Events
Department of Mathematics and Statistics
Texas Tech University
Since the emergence of the Calculus of Variations, obtaining equilibria for energies whose Lagrangians depend on geometric invariants have played an important role in Differential Geometry.
In this talk we will examine the history of these pioneer variational problems, their development toward the present, and some of their multiple applications, paying special attention to the speaker's own results.
This Job Colloquium is held in tandom with the PDGMP seminar group. Please watch online via this Zoom link.The relation between Hurwitz polynomials and some sequences of orthogonal polynomials is well known in the literature. More precisely, the even and odd parts of any Hurwitz polynomial can be related to an orthogonal polynomial and the associated second kind polynomial, respectively. In this talk we present several recent results that allow us to construct, by using perturbed sequences of orthogonal polynomials, families of Hurwitz polynomials (with one or more uncertain parameters) that are robustly stable. Some applications and open problems will be discussed.
This Job Colloquium is sponsored by the Applied Math seminar group. Please virtually attend via this Zoom link.
The development of probability theory was motivated by moral considerations in gambling without attempting to predict the outcomes. Making predictions with probability always assumes switching temporal and ensemble perspectives under the ergodic hypothesis. “No probability without ergodicity”, Nassim Taleb said. Famously, ergodicity is assumed in equilibrium statistical mechanics, which successfully describes the thermodynamic behavior of gases. However, in a wider context, many observables, such as the growth of personal fortune, don’t satisfy the ergodic hypothesis that makes probabilistic predictions nonsensical. The failure of the expected wealth model to describe actual human behavior is known as the St. Petersburg paradox and is treated in many textbooks on economics and probability theory. Following the recent works of Ole Peters, we address the question of non-ergodicity in the lottery and relate its degree of non-ergodicity with the degree of predictability studied by us. Finally, we present the recent numerical results of Veniamin Smirnov on non-ergodicity and predictability of the logistic map during its transition to chaos obtained with the use of supercomputing resources provided by the Texas Tech University's High Performance Computing Center.
Please virtually attend this talk Thursday the 9th at 3:30PM CDT via this Zoom link.
 | Tuesday
Sep. 7
3:30 PM
MATH 016
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| Real-Algebraic Geometry
Stalk
David Weinberg
Department of Mathematics and Statistics, Texas Tech University
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N/A
This series of talks will survey two joint papers with Dmitri Pavlov, which together prove a geometric enhancement of the cobordism hypothesis. As a special case, taking homotopy invariant geometric structures, our work provides the first rigorous proof of the topological cobordism hypothesis of Baez--Dolan, after Lurie's 2009 sketch. In this first talk, I will begin with motivation from quantum field theory and string theory. Then I will survey the basic mathematical structures that appear in the statement of the cobordism hypothesis and end with the statement of the main theorem.A discrete-time model of tumor-immune system interactions based on a published model is derived. The tumor is assumed to grow exponentially while the immune system can either slow down or shrink tumor growth. However, cancerous tumor cells have many mechanisms to impair the immune system. The mathematical model therefore incorporates immunotherapy of immune checkpoint inhibitors to study tumor dynamics. It is proven that the tumor can grow to unboundedly large if its intrinsic growth rate exceeds a critical value or if its i size is beyond a threshold when first detected. In addition, a region of initial conditions for which the tumor will eventually reach an unbounded size is derived when the tumor growth rate is smaller than the critical value. As a result, the immunotherapy fails to control the aggressive tumor. There are two positive equilibrium points when the tumor growth rate is smaller than the critical value. We verify that a saddle-node bifurcation occurs at the critical tumor growth rate whereas the equilibrium with the larger tumor size is always unstable. The equilibrium with the smaller tumor size may undergo a subcritical Neimark-Sacker bifurcation in certain parameter regimes. Parameter values derived from various clinical studies are simulated to illustrate our analytical findings.
See the colloquium at this link.Abstract: Two-way fixed -effects models are a popular tool for measuring the effect of policies using panel data. Recent methodological literature has emphasized two potential pitfalls of this technique. First, in the presence of treatment effect heterogeneity, two-way fixed-effects estimators may recover a weighted average of treatment effects in which some effects may receive negative weights. Second, the commonly employed cluster-robust standard error estimators may yield unreliable inference when clusters are heterogeneous. We review these recent advances and provide a guide for estimation and inference in two-way fixed-effects models.