The Department of Mathematics is committed to producing world-class research, providing high quality undergraduate, graduate and professional programs of study that attract the best students, and to attending to the mathematical needs of the University and the community.

Henry R. Monteith Building, home of the Department of Mathematics.

- Srinivasa Varadhan (NYU Courant) Delivers Distinguished Lecture on “The The Polaron Measure”
- Liang Xiao awarded NSF CAREER grant
- “Complex Math Visuals are This Researcher’s Handiwork” — work of David Nichols, graduate student in mathematics, profiled by UConn Today
- Students present at 2018 Spring Frontiers Exhibition
- Actuarial program named Center for Actuarial Excellence for eighth consecutive year
- Graduate student receives CETL Outstanding Teaching Award
- Damir Dzhafarov spotlighted by the Connecticut Institute for the Brain and Cognitive Science
- Lan-Hsuan Huang Huang and Damin Wu receive appointments at IAS; Prof. Huang awarded Simons and von Neumann Fellowships
- “Very Special Snowflakes” — the work of Vyron Vellis (Assistant Research Professor in Math) featured in UConn Today
- Talitha M. Washington, first African American to receive math PhD from UConn, writes about journey from student to math professor, in the AMS Notices

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PDE and Differential Geometry Seminar

Title: Reduced Basis Method: Recent improvements and a L1-based variant

Speaker: Yanlai Chen (UMass)

Time: Monday, October 15, 2018 at 2:30 pm

Place: MONT 214Abstract: Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized problem are desired in a fast/real-time fashion. Thanks to an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) can improve efficiency by several orders of magnitudes. The accuracy of the RBM solution is maintained through and the construction of the finite-dimensional RBM space is guided by an error estimation mechanism whose efficient development is critical. After giving a brief introduction of the RBM, this talk will present some of our recent applications, algorithmic improvements and innovations including a L1-based RBM variant that is significantly more efficient and applicable.

Title: The Ricci flow under almost non-negative curvature conditions

Speaker: Gunhee Cho (University of Connecticut)

Time: Monday, October 15, 2018 at 5:00 pm

Place: MONT 313 Abstract: This talk is aimed to understand the Theorem 1 in the paper https://arxiv.org/pdf/1707.03002.pdf. Based on R. Hamilton's Ricci flow program, now It is well known that the positivity condition of the certain curvature conditions of the initial metric is preserved under the Ricci flow, but positivity strongly restricts the underlying topology on the manifolds. This paper eliminates that topological restriction by allowing those curvature conditions to be the negative small value. The Theorem 1 is the essential block to generalize those known positivity curvature conditions to almost non-negative conditions in Theorem 2.

Title: Counting Solvable Extensions of Number Fields

Speaker: Brandon Alberts (University of Connecticut)

Time: Wednesday, October 17, 2018 at 11:15 am

Place: MONT 313Abstract: Fix a finite group $G$ and a number field $K$. How many $G$-extensions $L/K$ are there with $disc(L/K) < X$, taken as $X$ tends towards infinity? This is in general a difficult question, at least as hard as the inverse Galois problem. In this talk, I will outline the proof of an upper bound for this quantity when $G$ is solvable, which is conditional on the size of the $\ell$-torsion of class groups of number fields with fixed degree. The new conditional bounds give evidence in support of Malle's conjecture, and can be used to prove unconditional bounds which improve on previously known results when $G$ is 'nearly nilpotent'.

Title: Direct Integrals II

Speaker: Arthur Parzygnat (University of Connecticut)

Time: Wednesday, October 17, 2018 at 2:30 pm

Place: MONT 214Abstract: In the last talk, we provided motivation for the direct integral from the spectral theory of normal matrices and from solid state physics. In this talk, we will study Hilbert families and measurability structures. In talk III, I expect to define the direct integral and discuss some of its properties. The ultimate goal of this series of talks is to understand the mathematics behind band theory (topological and ordinary) from condensed matter physics. This talk is independent of last week's talk so newcomers are welcome!Comments: This talk is expository and will be at an introductory level assuming the audience knows some elementary measure theory.
If anyone is interested in giving a talk on something, please contact Arthur at arthur.parzygnat@uconn.edu. Graduate students are particularly encouraged to give talks.

Title: Acceleration of Series Convergence

Speaker: Anastasiia Minenkova (University of Connecticut)

Time: Wednesday, October 17, 2018 at 5:45 pm

Place: MONT 226Abstract: In calculus courses you learn how to test an infinite series for convergence or divergence, and perhaps error bounds for approximating a series by its partial sums, but how do you get good approximations to a series if its partial sums converge very slowly? The aim of this talk is to describe convergence acceleration methods, which can transform a slowly convergent series into a rapidly convergent series. We will discuss examples of such sequence transformations and their applications in numerical analysis. The talk will assume the audience has familiarity with the idea of a convergent infinite series.Comments: Free pizza and drinks!

Title: On the discrete Hilbert transform

Speaker: Rodrigo Banuelos (Purdue University)

Time: Thursday, October 18, 2018 at 4:00 pm

Place: MONT 214Abstract: The discrete Hilbert transform, acting on the space of (doubly infinite) sequences, was introduced by David Hilbert at the beginning of the 20th century. It is the discrete analogue of the continuous Hilbert transform acting on functions on the real line (conjugate function in the periodic case). In 1925, M. Riesz proved the $L^p$ boundedness, for $p$ larger than one and finite, of the continuous version, thereby solving a problem of considerable interest at the time. From this he deduced the same result for the discrete version. In 1926 E.C. Titchmarsh turned this around. He gave a direct proof of the boundedness of the discrete operator and from it deduced the same for the continuous version. Further, he showed that the discrete and continuous versions have the same $p$-norms. Unfortunately, the following year Titchmarsh pointed out that his argument for equality of the norms was incorrect. The problem of equality has been a long-standing conjecture since. In this talk the speaker describes, taking a historical point of view and avoiding technicalities as much as possible, some tools from probability theory that lead to a proof of this conjecture. The talk is based on joint work with Mateusz Kwasnicki of Wroclaw University, Poland.

Title: The Inverse Function Theorem in the Noncommutative Setting

Speaker: Mark E. Mancuso (Washington University in St. Louis)

Time: Friday, October 19, 2018 at 1:30 pm

Place: MONT 313Abstract: Classically, the inverse function theorem says that a $C^1$ function is locally invertible around a point of nonsingularity of its derivative. In this talk, we introduce the theory of noncommutative functions, which in the matrix case, are functions on a graded domain of tuples of matrices that preserve direct sums and similarities. We present inverse function-type theorems in this setting. Unlike in the classical case, noncommutative inverse function theorems are global: the derivative of a noncommutative function $f$ is invertible everywhere if and only if $f$ is invertible on its domain. We also discuss recent work on the inverse function theorem for operator noncommutative functions defined on domains sitting inside of $B(\mathcal{H})^d,$ for an infinite dimensional Hilbert space $\mathcal{H}$.

Title: On quantum cluster algebra from unpunctured surfaces: arbitrary coefficients and quantization.

Speaker: Min Huang (UniversitÃ© de Sherbrooke)

Time: Friday, October 19, 2018 at 3:00 pm

Place: MONT 313Abstract: In this talk, I will introduce a quantum Laurent expansion formula for quantum cluster algebras from unpunctured surfaces.

Title: Quantitative Fundamental Investing: Theory and Practice

Speaker: John McDermott (Fairfield University and Symmetry Partners)

Faculty Sponsor: Bin Zou

Time: Friday, October 19, 2018 at 3:30 pm

Place: MONT 214Abstract: Academic research in asset pricing is being successfully applied in investing strategies around the world. Come to learn about the strategies and the firms that employ them.

Title: Naber and Valtorta's Discrete Reifenberg Theorem

Speaker: Sean McCurdy (University of Washington)

Time: Friday, October 19, 2018 at 3:30 pm

Place: MONT 414Abstract: In Naber and Valtorta's 2017 Annals paper, one of the key tools of their work is a powerful new packing result, the so-called Discrete Reifenberg theorem, which yields a number of interesting results about rectifiability. This talk will focus not on the many applications of this theorem, but will focus on introducing the theorem and it's immediate implications, which can be seen as extensions of the Reifenberg Topological Disc theorem and the work of David-Toro. The focus will be on providing context with an outline of the proof ideas at the end, time willing. I promise to draw more pictures than equations.