2020 Talks
On the Spectra of Separable 2D almost Mathieu Operators
Alberto Takase
December 14th, 2020 - 1:00pm - 2:20pm - Virtual via Zoom
Abstract
We consider 2D discrete Schrödinger operators with separable potentials generated by 1D almost Mathieu operators. For fixed Diophantine frequencies we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J.\ Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to higher dimensional separable operators with quasiperiodic analytic potentials and Diophantine frequencies. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator, and utilizes the Newhouse Gap Lemma on sums of Cantor sets.
About the Speaker
Alberto is a 5th year graduate student.
Advisor and Collaborators
Alberto's advisor is Anton Gorodetski.
Optimal Hedging Under Slow-Scale Stochastic Volatility
Guangchu Yan
December 11th, 2020 - 4:00 - 4:50pm - Virtual via Zoom
Abstract
In recent years, fractional model on pricing option has been widely used to fit the observation in the market. The problem on this setting is that the fractional stochastic volatility model is neither a martingale nor a Markov process, which makes the derivation of the option price difficult. This talk focus on finding the optimal delta hedging strategy based on the result on option pricing under slow-scale stochastic volatility, where the fractional stochastic volatility model is used.
About the Speaker
Guangchu is a 4th year graduate student.
Advisor and Collaborators
Guangchu's advisor is Knut Solna.
Analytic Torsion of Circle Bundles of Symplectic Manifolds
David Clausen
November 30th, 2020 - 5:30 - 6:20pm - Virtual via Zoom
Abstract
For a symplectic manifold, we can study the symplectic structure of a manifold by encoding the symplectic form as the characteristic class of a circle bundle. The analytic torsion is an invariant of manifolds that can be used to distinguish different manifolds that appear to be homeomorphic, such as lens spaces. By studying the analytic torsion of the circle bundle, we hope to distinguish different symplectic structures and relate it to the chain complex of the primitive cohomology of the manifold.
About the Speaker
David is a 4th year graduate student working on Differential Topology and Symplectic Geometry.
Advisor and Collaborators
David's advisor is Li-Sheng Tseng. This is joint work with Li-Sheng Tseng and Xiang Tang.
Mathematical Representation of Data Visualization Methods
Kathryn Dover
September 8th, 2020 - 11:00 - 11:50am - Virtual via Zoom
Abstract
Data visualization often involves algorithms generating 2-dimensional representations of data to illustrate relationships between different data points. The standard algorithm used for data visualization is tSNE, a non-convex method that represents the differences between data points as weighted probabilities in high and low dimensions and then minimizes the "distance" between these distributions using the Kullback-Leibler divergence. Despite its wide success, there is still very little mathematical understanding of the algorithm. In this talk, we discuss tSNE and our attempt to better mathematically represent the algorithm in both theory and practice.
About the Speaker
Kat is a 4th year graduate student working on machine learning and probability and machine learning.
Advisor and Collaborators
Kat's advisor is Roman Vershynin.
Cohomology on Complements of Singular Hypersurfaces in P3 of A-D-E Singularities
Matthew Cheung
May 21st, 2020 - 1:00 - 1:50pm - Virtual via Zoom
Abstract
The number of solutions to a given hypersurface in projective space over finite fields is encoded in a generating function called the zeta functions of an algebraic variety over a finite field. Hence, to know the number of solutions to a given hypersurface, it suffices to determine the zeta function. The zeta function is given by the determinant of a matrix given by the actino of Frobenius on certain cohomology groups. Hence, the first step is to understand these cohomology groups. Counting points on smooth hypersurfaces has been well-studied so we move to singular hypersurfaces. In this talk, we explain about De Rham Cohomology on the Complement of a singular hypersurface of type A-D-E singularities over the complex numbers. We then explain how this result helps in determining the number of points over our singular hypersurface over finite fields and problems that still need to be answered.
About the Speaker
Matthew is a 3rd year graduate student working on number theory and algebraic geometry. In his spare time, he likes watching anime and playing anime games.
Advisor and Collaborators
Matthew's advisor is Vladimir Baranovsky.
Embracing Geometry: Or How I Learned to Stop Worrying and Solve Polynomials
Alex Sutherland
April 25th, 2020 - 3:45 - 4:35pm - Virtual via Zoom
Abstract (Part of the MGSC and AMS Math Graduate Student Conference)
A classical problem in mathematics is to describe the roots of a polynomial in terms of its coefficients. On the one hand, we have Abel's theorem (1824) that the generic polynomial of degree n is solvable in radicals if and only if n=1,2,3,4. On the other hand, the fundamental theorem of algebra yields that any complex polynomial has all of its roots in the complex numbers. In this colloquium-level talk, we will reconcile these two theorems and explore classical solutions of the quintic in a modern framework. If time allows, we may discuss resolvent degree and solutions of the sextic.
About the Speaker
Alex is a 4th year graduate student working on algebraic geometry and topology.
Advisor and Collaborators
Alex's advisor is Jesse Wolfson.
The Bergman Kernels of Domains in C^n and a Classification Theorem
John Treuer
April 25th, 2020 - 2:30 - 3:20pm - Virtual via Zoom
Abstract (Part of the MGSC and AMS Math Graduate Student Conference)
In complex analysis and several complex variables, the Bergman kernel is intimately connected with the existence of biholomorphic mappings. In this talk, I will explain what the Bergman kernel is, why it is important and state a classification theorem of Bergman kernels I proved in 2019. The talk will be accessible to graduate students who have taken Math 220.
About the Speaker
John is a 5th year graduate student working on complex geometry.
Advisor and Collaborators
John's advisor is Song-Ying Li.
Network Compression Techniques in Deep Learning and Applications to Image Processing
Fanghui Xue
April 25th, 2020 - 1:00 - 1:30pm - Virtual via Zoom
Abstract (Part of the MGSC and AMS Math Graduate Student Conference)
Convolutional neural networks (CNNs) have been widely used in image processing. Despite the high accuracy they have achieved in some tasks like image classification, the huge cost of computational resources can be unacceptable. In this talk, we will discuss some mathematical techniques for network compression. These may include regularization, threshoulding method, structured network pruning like channel pruning and some recent approaches in Nueral Architecture Search (NAS) such as gradient based methods. Some of the topics are from our recent work on variable splitting method and differentiable channel pruning algorithm.
About the Speaker
Fanghui Xue is a 1st year graduate student working on neural networks.
A Hybrid Stochastic-Deterministic Approach to Explore Multiple Infection and Evolution in HIV
Jesse Kreger
April 25th, 2020 - 12:15 - 12:45am - Virtual via Zoom
Abstract (Part of the MGSC and AMS Math Graduate Student Conference)
In vivo Human Immnuodeficiency Virus (HIV) infection dynamics are highly dependent on stochastic effects, such as mutation from the wild type into mutant strains that can escape from immune responses and/or drug treatment. Because of the large number of lymphocyte cells and the large number of different infected cell subpopulations in the context of many stains and multiple infection, stochastic algorithms such as the Gillespie method are extremely computationally inefficient. In this talk, we apply and further develop a general hybrid stochastic-deterministic algorithm (1) to HIV specific mathematical models (2). In particular, we use the hybrid approach to analyze the contributions of multiple infection and free virus versus synaptic cell-to-cell transmission on the evolution of an infection, and compare these results to deterministic predictions. (1) Rodriguez-Brenes IA, Komarova NL, Wodarz D. The role of shortening in carcinogenesis: A hybrid stochastic-deterministic approach. J theor Biol. 2019;460:144-152. (2) Kreger, J and Komarova, NL, and Wodarz, D. Effect of synaptic transmission of the evolution of double mutants in HIV. Journal of the Royal Society Interface. 2020;17.164.
About the Speaker
Jesse is a 4th year graduate student working on mathematical biology.
Advisor and Collaborators
Jesse's advisors are Natasha Komarova and Dominik Wodarz.
A Double Spherical Harmonic Solution to the Radiative Transport Equation
Sean Horan
April 25th, 2020 - 11:00 - 11:50am - Virtual via Zoom
Abstract (Part of the MGSC and AMS Math Graduate Student Conference)
The propagation of radiant energy inside a scattering and absorbing medium on mezzo to macro scales is governed by the radiative transport equation. Solutions to this integal-differential equation are of great use in biomedical optics, computer graphics, reactor physics and other application. These solutions have generally taken two forms: computationally expensive Monte Carlo simulations which provide high degrees of detail and more efficient, deterministic solutions which provide significantly less accuracy, often used to construct functionals of radiant energy rather than the energy itself. Here I present a new, deterministic solution to this equation based on a double spherical harmonic basis. This method provides a far higher level of detail in reconstructing radiant energy as a function of position and angle than previous deterministic methods. The accuracy of reconstruction in biologically relevant cases compares better to a Monte Carlo "gold standard" than any other fast deterministic method and computes its solutions tens of thousands to millions of times faster than these gold standards.
About the Speaker
Sean is a 5th year graduate student working on applied mathematics.
Advisor and Collaborators
Sean's mathematical advisor is John Lowengrub.
On a Calabi-type Estimate for Pluriclosed Flow
Josh Jordan
March 3rd, 2020 - 1:00 - 1:50pm - RH 340P
Abstract
The regularity theory for pluriclosed flow hinges on obtaining C^\alpha regularity for the metric assuming uniform equivalence to a background metric. This estimate was established by Streets in an adaptation of ideas from Evans-Krylov, the key input being a sharp differential inequality satisfied by the associated 'generalized metric' defined on T \oplus T^*. In this work, we give a sharpened form of this estimate with a simplified proof. To begin, we show that the generalized metric itself evolves by a natural curvature quantity, which leads quickly to an estimate on the associated Chern connections analogous to, and generalizing, Calabi-Yau's C^3 estimate for the complex Monge Ampere equation.
About the Speaker
Josh is a 3rd year graduate student working on geometric analysis with a focus on non-Kahler geometry.
Advisor and Collaborators
Josh's advisor is Jeffrey Streets, and this is joint work.
On the Geometry of Solutions of the Sextic in Two Variables
Alex Sutherland
January 27th, 2020 - 5:00 - 5:50pm - RH 306
Abstract
Abel's theorem (1824) that the generic polynomial of degree n is solvable in radicals if and only if n is less than 5 is well-known. However, the classical works of Bring (1786) and Klein (1884) give solutions of the generic quintic polynomial by allowing certain other "nice" algebraic functions of one variable. For the sextic, it is conjectured that any solution requires algebraic functions of two variables. In this talk, we will examine and relate the intrinsic geometries of the known solutions of the sextic in two variables, extending the work of Green (1978).
About the Speaker
Alex is a 4th year Ph.D candiate working at the intersection of topology and algebraic geometry. He is currently a 2019-2021 ARCS Foundation Scholar and was a 2019 Pedagogical Fellow.
Advisor and Collaborators
Alex's advisor is Jesse Wolfson.
