2024-2025 Mathematics Colloquium

Upcoming Talks: 

September 18  - 3:30pm - SDRICH 207

Brian Hall (Notre Dame)

• Title: TBA
• Abstract:  TBA

October 9  - 3:30pm - SDRICH 207

Kalyan B. Sinha (J. N. Centre for Advanced Scientific Research & Statistical Institute of Kolkata)

• Title: TBA
• Abstract:  TBA

October 23  - 3:30pm - SDRICH 207

Carlos Villegas Blas (Universidad Nacional Autonoma de Mexico)

• Title: TBA
• Abstract:  TBA

Recent Talks:

May 1  - 3:30pm - SDRICH 207

Sarah Witherspoon (Texas A&M)

• Title: Derivatives, Derivations, and Hochschild Cohomology
• Abstract:  Differentiation of functions on the real line satisfies the Leibniz (product) rule. More generally, linear operators on rings satisfying a similar product rule are called derivations. Generalizing yet further from linear to multilinear operators leads to Hochschild cohomology, which is useful in many settings, for example, in representation theory, in noncommutative geometry, and in algebraic deformation theory. In this talk, we will give a brief introduction to Hochschild cohomology and some of its uses, and we will explain some recent work on its structure.

April 24  - 3:30pm - SDRICH 207

Brian Simanek (Baylor)

• Title: Universality Limits for Orthogonal Polynomials
• Abstract:  We will consider the scaling limits of polynomial reproducing kernels for measures on the real line.  For many years there has been considerable research to find the weakest assumptions that one can place on a measure that allows one to prove that these rescaled kernels converge to the sinc kernel.  Our main result will provide the weakest conditions that have yet been found.  In particular, it will demonstrate that one only needs local conditions on the measure.  We will also settle a conjecture of Avila, Last, and Simon by showing that convergence holds at almost every point in the essential support of the absolutely continuous part of the measure.  This is joint work with Benjamin Eichinger and Milivoje Lukic.

March 6  - 3:30pm - SDRICH 207

Chris Thron (Texas A&M Central Texas)

• Title: Opportunities and pitfalls for mathematicians in the field in applied machine learning

• Abstract:  Machine learning, and particularly deep learning, has become a huge fad throughout engineering and the sciences. Researchers mention machine learning in paper titles to improve chances of acceptance;  RFP’s for industry and government-sponsored research often require machine learning based methodology.  Unfortunately, there is widespread misunderstanding as to the capabilities of machine learning and the conditions required for it to work well.  This situation poses an opportunity for mathematicians to make significant practical contributions,  both in developing and applying effective techniques in machine learning and in pointing out unrealistic expectations and inappropriate applications.

  This not-too-technical talk describes the speaker’s experience working in academics and industry supporting various  machine-learning based research  projects, mostly related to signal and image processing. Specific projects include: classification of astronomical image data; imputation of meteorological data; investment-insurance strategies; wireless communications channel estimation; and signal processing instrumentation calibration. The talk also suggests some conceptual guidelines which the speaker has found useful in  navigating the murky waters of applied machine learning.  

February 27  - 3:30pm - SDRICH 207

Victor Lie (Purdue University)

• Title: The LGC method: Recent progress on several problems in harmonic analysis

• Abstract:  Building on the (Rank I) LGC-methodology introduced by the speaker and on the novel perspective employed in the time-frequency discretization of the non-resonant bilinear Hilbert--Carleson operator (joint work with C. Benea, F. Bernicot and M. Vitturi), we design a new, versatile approach - referred to as Rank II LGC - that has as a consequence the advancement on several difficult problems within harmonic analysis. 
  
    One of the main difficulties in approaching many of these problems, is the lack of absolute summability for the associated (Rank I) LGC discretized model operators. In order to overcome this challenge, we design a so-called correlative time-frequency model whose control is achieved via the following interdependent elements: 
    · a sparse-unform decomposition of the input function(s) adapted to an appropriate time-frequency foliation of the phase-space, 
    · a structural analysis of suitable maximal ``joint Fourier coefficients", and 
    · a level set analysis with respect to the time-frequency correlation set. 
  
  Relying on the above ingredients, in this talk we discuss the following three problems: 
    ·(joint with my former postdoc Bingyang Hu) the boundedness of the n-linear Hilbert transform along the moment curve: 
   

  T_C(f₁,f₂,...,f_n)(x):= p.v.∫_R f₁(x-t)f₂(x+t²)...f_n(x+tⁿ) (1/t)dt,     x∈R.

   

  
    ·(joint with C. Benea and F. Bernicot) the boundedness of the hybrid trilinear Hilbert transform: 
   

  T_H(f₁,f₂,f₃)(x):= p.v. ∫_R f₁(x-t)f₂(x+t)f₃(x+t³)(1/t)dt,     x∈R. 

  
    ·(joint with my graduate student Martin Hsu) the boundedness of the 2D non-resonant Carleson-Radon transform: 
   

  CR(f)(x,y):= sup_{a∈ R}| p.v. ∫_R f(x-t^a₁,y-t^a₂)(e^aita₃)(t) dt|,     (x,y)∈R² 

  
  where here {a_j}_j=1³⊆R₊ are pairwise distinct.  

   

  February 6  - 3:30pm - SDRICH 207

Kabe Moen (University of Alabama)

• Title: A journey into the beauty of chess and mathematics
• Abstract:  This presentation explores the intersection of chess and mathematics, examining theoretical aspects of the game and chess compositions. We investigate upper bounds for game length and unique game possibilities. We then jump into the intricate world of chess compositions, surveying constructional tasks and combinatorial relationships.  While the mathematics in this talk will not be difficult, we hope to convey the logic and beauty of chess composition.  As a teaser, here is a mate in two moves (White to play up the board and checkmate Black on the second move).  

November 21  - 3:30pm - SDRICH 207

Wencai Liu (Texas A&M)

• Title: Algebraic geometry,analysis and combinatorics in spectral theory of Z^d-periodic graph operators
• Abstract:  In this talk, we will discuss the significant role that the algebraic and analytic properties of complex Bloch and Fermi varieties play in the study of periodic operators. I will begin by highlighting recent discoveries about these properties, especially their irreducibility. Then, I will show how we can use these findings, together with techniques from complex analysis and combinatorics, to study spectral and inverse spectral problems arising from Z^d-periodic operators. 

October 24  - 3:30pm - SDRICH 207

Melvin Leok (UCSD)

• Title: The Connections Between Discrete Geometric Mechanics, Information Geometry, Accelerated Optimization and Machine Learning
• Abstract:  Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-h flow map of a Hamiltonian system on the space of probability distributions. We will also discuss how time-adaptive Hamiltonian variational integrators can be used to discretize the Bregman Hamiltonian, whose flow generalizes the differential equation that describes the dynamics of the Nesterov accelerated gradient descent method.

September 12  - 3:30pm - SDRICH 207

David Blecher (University of Houston)

• Title: Real and complex structure, and beyond
• Abstract:  Real structure occurs naturally and crucially in very many areas of mathematics. Together with collaborators we have recently developed the theory of real operator spaces and (possibly nonselfadjoint) real operator algebras to a somewhat mature level. We begin by describing this theory and how standard constructions interact with the complexification. We characterize real structure in complex operator spaces, and characterize some of the most important objects in the subject.  There are some hard questions, some of which we have solved very recently with Neal, Peralta and Su.  Others involve duality.   Generalizing further, joint work with Mehrdad Kalantar gives a novel framework that contains the operator space complexification, as well as the less-studied quaternification, as special cases. It also may be viewed as the appropriate variant of Frobenius and Mackey's induced representations for the category of operator spaces.  

Click here to see a list of past colloquium talks.