The purpose is to set the schedule of possible guests for the semester.

Three faculty members of the Algebra group will give 10 minute presentations on their general area of research.

This is an opportunity for beginning grad students interested in Algebra to get a sense of what they might be working on and who their adviser could be.

The confirmed speakers are

Álvaro Lozano-Robledo (Number Theory and Arithmetic Geometry)

Andreas Malmendier (Algebraic Geometry, Mathematical Physics)

Kyu-Hwan Lee (Representation Theory)

Polynomials like \(x^4 + y^4\), which don't change when we exchange the variables, are called symmetric. We will show how a small number of basic symmetric polynomials (like \(x + y\) and \(xy\)) can be used to describe all possible symmetric polynomials. As an application, we will see how to generalize the discriminant \(b^2 - 4ac\) of a quadratic polynomial to higher-degree polynomials.


Note: Free refreshments. The talk starts at 5:45.

Join us for the first Logic Colloquium of the semester:

Rashed Ahmad:

"A Recipe for Paradox (A Better Schema than the Inclosure Schema)"


Abstract:

In this paper, we provide a recipe that not only captures the common structure between semantic paradoxes, but it also captures our intuitions regarding the relations between these paradoxes. Before we unveil our recipe, we first talk about a popular schema introduced by Graham Priest, namely, the inclosure schema. Without rehashing previous arguments against the inclosure schema, we contribute different arguments for the same concern that the inclosure schema bundles the wrong paradoxes together. That is, we will provide alternative arguments on why the inclosure schema is both too broad for including the Sorites paradox, and too narrow for excluding Curry’s paradox.

We then spell out our recipe. Our recipe consists of three ingredients: (1) a predicate that has two specific rules, (2) a simple method to find a partial negative modality, and (3) a diagonal lemma that would allow us to let sentences be their partial negative modalities. The recipe shows that all of the following paradoxes share the same structure: The Liar, Curry’s paradox, Validity Curry, Provability Liar, a paradox leading to Löb’s theorem, Knower’s paradox, Knower’s Curry, Grelling-Nelson’s paradox, Russell’s paradox in terms of extensions, alternative liar and alternative Curry, and other new paradoxes.

We conclude the paper by stating the lessons that we can learn from the recipe, and what kind of solutions does the recipe suggest if we want to adhere to the Principle of Uniform Solution.

https://logic.uconn.edu/calendar/

Abstract: Registered index-linked annuities (RILAs) are increasingly popular equity-based retirement savings products offered by U.S. life insurance companies. They combine features of fixed-index annuities and traditional variable annuities (TVAs), offering investors equity exposure with downside protection in a tax-deferred setting. This article
introduces RILAs to the academic literature by describing the products' key features, developing a general pricing model, and deriving the providers' hedging strategy by
decomposing their liabilities into short-term European options.
Numerical illustrations show that RILAs offer investors similar risk proles (in
the long run) as TVAs with maturity guarantees, and that many products currently
sold appear to be priced quite favorably for investors. For providers, RILAs may be a
preferable alternative or complement to TVAs as they greatly simplify the management
of the embedded equity risk and can naturally reduce the TVA capital requirements.
These features position RILAs as a viable long-term solution for this product space.

Webex Meeting link:
https://uconn-cmr.webex.com/uconn-cmr/j.php?MTID=mc219422cc5483292097f5e0b2841b48e
Meeting number:
120 035 2634
Password:
UConn

Speaker's bio: Thorsten is an Assistant Professor of actuarial science at Temple University in Philadelphia and co-director of the Actuarial Science M.S. program. His research focuses largely on the drivers of policyholder behavior in variable annuities with long-term guarantees, the optimal design of such products, and the hedging of these guarantees. Please visit his website for more information: https://sites.google.com/site/thorstenmoenig/home

When studying geometric stability questions it is common to obtain $L^p$ estimates for a sequence of Riemannian manifolds as a first step. As a next step one often would like to bootrstrap from $L^p$ estimates to Sormani-Wenger Intrinsic Flat (SWIF) and/or Gromov-Hausdorff (GH) convergence. We will discuss joint work with Raquel Perales and Christina Sormani which allows one to do just that. We will also present a Morrey type inequality which characterizes the $p$ for which one should expect to obtain both GH and SWIF convergence versus when one should only expect SWIF convergence. An example will be given which shows that $L^p$ convergence is strictly weaker than SWIF and GH convergence and we will give many examples which compare and contrast these notions that will provide intuition for all of the theorems discussed.

A complex manifold is Kobayashi hyperbolic (KH) if a certain pseudometric is in fact a metric. For projective manifolds, being KH is equivalent to all holomorphic functions from the complex plane to the manifold being constant, a condition called Brody hyperbolicity (BH).
For noncompact manifolds BH may be a weaker condition that KH.
Not many examples of KH (or BH) manifolds are known. The literature has focused on generic hypersurfaces and their complements in projective spaces, guided by a conjecture due to Green-Griffiths-Lang.
Here we focus on the case of complements in \(\mathbb P^3\) and improve on known bounds on the degree of the hypersurface that guarantee a weak version of hyperbolicity for its complement.
This is joint work with Gunhee Cho.

Game theory is the study of strategic interactions. The fate of every person in a "game" will usually depend on the decisions of all "players". Nash Equilibrium, developed by Nobel Prize Winner John Forbes Nash, says that in any finite game with a finite number of players, each player has a "best response", or best strategy, for that player to win the game. This week, together we will explore many different types of games, discuss possible strategies, and try some problems to help you start thinking about these interactions like a game theorist.

Note: Cookies will be provided.

Quiver representation theory is an important tool in Lie theory and
mathematical physics. It has produced a lot of interesting examples in
noncommutative geometry.

Deep learning shares a common setup with quiver theory. Namely, they both
concern about representations of a directed graph by vector spaces. On the
other hand, non-linear activation functions play a key role in deep
learning, but they were not found in quiver theory. It is natural to ask
for a unified point of view towards the two subjects.

In this talk, I will explain how to run deep learning over the moduli space
of quiver representations. Using uniformization of metrics, we will see
that the usual Euclidean setup is a special instance of our formulation.

Four faculty members of the Analysis and Probability group will give 10 minute talks on their areas of research.

This is intended to allow grad students interested in the field to get a feel for the research in the department and figure out who their advisors might be.

The speakers will be:

Fabrice Baudoin

Maria Gordina

Ambar Sengupta

Alexander Teplyaev

This is the mathematical analysis learning seminar: including but not limited to complex analysis, functional analysis, geometric analysis, harmonic analysis, numerical analysis, partial differential equations, probability theory, and real analysis. Regular and prospective seminar participants are encouraged to attend this short meeting to discuss the format for the seminar this year. What kinds of topics would you like to see? Would you like to give a talk? Come and let us know.

We consider the relativistic Euler equations with a physical vacuum boundary and an equation of state p(\varrho)=\varrho^\gamma$, $\gamma > 1$. We establish the following results. (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) we establish a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in $L^1_t Lip_x$ and a suitable weighted version of the density is at the same regularity level. This is joint work with Mihaela Ifrim and Daniel Tataru.

The probabilistic method is a technique for showing a mathematical object exists by considering a construction like it at random and showing the random construction has a positive probability of being the type of object you seek. The positive probability tells us the object has to exist, because if it did not exist then it could not have a positive probability of occurring. Examples of the probabilistic method will be shown from combinatorics and geometry.


Note: Free refreshments. The talk starts at 5:45.

In 1798 Gauss wrote Disquisitiones Arithmeticae, the first rigorous text in number theory. This book laid the groundwork for modern algebraic number theory and arithmetic geometry. Perhaps the most important contribution in the work is Gauss's theory of integral quadratic forms, which appears prominently in modern number theory (sums of squares, Galois theory, rational points on elliptic curves,L-functions, the Riemann Hypothesis, to name a few). Despite the plethora of modern developments in the field, Gauss’s first problem about quadratic forms has not been optimally resolved. Gauss's class number problem asks for the complete list of quadratic form discriminants with class number h. The difficulty is in effective computation, which arises from the fact that the Riemann Hypothesis remains open. To emphasize the subtlety of this problem, we note that the first case, where h=1, remained open until the 1970s. Its solution required deep work of Heegner and Stark, and the Fields medal theory of Baker on linear forms in logarithms. Unfortunately, these methods do not generalize to the cases where h>1. In the 1980s, Goldfeld, and Gross and Zagier famously obtained the first effective class number bounds by making use of deep results on the Birch and Swinnerton-Dyer Conjecture. This lecture will tell the story of Gauss’s class number problem, and will highlight new work by the speaker and Michael Griffin that offers new effective results by different (and also more elementary) means.

A fundamental identity connecting theta derivatives and theta constants with rational characteristics is proposed. The identity is applicable to arbitrary rational characteristics, and allows to find expressions through theta constants with rational characteristics for all theta derivatives with rational characteristics. An explicit formula for computing such an expression is derived, and illustrated with examples. Expressions for theta derivatives appear to be homogeneous of degree 3 with respect to theta constants.

Abstract: TBA

Speaker's short bio: Prof. Nair is D.A. Darling Professor of Statistics, Professor of Industrial and Operations Engineering at the University of Michigan. He has a broad range of interests in methodology, theory, and applications. He is a Fellow of the American Association for the Advancement of Science, Fellow of the American Statistical Association, Fellow of the Institute of Mathematical Statistics, and Elected Member of the International Statistical Institute. Please visit his website https://lsa.umich.edu/stats/people/faculty/vnn.html for more informaiton.

TBA

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