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Gromov-Witten Theory, Non-Archimedean Geometry, and Mirror Symmetry
No full text This thesis consists of three projects related to enumerative geometry and mirror symmetry, with an eye towards birational geometry.
The first project studies how certain non-archimedean Gromov-Witten invariants of log Calabi-Yau surfaces, called infinitesimal cylinder counts, behave under blowup. We discuss the case of primitive cylinders, and establish a formula that expresses cylinder counts on a blow up of a toric surface in terms of counts in a simpler surface. The proof of the formula uses non-archimedean geometry techniques in an essential way to produce suitable degenerations of the geometric objects enumerated by the counts.
The next two projects introduce and study the notion of F-bundle, a structure which can be used to formulate mirror symmetry type results using the language of differential geometry. Our spectral decomposition theorem provides a canonical decomposition for F-bundles satisfying a condition called maximality. We develop the theory of framing, and use it to obtain reconstruction theorems for isomorphisms between maximal F-bundles. As an application of this theory, we prove the uniqueness of certain decompositions of quantum cohomology related to birational geometry, complementing the existence results found in the literature. We also extend the framework of F-bundles to the setting of equivariant mirror symmetry, and prove an unfolding result which can be used to strengthen mirror symmetry statements from the small quantum cohomology to the big quantum cohomology. We apply this unfolding theorem to the equivariant mirror symmetry of general flag varieties, for which only the small quantum cohomology mirror symmetry was known until now. </p
Fresh Eyes for an Old Moon: ALMA and JWST Perspectives of Callisto
No full textAs though its surface were frozen in time, Jupiter's moon Callisto has seemingly done little more than collect and degrade impact features since its formation some ~4.5 billion years ago. One outcome of Callisto’s quiescence is that its geologic map retains only a few units, with large-scale landforms consisting of either enormous multi-ring impact basins or crater-laden plains. Despite this geologic simplicity, our knowledgebase of Callisto’s material surface properties and volatile ice distributions is limited compared to the other icy Galilean moons. Understanding how Callisto localizes its thermal properties and delicate volatile ices is essential to understanding how long-term particle bombardment, solar insolation, and extended impact damage has sculpted its aged surface. While Callisto’s much more active sibling satellites have scrubbed some or all of their surfaces free of the most ancient records, the oldest surface processes in the Galilean system remain visible today on Callisto. And with large telescope facilities such as the Atacama Large Millimeter/submillimeter Array (ALMA) and the James Web Space Telescope, these surface properties are now accessible to Earth-based observers.
In Chapters 2 and 3 of my dissertation, I complete the icy Galilean satellite ALMA catalogue with the full Callisto dataset which includes leading and trailing hemisphere images at ALMA Bands 7, 6, and 3, corresponding to 343, 223, and 97 GHz, respectively. At these frequencies, we sample the subsurface depths of order a few centimeters down to about half a meter. From these data, I demonstrate that Callisto’s’ subsurface thermal emission is much less susceptible to diurnal variation compared to the other icy satellites and that while Callisto’s largest craters are thermally consistent with much smaller ones, the warm surface anomalies tell a story of impact bombardment not recorded in current geologic maps. Moreover, I identify several cold anomalies associated with large impacts, as well as one that might be relevant to Callisto's tenuous and patchy CO₂ atmosphere.
In Chapter 4 of my dissertation, I present the results from a JWST NIRSpec 2.85–5.35 micron observing campaign that allowed us to inspect many of Callisto's volatile surface materials for the first time since the end of the Galileo mission in the early 2000s. In this work, I identify the Lofn/Heimdall impact region as Callisto's largest source of non-radiolytic CO₂. This particular crater suite may represent one of the best locations on Callisto to look for deep subsurface materials brought to the surface by the impact. Additionally, I propose Callisto's well-known radiolytic CO₂ trailing hemisphere bullseye is accompanied by a second bullseye in water ice exposure that may share a common origin.
Lastly, in Chapter 5, I offer a brief synthesis of the icy moon ALMA survey, an endeavor that fulfills scientific promises that pre-date the array itself.
Altogether, this dissertation offers the community two of the key Callisto datasets of the 2020s era of research. Now that ESA’s JUICE mission and NASA's Europa Clipper are en-route to the Jovian system, this research offers a timely complement to what is a blossoming era for Callisto and broader icy satellite exploration.</p
Metal Binding to Nsp1, a SARS-CoV-2 protein
Get PDFThe COVID-19 pandemic, caused by SARS-CoV-2, has underscored the need for novel antiviral strategies beyond vaccines. A key virulence factor in SARS-CoV-2 is nonstructural protein 1 (Nsp1), which suppresses host immune responses by degrading mRNA, inhibiting nuclear export, and binding to the 40S ribosomal subunit to block host translation. Its intrinsically disordered C-terminal domain complicates structure-based drug design, prompting exploration of alternative approaches.
This work investigates the use of transition metal coordination to target disordered regions of Nsp1. Copper(II) and cobalt(III) complexes were examined for their ability to bind histidine residues—particularly H165, critical for ribosome interaction. Biophysical techniques, including fluorescence spectroscopy, EPR, and ⁵⁹Co NMR, along with computational modeling, were used to characterize binding to Nsp1-derived peptides and the full-length protein.
Cu(II) displayed pH-dependent coordination through histidine and backbone amides, while oxidized Co(III) complexes formed stable, substitution-inert interactions. Multi-site binding and distinct kinetic profiles were observed. In vitro translation assays showed that metal complexes can affect translation, though selective inhibition of Nsp1 remains challenging.Overall, this work provides a foundation for targeting disordered viral proteins using coordination chemistry.</p
Invariant Combinatorics on Borel Equivalence Relations
No full textThis thesis comprises four independent parts and an appendix.
1. We define and study expansion problems on countable structures in the setting of descriptive combinatorics. We consider both expansions on countable Borel equivalence relations and on countable groups, in the Borel, measure, and category settings, and establish some basic correspondences between the two notions. We then explore in detail many examples, including finding spanning trees in graphs, finding monochromatic sets in Ramsey's Theorem, and linearizing partial orders.
2. Standard results in descriptive set theory provide sufficient conditions for a set P ⊆ ℕℕ × ℕℕ to admit a Borel uniformization, namely, when P has small or large sections. We consider an invariant analogue of these results with respect to a Borel equivalence relation E. Given E, we show that every such P admits an E-invariant Borel uniformization if and only if E is smooth. We also compute the definable complexity of counterexamples in the case where E is not smooth, using category, measure, and Ramsey-theoretic methods. We also show that the set of pairs (E, P) such that P has large sections and admits an E-invariant Borel uniformization is Σ12-complete.
3. Let E, F be Borel equivalence relations on X, Y, and P be an E-invariant Borel set whose sections contain countably many F-classes. We explore obstructions to the existence of Borel E-invariant uniformizing sets for P, i.e., sets choosing one F-class from every section. We survey known results, and prove new dichotomies for the case where P has σ-bounded finite sections. On the way, we prove a dichotomy characterizing the essential values of Borel cocycles into residually finite Polish groups.
4. We show that the Kechris–Solecki–Todorčević dichotomy implies the Harrington–Kechris–Louveau dichotomy. We also give a simple proof of a graph-theoretic dichotomy of Miller for doubly-indexed sequences of analytic graphs, and show that this dichotomy generalizes to finite-dimensional hypergraphs but not to ℵ0-dimensional hypergraphs.
5. An effective version of Nadkarni’s Theorem was proved in Ditzen’s unpublished Ph.D. thesis. The appendix contains a streamlined exposition of the proof and provides an alternative proof of the Effective Ergodic Decomposition Theorem for invariant measures (also originally proved by Ditzen). In addition, we show that the existence of an invariant Borel probability measure is not effective.</p
Exploring Versatility of Energy Metabolism and Dynamics of Anabolism and Growth in Anaerobic Methanotrophic Consortia
No full textTwo main questions are asked in this thesis, how environmental microorganisms respond and persist in an energy-limiting condition, and how we can investigate and disentangle the dynamics of these microbes’ activity and growth at a high spatiotemporal resolution. To the end, this thesis focuses on a symbiosis of methane oxidizing archaea (ANME) and sulfate reducing bacteria (SRB), who mediate the anaerobic oxidation of methane (AOM), an important process in the global methane cycle. For the former question, Chapter 2 first found that carbon monoxide (CO) was able to serve as an alternative electron donor for ANME-SRB, and notably CO can even reverse the direction of AOM in ANME archaea to produce methane by the reduction of CO₂. Chapter 3 then explored and verified from the other side the potential role of pyrite (nano)particles in supporting AOM via a predicted reaction at the pyrite-water interface to generate sulfate and iron oxides as electron sinks. For the latter question, Chapter 4 took advantage of stable isotopic probing combined with the high sensitivity and spatial resolution of nanometer-scale secondary ion mass spectrometry (nanoSIMS) approach, and proposed a pipeline of multi-isotope imaging to record and in situ read out the single cell activity in the past. Chapter 5 as an in progress work attempts to disentangle the native division process in the yet uncultured ANME archaea by means of serial block face electron microscopy (SBEM) and deep learning imaging analysis. Taken together, this work provides more evidence of the versatile energy metabolism for ANME-SRB symbiosis and at the same time offers solutions to capturing the dynamics of activity and growth in natural microorganisms for the field of environmental microbiology
From Hadley Cells to Heat Extremes: Novel Physical and Statistical Frameworks for Evaluating Climate Models
No full textAs climate models become increasingly central to projecting future climate impacts and guiding adaptation strategies, understanding the sources of intermodel spread is critical. This thesis investigates key drivers of model uncertainty through a process-based lens grounded in fundamental climate physics. It focuses on two major contributors to model spread: atmospheric circulation and land-surface processes, which together shape regional hydrological extremes and human-relevant climate outcomes.
First, I develop a zonal-mean energetic framework to explain variations in the meridional extent of the Hadley Circulation (HC) across seasonal, interannual, and multidecadal timescales. I show that changes in tropical net energy input and eddy energy export can be used to explain HC migrations in a variety of contexts. For example, this framework can be used to explain the larger migrations of the ascending HC branch relative to the descending branch over the seasonal cycle, and the contrasting HC migration observed during El Nino events and in response to global warming. This approach provides a physically grounded method for interpreting both observed short-term variability and projected future changes in the tropical atmospheric circulation.
Next, I address uncertainty in terrestrial water storage (TWS) trends across CMIP6 models. To circumvent the limitations of short observational records, I introduce a novel method that compares the seasonal cycle of TWS using empirical orthogonal functions derived from GRACE satellite data. This approach enables the ranking of models based on their fidelity to observed seasonal variability. Models with stronger agreement with GRACE observations exhibit more consistent and often larger trends in TWS, and share features such as deeper soils and spatially distinct precipitation patterns. These insights can guide model selection and development efforts.
Finally, I explore model spread in the temperature-humidity combinations that produce extreme wet-bulb temperatures (WBTs), a key metric of human heat stress. While WBT projections are relatively consistent across models, the partitioning between temperature and humidity—quantified using the "stickiness" metric—varies substantially. I show that these differences lead to large uncertainty in the physiological impacts of heat stress at a given WBT. This work underscores the importance of accurately capturing surface energy partitioning to constrain present and future heat stress risks.
Together, the chapters of this thesis offer several innovative approaches to evaluating uncertainty in climate projections. By applying simple theory, observational constraints, and novel metrics, this work demonstrates how physically grounded methods can identify more reliable models and improve the accuracy of future climate projections---particularly for hydrological extremes and human-centered impacts.</p
Synthesis of Strained Systems via Vinyl Carbocation Intermediates
Get PDFVinyl carbocations are a class of dicoordinated carbocations. Due to their challenging generation, they have been less studied compared to tricoordinated carbocations. This thesis reports multiple novel reactivities involving vinyl carbocation intermediates.
The first chapter reviews methods for generating vinyl carbocations and past reports of vinyl carbocation C–H insertion. It then introduces a field guide to assist researchers in using vinyl carbocation C–H insertion in their synthesis, providing detailed information and optimal reaction conditions developed in our laboratory.
The second chapter describes a catalytic method for forming medium-sized rings via intramolecular Friedel-Crafts reactions of vinyl carbocation intermediates. These reactive species are catalytically generated through the ionization of vinyl toluenesulfonates by a Lewis acidic lithium cation/weakly coordinating anion salt.
The third chapter details selective [2+2] cycloadditions between vinyl carbocations and terminal alkenes, using a LiHMDS-mediated approach. This method allows for the efficient synthesis of strained cyclobutene-containing bicycles under mild conditions, demonstrating the versatile application of vinyl carbocations in constructing complex strained organic structures.</p
Modal Analysis of Harmonically Forced Turbulent Flows with Application to Jets
Get PDFMany turbulent flows exhibit time-periodic statistics. These include flows in turbomachinery, the wakes of bluff bodies, and flows exposed to harmonic actuation. However, many existing techniques for identifying and modeling coherent structures, most notably spectral proper orthogonal decomposition (SPOD) and resolvent analysis, assume statistical stationarity. In this thesis, we develop extensions to study turbulent flows with periodic statistics. We focus on the application of turbulent jets and jet noise reduction through harmonic actuation, which is of interest for both commercial and military aviation due to its success in reducing noise by up to 5dB.
To analyze the coherent structures in harmonically forced flows, we develop the cyclostationary spectral proper orthogonal decomposition (CS-SPOD). We examine the resulting properties of CS-SPOD and develop a theoretical connection between CS-SPOD and harmonic resolvent analysis (HRA), thereby providing the theoretical basis for HRA to be used as a model for coherent structures of cyclostationary flows. We develop and validate a computationally efficient algorithm and then illustrate its efficacy using the linearized (complex) Ginzburg-Landau equation.
We next employ cyclostationary analysis to investigate the impact of an axisymmetric acoustic harmonic forcing on the mean, turbulence, and coherent structures of a round turbulent jet with a Mach number of 0.4 and a Reynolds number of 450000. We perform large-eddy simulations for four cases at two forcing frequencies and amplitudes. Both low-frequency (Strouhal number of 0.3) and high-frequency (Strouhal number of 1.5) forcing is found to generate an energetic, nonlinear, tonal response consisting of the rollup of vortices via the Kelvin-Helmholtz mechanism. However, the impact of forcing on the broadband turbulence and coherent structures is limited, particularly at the low forcing amplitude associated with jet-noise-reduction devices. Additionally, the dominant coherent structures for the forced jets are similar in their energy, structure, and mechanism. At high forcing amplitudes, phase-dependent features arise in the dominant coherent structures and are associated with coupling to the high-velocity/shear regions of the mean. Overall, our results support the existing hypotheses that jet noise reduction can be associated with the deformation of the mean flow field rather than through direct interaction between the forcing and the turbulence. Lastly, we find that HRA predicts the dominant coherent structures well. This shows that HRA can be used to develop models of forced jets in a similar manner to how resolvent is employed for natural jets, which may be useful to guide future sound-source models of jets subjected to active control.</p
Cloudy with a Chance of Microphysics: Modeling Droplet Collisions for the Climate Scale
Get PDFFeedbacks between a warming atmosphere, emission of aerosols, and clouds and precipitation are some of the most difficult aspects for climate models to accurately capture. While climate models operate at resolutions of tens or hundreds of kilometers, many of the physics that determine how and where clouds form or precipitate function at the micron droplet scale. Due to this disparity in physical scales, most of these cloud physics must be modeled with only a few approximate quantities and physical equations. These simplifications lead to large uncertainties about climate forcings such as the sensitivity of global warming to human-emitted aerosols.
This work presents several promising new techniques for modeling and understanding hydrometeors in the climate system, with a particular focus on processes that involve collisions between droplets. First, I extend a high-complexity high-fidelity Lagrangian microphysics method to represent the process of breakup, in which colliding droplets fragment upon collision. Next, I introduce two new methods which attempt to reduce the assumptions inherent to modeling droplet coalescence, in which colliding droplets combine to form a larger drop. The first method uses a spectral finite element approach, while the second generalizes this technique using a method of moments to create a fully flexible microphysics scheme. Finally, I turn to remote observations of clouds, aerosols, and lightning over busy shipping regions to offer new techniques for quantifying aerosol-cloud interactions from creative data resources. This combination of high-fidelity modeling tools, observational data, and efficient numerical methods offers a path toward improving our understanding of the role of cloud microphysics in our climate system.</p
The Neural Computation of Internal Affective States
Get PDFThe study of neural computation has long concentrated on our cognitive abilities, with extensive research dissecting the mechanisms of memory, decision-making, and navigation. In contrast, the realm of social innate behavior and emotion has often been treated as a simpler problem, overlooking the immense complexity and biological significance it entails. This thesis aims to bring neural computation into the domain of emotional or affective states, employing data-driven modeling methods that approximate neural activity as dynamical systems. The application of these methods has uncovered brain representations that encode key qualities of persistence and escalation associated with aggressive states, formalized as line attractors. These emergent features of neural circuits arise from the complex interplay of connectivity and network dynamics, challenging long-held notions of subcortical computation. This discovery led us to rigorously test various key properties of line attractor dynamics. Through closed-loop modeling and holographic neural activation, we demonstrate that the line attractor is intrinsic to the mammalian hypothalamus, providing some of the first causal evidence of this property for any continuous attractor. These experiments also suggest that functional connectivity within the hypothalamus underpins the stability of this attractor. Furthermore, using a new cell-type-specific gene-editing system, we show that the implementation of this line attractor depends on neuropeptides, indicating a non-canonical mechanism that contributes to the robustness of this innate attractor. Finally, we reveal that line attractors encode emotional states beyond aggression, including states of sexual receptivity in the female hypothalamus. Longitudinal recordings of neural data across the estrus cycle show that the line attractor disappears during non-estrus states, suggesting long-timescale modulation of attractor dynamics by hormones. Together, these studies present a new paradigm for understanding subcortical computation underlying internal states and suggest a canonical motif that the brain reuses to encode diverse internal affective states