeRoy E. (LCB) - 323
Seminar Type: Max Dehn Seminar

Novel Algebraic Perspectives on self-similarity: Group and Spectral Representations and Operator Algebras. Symmetries such as self-similarity are fundamental concepts that appear across mathematics, from probability and operator theory to mathematical physics. In this talk, I will explore how a novel and constructive algebraic perspective can deepen our understanding of these phenomena. I will begin by discussing how an alternative approach to the renormalization group method provides new insights into scaling limits and universality. I will proceed by presenting a unified framework for understanding self-similarity and Lie symmetries through the lens of group representation theory, operator algebras and spectral theory. This approach reveals the central role of the spectral representation in the Hilbert space of stochastic objects and offers a constructive strategy rooted in the celebrated Stone-Von Neumann-Mackey theorem. Within this framework, I will highlight the surprising roles of the Bessel operator and the Laplacian, which provide unexpected connections and insights.

Friday, February 14, 2025, 3 - 4pm
Cowles Building - LeRoy E. (LCB) - 215
Seminar Type: Stochastics Seminar

The Quantum Spectrum and Gamma Structure for Standard Flips In his 1998 ICM talk, Dubrovin conjectured that for a Fano manifold X, the derived category of X possesses a full exceptional collection if and only if the quantum cohomology of X is generically semi-simple, suggesting a deep connection between the derived category of X and its Gromov--Witten theory. This relationship has been clarified in recent years, with a prominent role being played by the quantum spectrum, that is, the eigenvalues of the operator c_1(X) *q - on H^*(X) where c_1(X) is the first Chern class of TX and *q denotes the quantum product at q. Kontsevich has conjectured that the quantum spectrum of X is closely related to semi-orthogonal decompositions of D^b(X). I will describe how, in the case of projective bundles, blow-ups, and standard flips, known semi-orthogonal decompositions of the derived category indeed correspond to the quantum spectrum at a special value of q. This is based on joint work with Yefeng Shen.

Tuesday, February 25, 2025, 3:30 - 4:30pm
Cowles Building - LeRoy E. (LCB) - 222
Seminar Type: Algebraic Geometry Seminar

Dynamical Symmetry

Thursday, February 27, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Thursday, March 6, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Isolated singularities in mixed characteristic

Friday, March 7, 2025, 2 - 3pm
Cowles Building - LeRoy E. (LCB) - 222
Seminar Type: Commutative Algebra Seminar

Thursday, March 20, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Tuesday, March 25, 2025, 4 - 5pm
Widtsoe Building - John A. (JWB) - 335
Seminar Type: Association for Women in Mathematics Speaker Series

Thursday, March 27, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Thursday, April 3, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Tuesday, April 8, 2025, 4 - 5pm
Widtsoe Building - John A. (JWB) - 335
Seminar Type: Association for Women in Mathematics Speaker Series

Thursday, April 10, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium

Thursday, April 24, 2025, 4 - 5pm
Cowles Building - LeRoy E. (LCB) - 219
Seminar Type: Departmental Colloquium